Analysis of a Hybrid Coupled System of ψ-Caputo Fractional Derivatives with Generalized Slit-Strips-Type Integral Boundary Conditions and Impulses

Abstract

In the current paper, we analyzed the existence and uniqueness of a solution for a coupled system of impulsive hybrid fractional differential equations involving $\psi$-Caputo fractional derivatives with generalized slit-strips-type integral boundary conditions. We also study the Ulam–Hyers stability for the considered system. For the existence and uniqueness of the solution, we use the Banach contraction principle. With the help of Schaefer’s fixed-point theorem and some assumptions, we also obtain at least one solution of the mentioned system. Finally, the main results are verified with an appropriate example.

1. Introduction

Extensive applications and significant contribution leads the popularity of fractional calculus, and operators of arbitrary order gives us more realistic and useful mathematical modeling of many phenomena (see [1,2,3]). In different fields of science and engineering, fractional-order nonlinear boundary value problems appear with different aspects. In recent studies the main focus is on the existence, uniqueness and stability of nonlinear arbitrary order differential equations with boundary conditions [4,5,6,7,8].

Coupled nonlinear fractional differential equations (FDEs), finds their uses in different applied and scientific models such as disease models [9,10], ecological models [11], synchronization of chaotic systems [12], etc. In recent years Hybrid differential equations of fractional-order is also very important area of research [13,14,15,16,17,18,19]. Ahmad et al. studied a new idea of slit-strips type conditions, which has very fundamental uses in acoustics [20] and imaging via strip-detectors [21].

In [22], the authors studied the following problem with slit-strips type condition:

$$\left\{\begin{array}{cc}& {}^c{\mathcal{D}}_0^{p}x\left(z\right)=f_1(z,x\left(z\right)), m-1<p\le m, z\in [0,1],m\in N,\hfill \\ & x\left(0\right)=0,x^{{}^{\prime }}\left(0\right)=0,...,x^{n-2}\left(0\right)=0,\hfill \\ & x\left(\xi \right)=a_1{\int }_0^{{\eta }_1}x\left(s\right)ds+a_2{\int }_{{\xi }_1}^{1}x\left(s\right)ds, 0<{\eta }_1<\xi <{\xi }_1<1,\hfill \end{array}\right.$$

where ${}^c{\mathcal{D}}_0^p$ denotes the Caputo fractional derivative (CFD) of order p, $f_1$ is a given function and $a_1$,$a_2$∈$\mathbb{R}$.

In 2017, Ahmad et al. [23] studied a coupled system of nonlinear FDEs

$$\left\{\begin{array}{cccc}& {}^c{\mathcal{D}}_0^{{\alpha }_1}x\left(z\right)\hfill & =& f_1(z,x\left(z\right),y\left(z\right)), z\in [0,1], 1<{\alpha }_1\le 2,\\ & {}^c{\mathcal{D}}_0^{{\beta }_1}y\left(z\right)\hfill & =& f_2(z,x\left(z\right),y\left(z\right)), z\in [0,1], 1<{\beta }_1\le 2,\end{array}\right.$$

supplemented with the coupled and uncoupled boundary conditions of the form:

$$\begin{array}{cc}& x\left(0\right)=0, x\left(\xi \right)=d_1{\int }_0^{\eta }y\left(s\right)ds+d_2{\int }_{{\xi }_1}^{1}y\left(s\right)ds, 0<\eta <\xi <{\xi }_1<1,\\ & y\left(0\right)=0, y\left(\xi \right)=d_1{\int }_0^{\eta }x\left(s\right)ds+d_2{\int }_{{\xi }_1}^{1}x\left(s\right)ds, 0<\eta <\xi <{\xi }_1<1,\\ & x\left(0\right)=0, x\left(\xi \right)=d_1{\int }_0^{\eta }x\left(s\right)ds+d_2{\int }_{{\xi }_1}^{1}x\left(s\right)ds, 0<\eta <\xi <{\xi }_1<1,\\ & y\left(0\right)=0, y\left(\xi \right)=d_1{\int }_0^{\eta }y\left(s\right)ds+d_2{\int }_{{\xi }_1}^{1}y\left(s\right)ds, 0<\eta <\xi <{\xi }_1<1,\end{array}$$

where ${}^c{\mathcal{D}}_0^{{\alpha }_1}$ and ${}^c{\mathcal{D}}_0^{{\beta }_1}$ denote the CFD of order ${\alpha }_1$ and ${\beta }_1$ respectively, $f_1,f_2:[0,1]\times \mathbb{R}\times \mathbb{R}⟶\mathbb{R}$ are given continuous functions and $d_1,d_2$ are real constants.

In 2019, Bashir Ahmad et al. [24] studied a coupled system of hybrid nonlinear FDEs

$$\begin{array}{cc}& {}^c{\mathcal{D}}_0^{{\gamma }_1}[u\left(z\right)-h_1(z,u\left(z\right),v\left(z\right))]={\theta }_1(z,u\left(z\right),v\left(z\right)), z\in [0,1], 1<{\gamma }_1\le 2,\\ & {}^c{\mathcal{D}}_0^{{\delta }_1}[v\left(z\right)-h_2(z,u\left(z\right),v\left(z\right))]={\theta }_2(z,u\left(z\right),v\left(z\right)), z\in [0,1], 1<{\delta }_1\le 2,\end{array}$$

equipped with coupled slit-strips type integral boundary conditions:

$$\begin{array}{cc}& u\left(0\right)=0, u\left({\eta }_1\right)={\omega }_1{\int }_0^{{\xi }_1}v\left(s\right)ds+{\omega }_2{\int }_{{\xi }_2}^{1}v\left(s\right)ds, 0<{\xi }_1<{\eta }_1<{\xi }_2<1,\\ & v\left(0\right)=0, v\left({\eta }_1\right)={\omega }_1{\int }_0^{{\xi }_1}u\left(s\right)ds+{\omega }_2{\int }_{{\xi }_2}^{1}u\left(s\right)ds, 0<{\xi }_1<{\eta }_1<{\xi }_2<1,\end{array}$$

where ${}^c{\mathcal{D}}_0^{{\gamma }_1}$ and ${}^c{\mathcal{D}}_0^{{\delta }_1}$ denotes the CFD of orders ${\gamma }_1$ and ${\delta }_1$ respectively, ${\theta }_i,h_i:[0,1]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are continuous functions with $h_i(0,u\left(0\right),v\left(0\right))=0$, $i=1,2$ and ${\omega }_1$, ${\omega }_2$ are real constants.

In this article, motivated from the aforementioned work, we study the coupled system of impulsive hybrid FDEs with generalized slit-strips type integral boundary conditions:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }[x\left(z\right)-f_1(z,x\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(z\right))]=g_1(z,x\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(z\right)), z\in (z_k,z_{k+1}], k=0,1,\dots ,p,\hfill & \hfill \\ {}^c{\mathcal{D}}_{z_k,z}^{{\beta }_1;\psi }[y\left(z\right)-f_2(z,y\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\beta }_1;\psi }y\left(z\right))]=g_2(z,y\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\beta }_1;\psi }y\left(z\right)), z\in (z_k,z_{k+1}], k=0,1,\dots ,p, \hfill & \hfill \\ x\left(0\right)=0, x\left(\eta \right)=a_1{\int }_{z_k}^{{\delta }_{2k}}x\left(\tau \right)d\tau +a_2{\int }_{{\delta }_{2k+1}}^{z_{k+1}}x\left(\tau \right)d\tau , z_k<{\delta }_{2k}<\eta <{\delta }_{2k+1}<z_{k+1},\hfill & \hfill \\ y\left(0\right)=0, y\left(\eta \right)=a_1{\int }_{z_k}^{{\delta }_{2k}}y\left(\tau \right)d\tau +a_2{\int }_{{\delta }_{2k+1}}^{z_{k+1}}y\left(\tau \right)d\tau , z_k<{\delta }_{2k}<\eta <{\delta }_{2k+1}<z_{k+1},\hfill & \hfill \\ \Delta x\left(z_k\right)=x\left(z_k^{+}\right)-x\left(z_k^{-}\right)=I_k\left(x\left(t_k\right)\right), \Delta x^{{}^{\prime }}\left(z_k\right)=x^{{}^{\prime }}\left(z_k^{+}\right)-x^{{}^{\prime }}\left(z_k^{-}\right)=J_k\left(x\left(z_k\right)\right), k=1,2,\dots ,p,\hfill & \hfill \\ \Delta y\left(z_k\right)=y\left(z_k^{+}\right)-y\left(z_k^{-}\right)=I_k^{*}\left(y\left(z_k\right)\right), \Delta y^{{}^{\prime }}\left(z_k\right)=y^{{}^{\prime }}\left(z_k^{+}\right)-y^{{}^{\prime }}\left(z_k^{-}\right)=J_k^{*}\left(y\left(z_k\right)\right), k=1,2,\dots ,p,\hfill & \hfill \end{array}\right.\end{array}$$

(1)

where ${}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }$ and ${}^c{\mathcal{D}}_{z_k,z}^{{\beta }_1;\psi }$ denote the $\psi$-CFDs with ${\alpha }_1,{\beta }_1\in (1,2]$, and $\mathcal{J}=[0,Z]$ with $Z>0$, $f_1,g_1,f_2,g_2:\mathcal{J}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are given continuous functions with $f_1(0,x\left(0\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(0\right))=0$, $f_2(0,y\left(0\right),{}^c{\mathcal{D}}_{z_k,z}^{{\beta }_1;\psi }y\left(0\right))=0$ and $a_1$, $a_2$ are real constants.

The main novelty of our paper is that, to the best of our knowledge, no one have studied the impulsive systems with slit strip boundary conditions. Here we introduced the generalized form of this boundary condition for each interval of the impulsive systems. In the problem (1), the integral boundary condition describes that the contribution due to finite strips of arbitrary lengths occupying the positions $(z_k,{\delta }_{2k})$ and $({\delta }_{2k+1},z_{k+1})$ on the intervals $(z_k,z_{k+1}],k=0,1,\dots ,p$, is related to the value of the unknown function at a nonlocal point $\eta$ on each impulsive interval $(z_k<{\delta }_{2k}<\eta <{\delta }_{2k+1}<z_{k+1})$, $k=0,1,\dots ,p$ located at an arbitrary position in the aperture (slit)—the region of the boundary off the strips. Examples of such boundary conditions include scattering by slits silicon strips detectors for scanned multi-slit X-ray imaging, acoustic impedance of baffled strips radiators, diffraction from an elastic knife-edge adjacent to a strip, sound fields of infinitely long strips, dielectric-loaded multiple slits in a conducting plane, lattice engineering [25,26,27,28,29,30,31,32,33].

2. Preliminaries and Notations

To overcome the vast number of Definitions of fractional derivatives and integrals [2], we can for instance consider general operators, from which choosing special kernels and some form of differential operator, we obtain the classical fractional integrals and derivatives. For example, for the kernel $k(x,t)=x-t$ and the differential operator d/dx, we obtain the Riemann-Liouville fractional derivative, and for $k(x,t)=ln\left(\frac{x}{t}\right)$ and the differential $x\frac{d}{dx}$, we obtain the Hadamard fractional derivative. But, in this case, most of the fundamental laws of the derivative operator can not be obtained, due to the arbitrariness of the kernel. The problems that arise from this approach are the natural limitations to the study of the basic properties of the fractional operators. To overcome this issue, Almedia [34] considered the special case when the kernel is of the type $k(x,t)=\psi \left(x\right)-\psi \left(t\right)$ and the derivative operator is of the form $\frac{1}{{\psi }^{\prime }\left(x\right)}\frac{d}{dx}$. Although the kernel is still unknown, involving the function $\psi$, he deduced some properties for the fractional operator known as $\psi$-Caputo fractional derivative. Here we recall few definitions and lemmas for $\psi$-Caputo fractional derivative.

Definition 1

(see [34]). The left-sided ψ-Riemann-Liouville (RL) fractional integral of order ${\alpha }_1(>0)$ for an integrable function $h\left(z\right):[0,1]\to \mathbb{R}$ with respect to another increasing differentiable function $\psi :[a,b]\to \mathbb{R}$ such that ${\psi }^{{}^{\prime }}\left(z\right)$ ≠ 0 for all z ∈ [a,b] is defined by

$${\mathcal{I}}_{a+}^{{\alpha }_1;\psi }h\left(z\right)=\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_a^z{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z\right)-\psi \left(s\right))}^{{\alpha }_1-1}h\left(s\right)ds$$

where Γ is the Euler Gamma function.

Definition 2

(see [34]). Let $n\in N$ and ψ(z), h(z) ∈$C^n([a,b],\mathbb{R})$ be two functions such that $\psi \left(z\right)$ is increasing and ${\psi }^{{}^{\prime }}\left(z\right)$ ≠ 0 for all z ∈ [a,b]. The left-sided ψ-RL fractional derivative of a function h(z) of order ${\alpha }_1$ is defined by

$$\begin{array}{ccc}\hfill {\mathcal{D}}_{a+}^{{\alpha }_1;\psi }h\left(z\right)& =& {\left(\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}\right)}^n{\mathcal{I}}_{a+}^{n-{\alpha }_1;\psi }h\left(z\right)\hfill \\ & =& \frac{1}{\Gamma (n-{\alpha }_1)}{\left(\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}\right)}^n{\int }_a^z{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z\right)-\psi \left(s\right))}^{n-{\alpha }_1-1}h\left(s\right)ds\hfill \end{array}$$

where $n=\left[{\alpha }_1\right]+1$ and $\left[{\alpha }_1\right]$ denotes the integer part of the real number ${\alpha }_1$.

Definition 3

(see [34]). Let $n-1<{\alpha }_1<n$, $n\in N$ and ψ(z), h(z) ∈$C^n([a,b],\mathbb{R})$ be two functions such that $\psi \left(z\right)$ is increasing and ${\psi }^{{}^{\prime }}\left(z\right)\ne 0$ for all z ∈ [a,b]. The left-sided ψ-CFD of a function h(z) of order ${\alpha }_1$ is defined by

$${}^c{\mathcal{D}}_{a+}^{{\alpha }_1;\psi }h\left(z\right)={\mathcal{D}}_{a+}^{{\alpha }_1;\psi }[h\left(z\right)-\sum _{l=0}^{n-1}\frac{h_{\psi }^{\left[l\right]\left(a\right)}}{l!}{(\psi \left(z\right)-\psi \left(a\right))}^l],$$

where $h_{\psi }^{\left[l\right]}\left(z\right)={\left(\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}\right)}^{l}h\left(z\right)$ and $n=\left[{\alpha }_1\right]+1$ for ${\alpha }_1\notin N$, $n={\alpha }_1$ for ${\alpha }_1\in N.$

Further if h(z) ∈ $C^n([a,b],\mathbb{R})$ and ${\alpha }_1\notin N$, then

$$\begin{array}{ccc}\hfill {}^c{\mathcal{D}}_{a+}^{{\alpha }_1;\psi }h\left(z\right)& =& {\mathcal{I}}_{a+}^{n-{\alpha }_1;\psi }{\left(\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}\right)}^{n}h\left(z\right)\hfill \\ & =& \frac{1}{\Gamma (n-{\alpha }_1)}{\int }_a^z{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z\right)-\psi \left(s\right))}^{n-{\alpha }_1-1}{h}_{\psi }^{\left[n\right]}\left(s\right)ds.\hfill \end{array}$$

Thus if ${\alpha }_1=n\in N$, then ${}^c{\mathcal{D}}_{a+}^{{\alpha }_1;\psi }h\left(z\right)$ = $h_{\psi }^{\left[n\right]}\left(z\right)$.

Lemma 1

(see [34]). Let ${\alpha }_1>0$, and the following holds:

If $h\left(z\right)\in$$C\left(\right[a,b],\mathbb{R})$, then ${}^c{\mathcal{D}}_{a+}^{{\alpha }_1;\psi }{\mathcal{I}}_{a+}^{{\alpha }_1;\psi }h\left(z\right)=h\left(z\right), z\in [a,b].$

If h(z) $\in C^n([a,b],\mathbb{R})$, $n-1<{\alpha }_1<n$, then

$${\mathcal{I}}_{a+}^{{\alpha }_1;\psi }{}^c{\mathcal{D}}_{a+}^{{\alpha }_1;\psi }h\left(z\right)=h\left(z\right)-\sum _{l=0}^{n-1}{c}_l{(\psi \left(z\right)-\psi \left(a\right))}^l, z\in [a,b],$$

where $c_l=\frac{h_{\psi }^{\left[l\right]\left(a\right)}}{l!}.$

Lemma 2

(Schaefer’s fixed point theorem [35]). Let $\tau :E\to E$ be a completely continuous operator (i.e a map that restricted to any bounded set in E is compact). Let $S\left(\tau \right)=\{x\in E:x=\nu \tau (x), \mathrm{for} \mathrm{some} 0<\nu <1\}$. Then either the set $S\left(\tau \right)$ is unbounded or τ has at least one fixed point.

Definition 4

(see [36]). A differential equation

$$\frac{dx}{dz}=f(z,x)$$

(2)

is said to be Ulam-Hyers stable, if there exists a constant $l_f\in {\mathbb{R}}^{+}$, such that for every ${ϵ}^{{}^{\prime }}>0$ and any solution $y\left(z\right)$ of the inequality:

$$|\frac{dy}{dz}-f(z,y)|\le {ϵ}^{{}^{\prime }},$$

there exists a solution $x_1\left(z\right)$ of (2), such that,

$$|y\left(z\right)-x_1\left(z\right)|\le l_f{ϵ}^{{}^{\prime }}.$$

3. Main Results

For $z_k\in {\mathcal{J}}_k$, such that $0=z_0<z_1<z_2<\cdots <z_p=Z$ and $\mathcal{J}={\mathcal{J}}_0\cup {\mathcal{J}}_1\cup \cdots \cup {\mathcal{J}}_p$, where ${\mathcal{J}}_0=(0,z_1], {\mathcal{J}}_1=(z_1,z_2],\cdots ,{\mathcal{J}}_p=[z_p,z_{p+1}]$ and ${\mathcal{J}}^{\prime }=\mathcal{J}-\{z_0,z_1,z_2,\cdots ,z_p\}$. We define the space $X^{{}^{\prime }}=\{x:\mathcal{J}\to \mathbb{R}\mid x\in PC\left([\mathcal{J},\mathbb{R}]\right)$, such that $\mathrm{the} \mathrm{right} \mathrm{limits} x\left(z_k^{+}\right), x^{\prime }\left(z_k^{+}\right) \mathrm{and} \mathrm{left} \mathrm{limits} x\left(z_k^{-}\right), x^{\prime }\left(z_k^{-}\right) \mathrm{exists} \mathrm{and} \Delta x\left(z_k\right)=x\left(z_k^{+}\right)-x\left(z_k^{-}\right), \Delta x^{\prime }\left(z_k\right)=x^{\prime }\left(z_k^{+}\right)-x^{\prime }\left(z_k^{-}\right), k=1,2,\cdots ,p\}$. Then clearly, $X^{{}^{\prime }}$ is a Banach space equipped with the norm $\parallel x\left(z\right)\parallel ={max}_{z\in \mathcal{J}}|x\left(z\right)|$. Similarly, define the space $Y^{{}^{\prime }}=\{y:\mathcal{J}\to \mathbb{R}\mid y\in PC\left([\mathcal{J},\mathbb{R}]\right),\mathrm{right} \mathrm{limits} y\left(z_k^{+}\right), y^{\prime }\left(z_k^{+}\right) \mathrm{left} \mathrm{limits} y\left(z_k^{-}\right), y^{\prime }\left(z_k^{-}\right) \mathrm{exists} \mathrm{and} \Delta y\left(z_k\right)=y\left(z_k^{+}\right)-y\left(z_k^{-}\right), \Delta y^{\prime }\left(z_k\right)=y^{\prime }\left(z_k^{+}\right)-y^{\prime }\left(z_k^{-}\right) k=1,2,\cdots ,p\}$. Then clearly, $Y^{{}^{\prime }}$ is a Banach space equipped with the norm $\parallel y\left(z\right)\parallel ={max}_{z\in \mathcal{J}}|y\left(z\right)|$.

Lemma 3.

The solution x(z) ∈$PC(J,\mathbb{R})$ of the impulsive FDEs with slit-strips type integral boundary condition

$$\left\{\begin{array}{cc}& {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }[x\left(z\right)-f_1\left(z\right)]=g_1\left(z\right), z\in \mathcal{J}, z\ne z_k, 1<{\alpha }_1\le 2, k=0,1,\cdots ,p,\hfill \\ & x\left(0\right)=0, x\left(\eta \right)=a_1{\int }_{z_k}^{{\delta }_{2k}}x\left(\tau \right)d\tau +a_2{\int }_{{\delta }_{2k+1}}^{z_{k+1}}x\left(\tau \right)d\tau ,\hfill \\ & \Delta x\left(z_k\right)=x\left(z_k^{+}\right)-x\left(z_k^{-}\right)=I_k\left(x\left(z_k\right)\right), \Delta x^{{}^{\prime }}\left(z_k\right)=x^{{}^{\prime }}\left(z_k^{+}\right)-x^{{}^{\prime }}\left(z_k^{-}\right)=J_k\left(x\left(z_k\right)\right), k=1,2,\cdots ,p,\hfill \end{array}\right.$$

(3)

is given by

$$\begin{array}{ccc}\hfill x\left(z\right)& =& f_1\left(z\right)+\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^z{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1\left(s\right)ds+\frac{1}{\Delta }[f_1\left(\eta \right)+\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\eta \right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1\left(s\right)ds\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1\left(s\right)ds\hfill \\ \hfill & & +\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}{g}_1\left(s\right)ds\hfill \\ \hfill & & +I_{i}x\left(z_i\right)+\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_{i}x\left(z_i\right))-a_1{\int }_{z_k}^{{\delta }_{2k}}(f_1\left(\tau \right)+\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\tau \right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1\left(s\right)ds\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1\left(s\right)ds\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}{g}_1\left(s\right)ds\hfill \\ \hfill & & +I_{i}x\left(z_i\right)+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_{i}x\left(z_i\right)\left)\right)d\tau -a_2{\int }_{{\delta }_{2k+1}}^{z_{k+1}}(f_1\left(\tau \right)+\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\tau \right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1\left(s\right)ds\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1\left(s\right)ds\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}{g}_1\left(s\right)ds\hfill \\ \hfill & & +I_{i}x\left(z_i\right)+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_{i}x\left(z_i\right)\left)\right)d\tau ](\psi \left(z\right)-\psi \left(z_k\right)\hfill \\ \hfill & & +\sum _{i=1}^p(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))+\sum _{i=1}^p\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_1}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1\left(s\right)ds\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_1}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}{g}_1\left(s\right)ds+\sum _{i=1}^p{I}_{i}x\left(z_i\right)\hfill \\ & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_{i}x\left(z_i\right), for z\in (z_k,z_{k+1}], k=1,2,\cdots ,p,\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill \Delta & =& a_1{\int }_{z_k}^{{\delta }_{2k}}(\psi \left(\tau \right)-\psi \left(z_k\right)+\sum _{i=1}^p(\psi \left(z_i\right)-\psi \left(z_{i-1}\right)))d\tau \hfill \\ & & +a_2{\int }_{{\delta }_{2k+1}}^{z_{k+1}}(\psi \left(\tau \right)-\psi \left(z_k\right)+\sum _{i=1}^p(\psi \left(z_i\right)-\psi \left(z_{i-1}\right)))d\tau \hfill \\ & & -(\psi \left(\eta \right)-\psi \left(z_k\right)+\sum _{i=1}^p(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))).\hfill \end{array}$$

Proof. 

Let

$${}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }[x\left(z\right)-f_1\left(z\right)]=g_1\left(z\right).$$

Then applying Lemma 1 to the differential Equation (3), for any $z\in {\mathcal{J}}_0$, there exist constants $c_0, c_1\in \mathbb{R}$, such that

$$x\left(z\right)=f_1\left(z\right)+\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_0}^z{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1\left(s\right)ds+c_0+c_1(\psi \left(z\right)-\psi \left(z_0\right)).$$

(4)

Using initial condition $x\left(0\right)=0$, Equation (4) yields that

$$c_0=0.$$

Therefore, Equation (4) takes the form

$$x\left(z\right)=f_1\left(z\right)+\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_0}^z{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1\left(s\right)ds+c_1(\psi \left(z\right)-\psi \left(z_0\right)).$$

Furthermore, we obtain

$$x^{{}^{\prime }}\left(z\right)=f_1^{{}^{\prime }}\left(z\right)+\frac{1}{\Gamma ({\alpha }_1-1)}{\int }_{z_0}^z{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z\right)-\psi \left(s\right))}^{{\alpha }_1-2}{g}_1\left(s\right)ds+c_1{\psi }^{{}^{\prime }}\left(z\right).$$

For $z\in (z_1,z_2]$, there are $d_0, d_1\in \mathbb{R}$ such that

$$\left\{\begin{array}{c}x\left(z\right)=f_1\left(z\right)+\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_1}^z{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1\left(s\right)ds+d_0+d_1(\psi \left(z\right)-\psi \left(z_1\right)),\hfill \\ \\ x^{{}^{\prime }}\left(z\right)=f_1^{{}^{\prime }}\left(z\right)+\frac{1}{\Gamma ({\alpha }_1-1)}{\int }_{z_1}^z{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z\right)-\psi \left(s\right))}^{{\alpha }_1-2}{g}_1\left(s\right)ds+d_1{\psi }^{{}^{\prime }}\left(z\right).\hfill \end{array}\right.$$

Hence it follows that

$$\left\{\begin{array}{c}x\left(z_1^{-}\right)=f_1\left(z\right)+\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_0}^{z_1}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_1\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1\left(s\right)ds+c_1(\psi \left(z_1\right)-\psi \left(z_0\right)),\hfill \\ x\left(z_1^{+}\right)=f_1\left(z\right)+d_0,\hfill \\ x^{{}^{\prime }}\left(z_1^{-}\right)=f_1^{{}^{\prime }}\left(z\right)+\frac{1}{\Gamma ({\alpha }_1-1)}{\int }_{z_0}^{z_1}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_1\right)-\psi \left(s\right))}^{{\alpha }_1-2}{g}_1\left(s\right)ds+c_1{\psi }^{{}^{\prime }}\left(z_1\right),\hfill \\ x^{{}^{\prime }}\left(z_1^{+}\right)=f_1^{{}^{\prime }}\left(z\right)+d_1{\psi }^{{}^{\prime }}\left(z_1\right).\hfill \end{array}\right.$$

Using

$$\left\{\begin{array}{c}\Delta x\left(z_1\right)=x\left(z_1^{+}\right)-x\left(z_1^{-}\right)=I_{1}x\left(z_1\right),\hfill \\ \Delta x^{\prime }\left(z_1\right)=x^{\prime }\left(z_1^{+}\right)-x^{\prime }\left(z_1^{-}\right)=J_{1}x\left(z_1\right),\hfill \end{array}\right.$$

we obtain

$$\left\{\begin{array}{c}{d}_0=\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_0}^{z_1}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_1\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1\left(s\right)ds+c_1(\psi \left(z_1\right)-\psi \left(z_0\right))+I_{1}x\left(z_1\right),\hfill \\ d_1=\frac{1}{{\psi }^{(}{z}_1)\Gamma ({\alpha }_1-1)}{\int }_{z_0}^{z_1}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_1\right)-\psi \left(s\right))}^{{\alpha }_1-2}{g}_1\left(s\right)ds+c_1+\frac{1}{{\psi }^{{}^{\prime }}\left(z_1\right)}{J}_{1}x\left(z_1\right).\hfill \end{array}\right.$$

Thus

$$\begin{array}{ccc}\hfill x\left(z\right)& =& f_1\left(z\right)+\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_1}^z{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1\left(s\right)ds\hfill \\ & & +\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_0}^{z_1}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_1\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1\left(s\right)ds\hfill \\ & & +c_1(\psi (z_1-\psi \left(z_0\right))+I_{1}x\left(z_1\right)+c_1(\psi \left(z\right)-\psi \left(z_1\right))\hfill \\ & & +\frac{\psi \left(z\right)-\psi \left(z_1\right)}{{\psi }^{{}^{\prime }}\left(z_1\right)\Gamma ({\alpha }_1-1)}{\int }_{z_0}^{z_1}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_1\right)-\psi \left(s\right))}^{{\alpha }_1-2}{g}_1\left(s\right)ds\hfill \\ & & +\frac{\psi \left(z\right)-\psi \left(z_1\right)}{{\psi }^{{}^{\prime }}\left(z_1\right)}{J}_{1}x\left(z_1\right), z\in (z_1,z_2].\hfill \end{array}$$

Similarly, we have

$$\begin{array}{ccc}\hfill x\left(z\right)& =& f_1\left(z\right)+\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^z{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1\left(s\right)ds+c_1(\psi \left(z\right)-\psi \left(z_k\right)\hfill \\ & & +\sum _{i=1}^p(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))+\sum _{i=1}^p\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_1}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1\left(s\right)ds\hfill \\ & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_1}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}{g}_1\left(s\right)ds+\sum _{i=1}^p{I}_{i}x\left(z_i\right)\hfill \\ & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_{i}x\left(z_i\right), for z\in (z_k,z_{k+1}], k=1,2,\cdots ,p.\hfill \end{array}$$

(5)

Finally, after applying $x\left(\eta \right)=a_1{\int }_{z_k}^{{\delta }_{2k}}x\left(\tau \right)d\tau +a_2{\int }_{{\delta }_{2k+1}}^{z_{k+1}}x\left(\tau \right)d\tau ,$ to (5) and calculating the value of $c_1$, we obtain

$$\begin{array}{ccc}\hfill c_1& =& \frac{1}{\Delta }[f_1\left(\eta \right)+\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\eta \right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1\left(s\right)ds\hfill \\ & & +\sum _{i=1}^p(\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1\left(s\right)ds\hfill \\ & & +\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}{g}_1\left(s\right)ds\hfill \\ & & +I_{i}x\left(z_i\right)+\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_{i}x\left(z_i\right))-a_1{\int }_{z_k}^{{\delta }_{2k}}(f_1\left(\tau \right)\hfill \\ & & +\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\tau \right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1\left(s\right)ds\hfill \\ & & +\sum _{i=1}^p(\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1\left(s\right)ds\hfill \\ & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}{g}_1\left(s\right)ds\hfill \\ & & +I_{i}x\left(z_i\right)+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_{i}x\left(z_i\right)\left)\right)d\tau -a_2{\int }_{{\delta }_{2k+1}}^{z_{k+1}}(f_1\left(\tau \right)\hfill \\ & & +\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\tau \right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1\left(s\right)ds\hfill \\ & & +\sum _{i=1}^p(\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1\left(s\right)ds\hfill \\ & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}{g}_1\left(s\right)ds\hfill \\ & & +I_{i}x\left(z_i\right)+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_{i}x\left(z_i\right)\left)\right)d\tau ],\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill \Delta & =& a_1{\int }_{z_k}^{{\delta }_{2k}}(\psi \left(\tau \right)-\psi \left(z_k\right)+\sum _{i=1}^p(\psi \left(z_i\right)-\psi \left(z_{i-1}\right)))d\tau \hfill \\ & & +a_2{\int }_{{\delta }_{2k+1}}^{z_{k+1}}(\psi \left(\tau \right)-\psi \left(z_k\right)+\sum _{i=1}^p(\psi \left(z_i\right)-\psi \left(z_{i-1}\right)))d\tau \hfill \\ & & -(\psi \left(\eta \right)-\psi \left(z_k\right)+\sum _{i=1}^p(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))).\hfill \end{array}$$

Put the value of $c_1$ in Equation (5), we get Equation (4). □

4. Existence and Uniqueness Results for the Problem (1)

In this section we study the existence of solution for the considered problem. Here we consider some hypothesis, which will be used in our results.

$\left(G_1\right)$

For each z ∈J and $x_1$,$x_2$∈${\mathbb{X}}^{{}^{\prime }}$, there exist positive constants $M_{f_1}$, $N_{f_1}$ such that

$$|f_1(z,x_1\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }{x}_1\left(z\right))-f_1(z,x_2\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1}{x}_2\left(z\right))|\le M_{f_1}|x_1\left(z\right)-x_2\left(z\right)|+N_{f_1}|{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }{x}_1\left(z\right)-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }{x}_2\left(z\right)|.$$

There exist positive constants $L_{g_1}$, $K_{g_1}$ such that

$$|g_1(z,x_1\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }{x}_1\left(z\right))-g_1(z,x_2\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }{x}_2\left(z\right))|\le L_{g_1}|x_1\left(z\right)-x_2\left(z\right)|+K_{g_1}|{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }{x}_1\left(z\right)-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }{x}_2\left(z\right)|.$$

$\left(G_2\right)$

For each z ∈J and $y_1$,$y_2$∈${\mathbb{Y}}^{{}^{\prime }}$, there exist positive constants $M_{f_2}$, $N_{f_2}$ such that

$$|f_2(z,y_1\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\beta }_1;\psi }{y}_1\left(z\right))-f_2(z,y_2\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\beta }_1;\psi }{y}_2\left(z\right))|\le M_{f_2}|y_1\left(z\right)-y_2\left(z\right)|+N_{f_2}|{}^c{\mathcal{D}}_{z_k,z}^{{\beta }_1;\psi }{y}_1\left(z\right)-{}^c{\mathcal{D}}_{z_k,z}^{{\beta }_1;\psi }{y}_2\left(z\right)|.$$

There exist positive constants $L_{g_2}$, $K_{g_2}$ such that

$$|g_2(z,y_1\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\beta }_1;\psi }{y}_1\left(z\right))-g_2(z,y_2\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\beta }_1;\psi }{y}_2\left(z\right))|\le L_{g_2}|y_1\left(z\right)-y_2\left(z\right)|+K_{g_2}|{}^c{\mathcal{D}}_{z_k,z}^{{\beta }_1;\psi }{y}_1\left(z\right)-{}^c{\mathcal{D}}_{z_k,z}^{{\beta }_1;\psi }{y}_2\left(z\right)|.$$

$\left(G_3\right)$

For every $x_1,x_2\in {\mathbb{X}}^{{}^{\prime }}$ and there exist constants $A_1,A_2>0$ such that

$$|I_k\left(x_1\left(z_k\right)\right)-I_k\left(x_2\left(z_k\right)\right)|\le A_1|x_1\left(z_k\right)-x_2\left(z_k\right)|,$$

$$|J_k\left(x_1\left(z_k\right)\right)-J_k\left(x_2\left(z_k\right)\right)|\le A_2|x_1\left(z_k\right)-x_2\left(z_k\right)|.$$

For every $y_1,y_2\in {\mathbb{Y}}^{{}^{\prime }}$ and there exist constants $A_3,A_4>0$ such that

$$|I_k^{*}\left(y_1\left(z_k\right)\right)-I_k^{*}\left(y_2\left(z_k\right)\right)|\le A_3|y_1\left(z_k\right)-y_2\left(z_k\right)|,$$

$$|J_k^{*}\left(y_1\left(z_k\right)\right)-J_k^{*}\left(y_2\left(z_k\right)\right)|\le A_4|y_1\left(z_k\right)-y_2\left(z_k\right)|.$$

$\left(G_4\right)$

There exist constants ${\theta }_0$, ${\theta }_1$ and ${\theta }_2$ such that

$$|f_1(z,x\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(z\right))|\le {\theta }_0\left(z\right)+{\theta }_1\left(z\right)|x\left(z\right)|+{\theta }_2\left(z\right)|{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(z\right)|,$$

with ${sup}_{z\in J}{\theta }_0\left(z\right)={\theta }_0^{🟉}$, ${sup}_{z\in J}{\theta }_1\left(z\right)={\theta }_1^{🟉}$ and ${sup}_{z\in J}{\theta }_2\left(z\right)={\theta }_2^{🟉}$.

There exist constants ${\theta }_3$, ${\theta }_4$ and ${\theta }_5$ such that

$$|g_1(z,x\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(z\right))|\le {\theta }_3\left(z\right)+{\theta }_4\left(z\right)|x\left(z\right)|+{\theta }_5\left(z\right)|{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(z\right)|,$$

with ${sup}_{z\in J}{\theta }_3\left(z\right)={\theta }_3^{🟉}$, ${sup}_{z\in J}{\theta }_4\left(z\right)={\theta }_4^{🟉}$ and ${sup}_{z\in J}{\theta }_5\left(z\right)={\theta }_5^{🟉}$.

$\left(G_5\right)$

For each $x\left(z\right)\in \mathbb{R}$ there exist constants $A_1,N_1>0$ and $A_2,N_2>0$ such that the functions $I_k,J_k:\mathbb{R}\to \mathbb{R}$ are continuous and satisfy the inequalities:

$$|I_{k}x\left(z_k\right)|\le A_1|x\left(z\right)|+N_1, |J_{k}x\left(z_k\right)|\le A_2|x\left(z\right)|+N_2, k=1,2,\cdots ,p.$$

For each $y\left(z\right)\in$ ${\mathbb{Y}}^{{}^{\prime }}$ there exist constants $A_3,N_3>0$ and $A_4,N_4>0$ such that the functions $I_k^{*},J_k^{*}:\mathbb{R}\to \mathbb{R}$ are continuous and satisfy the inequalities:

$$|I_k^{*}y\left(z_k\right)|\le A_3|y\left(z\right)|+N_3, |J_k^{*}v\left(z_k\right)|\le A_4|y\left(z\right)|+N_4, k=1,2,\cdots ,p.$$

Let us define an operator $P:{\mathbb{X}}^{{}^{\prime }}\times {\mathbb{Y}}^{{}^{\prime }}⟶{\mathbb{X}}^{{}^{\prime }}\times {\mathbb{Y}}^{{}^{\prime }}$ such that

$$P(x,y)\left(z\right)=(P_1(x,y)\left(z\right),P_2(x,y)\left(z\right)),$$

where

$$\begin{array}{ccc}\hfill P_1(x,y)\left(z\right)& =& f_1(z,x\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(z\right))\hfill \\ \hfill & & +\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^z{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +\frac{1}{\Delta }[f_1(\eta ,x\left(\eta \right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(\eta \right))+\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\eta \right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +I_{i}x\left(z_i\right)+\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_{i}x\left(z_i\right))-a_1{\int }_{z_k}^{{\delta }_{2k}}\hfill \\ \hfill & & (f_1(\tau ,x\left(\tau \right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(\tau \right))+\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\tau \right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +I_{i}x\left(z_i\right)+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_{i}x\left(z_i\right)\left)\right)d\tau -a_2{\int }_{{\delta }_{2k+1}}^{z_{k+1}}\hfill \\ \hfill & & (f_1(\tau ,x\left(\tau \right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(\tau \right))+\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\tau \right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +I_{i}x\left(z_i\right)+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_{i}x\left(z_i\right)\left)\right)d\tau ](\psi \left(z\right)-\psi \left(z_k\right))\hfill \\ & & +\sum _{i=1}^p(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))+\sum _{i=1}^p\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_1}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_1}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds+\sum _{i=1}^p{I}_{i}x\left(z_i\right)\hfill \\ & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_{i}x\left(z_i\right),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill P_2(x,y)\left(z\right)& =& f_2(z,x\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(z\right))+\frac{1}{\Gamma \left({\beta }_1\right)}{\int }_{z_k}^z{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z\right)-\psi \left(s\right))}^{{\beta }_1-1}{g}_2(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +\frac{1}{\Delta }[f_2(\eta ,x\left(\eta \right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(\eta \right))+\frac{1}{\Gamma \left({\beta }_1\right)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\eta \right)-\psi \left(s\right))}^{{\beta }_1-1}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{1}{\Gamma \left({\beta }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\beta }_1-1}{g}_2(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\beta }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\beta }_1-2}{g}_2(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +I_i^{*}y\left(z_i\right)+\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_i^{*}y\left(z_i\right))\hfill \\ \hfill & & -b_1{\int }_{z_k}^{{\delta }_{2k}}(f_2(\tau ,x\left(\tau \right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(\tau \right))+\frac{1}{\Gamma \left({\beta }_1\right)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\tau \right)-\psi \left(s\right))}^{{\beta }_1-1}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{1}{\Gamma \left({\beta }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\beta }_1-1}{g}_2(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\beta }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\beta }_1-2}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +I_i^{*}y\left(z_i\right)+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_i^{*}y\left(z_i\right)\left)\right)d\tau \hfill \\ \hfill & & -b_2{\int }_{{\delta }_{2k+1}}^{z_{k+1}}(f_2(\tau ,x\left(\tau \right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(\tau \right))+\frac{1}{\Gamma \left({\beta }_1\right)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\tau \right)-\psi \left(s\right))}^{{\beta }_1-1}{g}_2(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{1}{\Gamma \left({\beta }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\beta }_1-1}{g}_2(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\beta }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\beta }_1-2}{g}_2(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +I_i^{*}y\left(z_i\right)+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_i^{*}y\left(z_i\right)\left)\right)d\tau ](\psi \left(z\right)-\psi \left(z_k\right))\hfill \\ & & +\sum _{i=1}^p(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))\hfill \\ \hfill & & +\sum _{i=1}^p\frac{1}{\Gamma \left({\beta }_1\right)}{\int }_{z_{i-1}}^{z_1}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\beta }_1-1}{g}_2(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\beta }_1-1)}{\int }_{z_{i-1}}^{z_1}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\beta }_1-2}{g}_2(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds+\sum _{i=1}^p{I}_i^{*}y\left(z_i\right)\hfill \\ & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_i^{*}y\left(z_i\right).\hfill \end{array}$$

Our first result is stated as follows.

Theorem 1.

Assume that the conditions $\left(G_1\right)-\left(G_3\right)$ are satisfied, and

$$Z^{*}=max\{Z_1,Z_2\}<1,$$

(6)

where $Z_1$ and $Z_2$ are provided in the proof, then the coupled system (1) has a unique solution.

Proof. 

Let $(x,y),(\overline{x},\overline{y})$ $\in {\mathbb{X}}^{{}^{\prime }}\times {\mathbb{Y}}^{{}^{\prime }}$ then:

$$\begin{array}{ccc}& & |P_1(x,y)\left(z\right)-P_1(\overline{x},\overline{y})\left(z\right)|\hfill \\ & \le & |f_1(z,x\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(z\right))-f_1(z,\overline{x}\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\left(z\right))|\hfill \\ \hfill & & +\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^z{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z\right)-\psi \left(s\right))}^{{\alpha }_1-1}(|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))-g_1(s,\overline{x}\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\left(s\right))|)ds\hfill \\ \hfill & & +\frac{1}{|\Delta |}[|f_1(\eta ,x\left(\eta \right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(\eta \right))-f_1(\eta ,\overline{x}\left(\eta \right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\left(\eta \right))|\hfill \\ \hfill & & +\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\eta \right)-\psi \left(s\right))}^{{\alpha }_1-1}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))-g_1(s,\overline{x}\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\left(s\right))|ds\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))-g_1(s,\overline{x}\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\left(s\right))|ds\hfill \\ \hfill & & +\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))-g_1(s,\overline{x}\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\left(s\right))|ds\hfill \\ \hfill & & +|I_{i}x\left(z_i\right)-I_i\overline{x}\left(z_i\right)|+\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}|J_{i}x\left(z_i\right)-J_i\overline{x}\left(z_i\right)|)\hfill \\ \hfill & & +|a_1|{\int }_{z_k}^{{\delta }_{2k}}(|f_1(\tau ,x\left(\tau \right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(\tau \right))-f_1(\tau ,\overline{x}\left(\tau \right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\left(\tau \right))|+\frac{1}{\Gamma \left({\alpha }_1\right)}\hfill \\ \hfill & & \times {\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\tau \right)-\psi \left(s\right))}^{{\alpha }_1-1}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))-g_1(s,\overline{x}\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\left(s\right))|ds\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))-g_1(s,\overline{x}\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\left(s\right))|ds\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))-g_1(s,\overline{x}\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\left(s\right))|ds\hfill \\ \hfill & & +|I_{i}x\left(z_i\right)-I_i\overline{x}\left(z_i\right)|+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}|J_{i}x\left(z_i\right)-J_i\overline{x}\left(z_i\right)|\left)\right)d\tau \hfill \\ \hfill & & +|a_2|{\int }_{{\delta }_{2k+1}}^{z_{k+1}}(|f_1(\tau ,x\left(\tau \right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(\tau \right))-f_1(\tau ,\overline{x}\left(\tau \right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\left(\tau \right))|+\frac{1}{\Gamma \left({\alpha }_1\right)}\hfill \\ \hfill & & \times {\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\tau \right)-\psi \left(s\right))}^{{\alpha }_1-1}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))-g_1(s,\overline{x}\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\left(s\right))|ds\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))-g_1(s,\overline{x}\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\left(s\right))|ds\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))-g_1(s,\overline{x}\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\left(s\right))|ds\hfill \\ \hfill & & +|I_{i}x\left(z_i\right)-I_i\overline{x}\left(z_i\right)|+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}|J_{i}x\left(z_i\right)-J_i\overline{x}\left(z_i\right)|\left)\right)d\tau ](\psi \left(z\right)-\psi \left(z_k\right))\hfill \\ \hfill & & +\sum _{i=1}^p\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_1}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))\hfill \\ \hfill & & -g_1(s,\overline{x}\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\left(s\right))|ds\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_1}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))\hfill \\ \hfill & & -g_1(s,\overline{x}\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\left(s\right))|ds+\sum _{i=1}^p|I_{i}x\left(z_i\right)-I_i\overline{x}\left(z_i\right)|\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}|J_{i}x\left(z_i\right)-J_i\overline{x}\left(z_i\right)|.\hfill \end{array}$$

$$\begin{array}{ccc}\hfill |P_1(x,y)\left(z\right)-P_1(\overline{x},\overline{y})\left(z\right)|& \le & M_{f_1}\parallel x-\overline{x}\parallel +N_{f_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel \hfill \\ \hfill & & +\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}(L_{g_1}\parallel x-\overline{x}\parallel +K_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel )\hfill \\ \hfill & & +\frac{1}{|\Delta |}[M_{f_1}\parallel x-\overline{x}\parallel +N_{f_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel \hfill \\ \hfill & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}(L_{g_1}\parallel x-\overline{x}\parallel +K_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel )\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}(L_{g_1}\parallel x-\overline{x}\parallel +K_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel )+\hfill \\ \hfill & & +\frac{(\psi \left(\eta \right)-\psi \left(z_i\right)){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)}(L_{g_1}\parallel x-\overline{x}\parallel \hfill \\ \hfill & & +K_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel )+A_1|x\left(z_i\right)-\overline{x}\left(z_i\right)|+\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{A}_2|x\left(z_i\right)-\overline{x}\left(z_i\right)|)\hfill \\ \hfill & & +|a_1|{\int }_{z_k}^{{\delta }_{2k}}(M_{f_1}\parallel x-\overline{x}\parallel +N_{f_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel \hfill \\ \hfill & & +\frac{{(\psi \left(\tau \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}(L_{g_1}\parallel x-\overline{x}\parallel +K_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel )\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}(L_{g_1}\parallel x-\overline{x}\parallel +K_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel )\hfill \\ \hfill & & +\frac{(\psi \left(\tau \right)-\psi \left(z_i\right)){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)}(L_{g_1}\parallel x-\overline{x}\parallel +K_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel )\hfill \\ \hfill & & +A_1|x\left(z_1\right)-\overline{x}\left(z_i\right)|+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{A}_2|x\left(z_i\right)-\overline{x}\left(z_i\right)|\left)\right)d\tau \hfill \\ \hfill & & +|a_2|{\int }_{{\delta }_{2k+1}}^{z_{k+1}}((M_{f_1}\parallel x-\overline{x}\parallel +N_{f_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel )\hfill \\ \hfill & & \frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma \left({\alpha }_1\right)}(L_{g_1}\parallel x-\overline{x}\parallel +K_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel )\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}(L_{g_1}\parallel x-\overline{x}\parallel +K_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel )\hfill \\ \hfill & & +\frac{(\psi \left(\tau \right)-\psi \left(z_i\right)){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }\left(z_i\right)\Gamma \left({\alpha }_1\right)}}(L_{g_1}\parallel x-\overline{x}\parallel +K_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel )\hfill \\ \hfill & & +A_1|x\left(z_i\right)-\overline{x}\left(z_i\right)|+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{A}_2\parallel x-\overline{x}\parallel \left)\right)d\tau ]|\psi \left(z\right)-\psi \left(z_k\right)|\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{\alpha }}{\Gamma ({\alpha }_1+1)}(L_{g_1}\parallel x-\overline{x}\parallel \hfill \\ \hfill & & +K_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel )+\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{\alpha -1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)}(L_{g_1}\parallel x-\overline{x}\parallel \hfill \\ \hfill & & +K_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel )+\sum _{i=1}^p{A}_1|x\left(z_i\right)-\overline{x}\left(z_i\right)|\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{A}_2|x\left(z_i\right)-\overline{x}\left(z_i\right)|\hfill \end{array}$$

$$\begin{array}{ccc}\hfill |P_1(x,y)\left(z\right)-P_1(\overline{x},\overline{y})\left(z\right)|& \le & M_{f_1}\parallel x-\overline{x}\parallel +N_{f_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel +\hfill \\ \hfill & & +\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}{L}_{g_1}\parallel x-\overline{x}\parallel +\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}{K}_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel )\hfill \\ \hfill & & +(\frac{M_{f_1}}{|\Delta |}\parallel x-\overline{x}\parallel +\frac{N_{f_1}}{|\Delta |}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel \hfill \\ \hfill & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}{L}_{g_1}\parallel x-\overline{x}\parallel +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}{K}_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel \hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}{L}_{g_1}\parallel x-\overline{x}\parallel +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}{K}_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel \hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}{L}_{g_1}\parallel x-\overline{x}\parallel \hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}{K}_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel \hfill \\ \hfill & & +\sum _{i=1}^p\frac{A_1}{|\Delta |}\parallel x-\overline{x}\parallel +\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}{A}_2\parallel x-\overline{x}\parallel \hfill \\ \hfill & & +\frac{|a_1|}{|\Delta |}({\delta }_{2k}-z_k)M_{f_1}\parallel x-\overline{x}\parallel +\frac{|a_1|}{|\Delta |}({\delta }_{2k}-z_k)N_{f_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel \hfill \\ \hfill & & +\frac{{(\psi \left(\tau \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)L_{g_1}\parallel x-\overline{x}\parallel \hfill \\ \hfill & & +\frac{{(\psi \left(\tau \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)K_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel \hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)L_{g_1}\parallel x-\overline{x}\parallel \hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)K_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel \hfill \\ \hfill & & +\frac{(\psi \left(\tau \right)-\psi \left(z_i\right)){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)L_{g_1}\parallel x-\overline{x}\parallel \hfill \\ \hfill & & +\frac{(\psi \left(\tau \right)-\psi \left(z_i\right)){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)K_{g_1}\hfill \\ \hfill & & \times \parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel \hfill \\ \hfill & & +p\frac{A_1}{|\Delta |}|a_1|({\delta }_{2k}-z_k)\parallel x-\overline{x}\parallel \hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)A_2\parallel x-\overline{x}\parallel \hfill \\ \hfill & & +\frac{|a_2|}{|\Delta |}(z_{k+1}-{\delta }_{2k+1})M_{f_1}\parallel x-\overline{x}\parallel \hfill \\ \hfill & & +\frac{|a_2|(z_{k+1}-{\delta }_{2k+1})}{|\Delta |}{N}_{f_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel \hfill \\ \hfill & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma \left({\alpha }_1\right)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})L_{g_1}\parallel x-\overline{x}\parallel \hfill \\ \hfill & \end{array}$$

$$\begin{array}{ccc}& & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma \left({\alpha }_1\right)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})K_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel \hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})L_{g_1}\parallel x-\overline{x}\parallel \hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})K_{g_1}\hfill \\ \hfill & & \times \parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel \hfill \\ \hfill & & +p\frac{A_1}{|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})\parallel x-\overline{x}\parallel \hfill \\ \hfill & & +\sum _{i=1}^p\frac{A_2}{|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})\parallel x-\overline{x}\parallel )\left(\psi \left(z\right)-\psi \left(z_k\right)\right)\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{\alpha }}{\Gamma ({\alpha }_1+1)}{L}_{g_1}\parallel x-\overline{x}\parallel \hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}{K}_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel \hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)}{L}_{g_1}\parallel x-\overline{x}\parallel \hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)}{K}_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel \hfill \\ \hfill & & +p{A}_1\parallel x-\overline{x}\parallel +A_2\sum _{i=1}^p\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\parallel x-\overline{x}\parallel .\hfill \end{array}$$

$$\begin{array}{ccc}\hfill |P_1(x,y)\left(z\right)-P_1(\overline{x},\overline{y})\left(z\right)|& \le & (M_{f_1}+\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}{L}_{g_1}\hfill \\ \hfill & & +(\frac{M_{f_1}}{|\Delta |}+\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}{L}_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}{L}_{g_1}+\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}{L}_{g_1}\hfill \\ \hfill & & +p\frac{A_1}{|\Delta |}+\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}{A}_2\hfill \\ \hfill & & +\frac{|a_1|}{|\Delta |}({\delta }_{2k}-z_k)M_{f_1}\hfill \\ \hfill & & +\frac{{(\psi \left(\tau \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)L_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)L_{g_1}\hfill \\ \hfill & & +\frac{(\psi \left(\tau \right)-\psi \left(z_i\right)){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)L_{g_1}+p\frac{A_1}{|\Delta |}|a_1|({\delta }_{2k}-z_k)\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)A_2+\frac{|a_2|}{|\Delta |}(z_{k+1}-{\delta }_{2k+1})M_{f_1}\hfill \\ \hfill \end{array}$$

$$\begin{array}{ccc}& & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma \left({\alpha }_1\right)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})L_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})L_{g_1}\hfill \\ \hfill \\ & & +p\frac{A_1}{|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})\hfill \\ \hfill & & +\sum _{i=1}^p\frac{A_2}{|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\left)\right(\psi \left(z\right)-\psi \left(z_k\right))\hfill \\ \hfill \\ & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{\alpha }}{\Gamma ({\alpha }_1+1)}{L}_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)}{L}_{g_1}\hfill \\ \hfill & & +p{A}_1+A_2\sum _{i=1}^p\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)})\parallel x-\overline{x}\parallel \hfill \\ & & +(N_{f_1}+\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}{K}_{g_1}\hfill \\ \hfill & & +(\frac{N_{f_1}}{|\Delta |}+\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}{K}_{g_1}+\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}{K}_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}{K}_{g_1}+\frac{|a_1|}{|\Delta |}({\delta }_{2k}-z_k)N_{f_1}\hfill \\ \hfill & & +\frac{{(\psi \left(\tau \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)K_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)K_{g_1}\hfill \\ \hfill & & +\frac{(\psi \left(\tau \right)-\psi \left(z_i\right)){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)K_{g_1}+\frac{|a_2|(z_{k+1}-{\delta }_{2k+1})}{|\Delta |}{N}_{f_1}\hfill \\ \hfill & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma \left({\alpha }_1\right)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})K_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})K_{g_1}\left)\right(\psi \left(z\right)-\psi \left(z_k\right))\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}{K}_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)}{K}_{g_1})\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel .\hfill \end{array}$$

Let

$$\begin{array}{ccc}\hfill {\Omega }_1& =& M_{f_1}+\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}{L}_{g_1}\hfill \\ \hfill & & +(\frac{M_{f_1}}{|\Delta |}+\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}{L}_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}{L}_{g_1}+\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}{L}_{g_1}\hfill \\ \hfill & & +p\frac{A_1}{|\Delta |}+\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}{A}_2\hfill \\ \hfill & & +\frac{|a_1|}{|\Delta |}({\delta }_{2k}-z_k)M_{f_1}\hfill \\ \hfill & & +\frac{{(\psi \left(\tau \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)L_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)L_{g_1}\hfill \\ \hfill & & +\frac{(\psi \left(\tau \right)-\psi \left(z_i\right)){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)L_{g_1}+p\frac{A_1}{|\Delta |}|a_1|({\delta }_{2k}-z_k)\hfill \\ \hfill & & +A_2\sum _{i=1}^p\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)+\frac{|a_2|}{|\Delta |}(z_{k+1}-{\delta }_{2k+1})M_{f_1}\hfill \\ \hfill & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma \left({\alpha }_1\right)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})L_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})L_{g_1}\hfill \\ \hfill & & +p\frac{A_1}{|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})\hfill \\ \hfill & & +\frac{A_2}{|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})\sum _{i=1}^p\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\left)\right(\psi \left(z\right)-\psi \left(z_k\right))\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{\alpha }}{\Gamma ({\alpha }_1+1)}{L}_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)}{L}_{g_1}\hfill \\ \hfill & & +p{A}_1+A_2\sum _{i=1}^p\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)})\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\Omega }_2& =& N_{f_1}+\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}{K}_{g_1}\hfill \\ \hfill & & +(\frac{N_{f_1}}{|\Delta |}+\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}{K}_{g_1}+\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}{K}_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}{K}_{g_1}+\frac{|a_1|}{|\Delta |}({\delta }_{2k}-z_k)N_{f_1}\hfill \\ \hfill & & +\frac{{(\psi \left(\tau \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)K_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)K_{g_1}\hfill \\ \hfill & & +\frac{(\psi \left(\tau \right)-\psi \left(z_i\right)){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)K_{g_1}+\frac{|a_2|(z_{k+1}-{\delta }_{2k+1})}{|\Delta |}{N}_{f_1}\hfill \\ \hfill & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma \left({\alpha }_1\right)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})K_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})K_{g_1}\left)\right(\psi \left(z\right)-\psi \left(z_k\right))\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}{K}_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)}{K}_{g_1}.\hfill \end{array}$$

Thus we have,

$$\begin{array}{ccc}\hfill \parallel P_1(x,y)-P_1(\overline{x},\overline{y})\parallel & \le & {\Omega }_1\parallel x-\overline{x}\parallel +{\Omega }_2\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel .\hfill \end{array}$$

But

$$\begin{array}{c}\\ \hfill \parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel & \le & \frac{\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\alpha }_1}}{\Gamma \left({\alpha }_1\right)}\left(\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}\right)M_{f_1}+L_{g_1}}{1-(\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\alpha }_1}}{\Gamma \left({\alpha }_1\right)}\left(\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}\right)N_{f_1}+K_{g_1})}\parallel x-\overline{x}\parallel .\hfill \end{array}$$

Then,

$$\begin{array}{ccc}\hfill \parallel P_1(x,y)-P_1(\overline{x},\overline{y})\parallel & \le & \left({\Omega }_1+{\Omega }_2\frac{\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\alpha }_1}}{\Gamma \left({\alpha }_1\right)}\left(\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}\right)M_{f_1}+L_{g_1}}{1-(\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\alpha }_1}}{\Gamma \left({\alpha }_1\right)}\left(\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}\right)N_{f_1}+K_{g_1})}\right)\parallel x-\overline{x}\parallel .\hfill \end{array}$$

Provided that,

$$\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\alpha }_1}}{\Gamma \left({\alpha }_1\right)}\left(\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}\right)N_{f_1}+K_{g_1}<1.$$

Similarly,

$$\begin{array}{ccc}\hfill \parallel P_2(x,y)-P_2(\overline{x},\overline{y})\parallel & \le & \left({\Omega }_3+{\Omega }_4\frac{\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\beta }_1}}{\Gamma \left({\beta }_1\right)}\left(\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}\right)M_{f_2}+L_{g_2}}{1-(\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\beta }_1}}{\Gamma \left({\beta }_1\right)}\left(\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}\right)N_{f_2}+K_{g_2})}\right)\parallel y-\overline{y}\parallel .\hfill \end{array}$$

Provided that,

$$\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\beta }_1}}{\Gamma \left({\beta }_1\right)}\left(\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}\right)N_{f_2}+K_{g_2}<1.$$

Where

$$\begin{array}{ccc}\hfill {\Omega }_3& =& M_{f_2}+\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{{\beta }_1}}{\Gamma ({\beta }_2+1)}{L}_{g_2}\hfill \\ \hfill & & +(\frac{M_{f_2}}{|\Delta |}+\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\beta }_2}}{\Gamma ({\beta }_2+1)|\Delta |}{L}_{g_2}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_2}}{\Gamma ({\beta }_2+1)|\Delta |}{L}_{g_2}+\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_2-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\beta }_2\right)|\Delta |}{L}_{g_2}\hfill \\ \hfill & & +p\frac{A_1}{|\Delta |}+\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}{A}_2\hfill \\ \hfill & & +\frac{|a_2|}{|\Delta |}({\delta }_{2k}-z_k)M_{f_2}\hfill \\ \hfill & & +\frac{{(\psi \left(\tau \right)-\psi \left(z_k\right))}^{{\beta }_2}}{\Gamma ({\beta }_2+1)|\Delta |}|a_2|({\delta }_{2k}-z_k)L_{g_2}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_2}}{\Gamma ({\beta }_2+1)|\Delta |}|a_2|({\delta }_{2k}-z_k)L_{g_2}\hfill \\ \hfill & & +\frac{(\psi \left(\tau \right)-\psi \left(z_i\right)){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_2-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\beta }_2\right)|\Delta |}|a_2|({\delta }_{2k}-z_k)L_{g_2}+\frac{A_1}{|\Delta |}|a_2|({\delta }_{2k}-z_k)\hfill \\ \hfill & & +\sum _{i=1}^{}\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}|a_2|({\delta }_{2k}-z_k)A_2+\frac{|a_2|}{|\Delta |}(z_{k+1}-{\delta }_{2k+1})M_{f_2}\hfill \\ \hfill & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\beta }_2}}{\Gamma \left({\beta }_2\right)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})L_{g_2}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_2}}{\Gamma ({\beta }_2+1)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})L_{g_2}\hfill \\ \hfill & & +\frac{A_1}{|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})\hfill \\ \hfill & & +\frac{A_2}{|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})\sum _{i=1}^p\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\left)\right(\psi \left(z\right)-\psi \left(z_k\right))\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{\alpha }}{\Gamma ({\beta }_2+1)}{L}_{g_2}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_2-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\beta }_2\right)}{L}_{g_2}\hfill \\ \hfill & & +p{A}_1+A_2\sum _{i=1}^p\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)})\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\Omega }_4& =& N_{f_2}+\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{{\beta }_2}}{\Gamma ({\beta }_2+1)}{K}_{g_2}\hfill \\ \hfill & & +(\frac{N_{f_2}}{|\Delta |}+\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\beta }_2}}{\Gamma ({\beta }_2+1)|\Delta |}{K}_{g_2}+\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_2}}{\Gamma ({\beta }_2+1)|\Delta |}{K}_{g_2}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_2-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\beta }_2\right)|\Delta |}{K}_{g_2}+\frac{|a_2|}{|\Delta |}({\delta }_{2k}-z_k)N_{f_2}\hfill \\ \hfill & & +\frac{{(\psi \left(\tau \right)-\psi \left(z_k\right))}^{{\beta }_2}}{\Gamma ({\beta }_2+1)|\Delta |}|a_2|({\delta }_{2k}-z_k)K_{g_2}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_2}}{\Gamma ({\beta }_2+1)|\Delta |}|a_2|({\delta }_{2k}-z_k)K_{g_2}\hfill \\ \hfill & & +\frac{(\psi \left(\tau \right)-\psi \left(z_i\right)){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_2-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\beta }_2\right)|\Delta |}|a_2|({\delta }_{2k}-z_k)K_{g_2}+\frac{|a_2|(z_{k+1}-{\delta }_{2k+1})}{|\Delta |}{N}_{f_2}\hfill \\ \hfill & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\beta }_2}}{\Gamma \left({\beta }_2\right)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})K_{g_2}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_2}}{\Gamma ({\beta }_2+1)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})K_{g_2}\left)\right(\psi \left(z\right)-\psi \left(z_k\right))\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_2}}{\Gamma ({\beta }_2+1)}{K}_{g_2}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_2-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\beta }_2\right)}{K}_{g_2}.\hfill \end{array}$$

Let

$$\begin{array}{ccc}\hfill Z_1& =& {\Omega }_1+{\Omega }_2\frac{\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\alpha }_1}}{\Gamma \left({\alpha }_1\right)}\left(\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}\right)M_{f_1}+L_{g_1}}{1-(\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\alpha }_1}}{\Gamma \left({\alpha }_1\right)}\left(\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}\right)N_{f_1}+K_{g_1})}.\hfill \end{array}$$

Then we obtain

$$\parallel P_1(x,y)-P_1(\overline{x},\overline{y})\parallel \le Z_1\parallel x-\overline{x}\parallel .$$

Similarly,

$$\parallel P_2(x,y)-P_2(\overline{x},\overline{y})\parallel \le Z_2\parallel y-\overline{y}\parallel ,$$

where

$$\begin{array}{ccc}\hfill Z_2& =& {\Omega }_3+{\Omega }_4\frac{\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\beta }_1}}{\Gamma \left({\beta }_1\right)}\left(\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}\right)M_{f_2}+L_{g_2}}{1-(\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\beta }_1}}{\Gamma \left({\beta }_1\right)}\left(\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}\right)N_{f_2}+K_{g_2})}.\hfill \end{array}$$

As it is assumed that

$$max\{Z_1,Z_2\}=Z^{*}<1.$$

(7)

So we have

$$\parallel P(x,y)-P(\overline{x},\overline{y})\parallel \le Z^{*}(\parallel x-\overline{x}\parallel +\parallel y-\overline{y}\parallel ).$$

Then from above inequality we can say that P is a contraction mapping and by Banach contraction principle P has a unique fixed point. □

Theorem 2.

Assume that the conditions $\left(G_1\right)$–$\left(G_5\right)$ are satisfied, then the coupled system (1) has at least one solution.

Proof. 

To prove that the coupled system (1) has at least one solution, we use the Schaefer’s fixed point theorem. As $f_1,g_1,I,J$ are continuous functions, so $P_1$ is continuous. Also from the continuity of $f_2,g_2$ and $I^{*},J^{*}$ the operator $P_2$ is continuous. This shows that P is continuous.

Consider a set:

$$Q_r=\{(x,y)\in {\mathbb{X}}^{\prime }\times {\mathbb{Y}}^{\prime }:\parallel (x,y)\parallel \le r\}.$$

For any $z\in [0,Z]$, we have

$$\begin{array}{ccc}\hfill |P_1(x,y)(z)|& \le & |f_1(z,x(z),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x(z))|+\frac{1}{\Gamma ({\alpha }_1)}{\int }_{z_k}^z{\psi }^{{}^{\prime }}(s){(\psi (z)-\psi (s))}^{{\alpha }_1-1}|g_1(s,x(s),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x(s))|ds\hfill \\ & & +\frac{1}{|\Delta |}[|f_1(\eta ,x\left(\eta \right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(\eta \right))|+\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\eta \right)-\psi \left(s\right))}^{{\alpha }_1-1}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))|ds\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))|ds\hfill \\ \hfill & & +\frac{\psi (\eta )-\psi (z_i)}{{\psi }^{{}^{\prime }}(z_i)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}(s){(\psi (z_i)-\psi (s))}^{{\alpha }_1-2}|g_1(s,x(s),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x(s))|ds\hfill \\ & & +|I_{i}x\left(z_i\right)|+\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}|J_{i}x\left(z_i\right)|)\hfill \\ & & +|a_1|{\int }_{z_k}^{{\delta }_{2k}}(|f_1(\tau ,x(\tau ),{}^c{\mathcal{D}}_{z_k,\tau }^{{\alpha }_1;\psi }x(\tau ))|+\frac{1}{\Gamma ({\alpha }_1)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}(s){(\psi (\tau )-\psi (s))}^{{\alpha }_1-1}|g_1(s,x(s),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x(s))|ds\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{1}{\Gamma ({\alpha }_1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}(s){(\psi (z_i)-\psi (s))}^{{\alpha }_1-1}|g_1(s,x(s),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x(s))|ds\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))|ds\hfill \\ \hfill & & +|I_{i}x\left(z_i\right)|+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}|J_{i}x\left(z_i\right)|\left)\right)d\tau \hfill \\ & & +|a_2|{\int }_{{\delta }_{2k+1}}^{z_{k+1}}(|f_1(\tau ,x\left(\tau \right),{}^c{\mathcal{D}}_{z_k,\tau }^{{\alpha }_1;\psi }x\left(\tau \right))|+\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\tau \right)-\psi \left(s\right))}^{{\alpha }_1-1}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))|ds\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))|ds\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))|ds\hfill \\ \hfill & & +|I_{i}x(z_i|)+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}|J_{i}x(z_i|)\left)\right)d\tau ](\psi \left(z\right)-\psi \left(z_k\right))\hfill \\ & & +\sum _{i=1}^p(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))+\sum _{i=1}^p(\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_1}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))|ds\hfill \\ \hfill & & +\frac{\psi \left(z\right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_1}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}{g}_1\left(s\right)ds+|I_{i}x\left(z_i\right)|\hfill \\ & & +\frac{\psi \left(z\right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}|J_{i}x\left(z_i\right)|)\hfill \end{array}$$

$$\begin{array}{ccc}& \le & {\theta }_0\left(z\right)+{\theta }_1\left(z\right)|x\left(z\right)|+{\theta }_2\left(z\right)|{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1,\psi }x\left(z\right)|+\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3\left(z\right)+{\theta }_4\left(z\right)|x\left(z\right)|+{\theta }_5\left(z\right)|{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1,\psi }x\left(z\right)|)\hfill \\ & & +\frac{1}{|\Delta |}[{\theta }_0\left(\eta \right)+{\theta }_1\left(\eta \right)|x\left(\eta \right)|+{\theta }_2\left(\eta \right)|{}^c{\mathcal{D}}_{z_k,\eta }^{{\alpha }_1,\psi }x\left(\eta \right)|\hfill \\ & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3\left(\eta \right)+{\theta }_4\left(\eta \right)|x\left(\eta \right)|+{\theta }_5\left(\eta \right)|{}^c{\mathcal{D}}_{z_k,\eta }^{{\alpha }_1,\psi }x\left(\eta \right)|)\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3\left(z_i\right)+{\theta }_4\left(z_i\right)|x\left(z_i\right)|+{\theta }_5\left(z_i\right)|{}^c{\mathcal{D}}_{z_{i-1},z_i}^{{\alpha }_1,\psi }x\left(z_i\right)|)\hfill \\ \hfill & & +\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\times \frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}({\theta }_3\left(z_i\right)+{\theta }_4\left(z_i\right)|x\left(z_i\right)|+{\theta }_5\left(z_i\right)|{}^c{\mathcal{D}}_{z_{i-1},z_i}^{{\alpha }_1,\psi }x\left(z_i\right)|)\hfill \\ & & +A_1|x\left(z\right)|+N_1+\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}(A_2|x\left(z\right)|+N_2))\hfill \\ & & +|a_1|{\int }_{z_k}^{{\delta }_{2k}}({\theta }_0\left(\tau \right)+{\theta }_1\left(\tau \right)|x\left(\eta \right)|+{\theta }_2\left(\tau \right)|{}^c{\mathcal{D}}_{z_k,\tau }^{{\alpha }_1,\psi }x\left(\tau \right)|\hfill \\ & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3\left(\eta \right)+{\theta }_4\left(\eta \right)|x\left(\eta \right)|+{\theta }_5\left(\eta \right)|{}^c{\mathcal{D}}_{z_k,\eta }^{{\alpha }_1,\psi }x\left(\eta \right)|)\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3\left(z_i\right)+{\theta }_4\left(z_i\right)|x\left(z_i\right)|+{\theta }_5\left(z_i\right)|{}^c{\mathcal{D}}_{z_{i-1},z_i}^{{\alpha }_1,\psi }x\left(z_i\right)|)\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\times \frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}({\theta }_3\left(z_i\right)+{\theta }_4\left(z_i\right)|x\left(z_i\right)|+{\theta }_5\left(z_i\right)|{}^c{\mathcal{D}}_{z_{i-1},z_i}^{{\alpha }_1,\psi }x\left(z_i\right)|)\hfill \\ & & +A_1|x\left(z\right)|+N_1+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}(A_2|x\left(z\right)|+N_2)\left)\right)d\tau \hfill \\ & & +|a_2|{\int }_{{\delta }_{2k+1}}^{z_{k+1}}({\theta }_0\left(\tau \right)+{\theta }_1\left(\tau \right)|x\left(\eta \right)|+{\theta }_2\left(\tau \right)|{}^c{\mathcal{D}}_{z_k,\tau }^{{\alpha }_1,\psi }x\left(\tau \right)|\hfill \\ & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3\left(\eta \right)+{\theta }_4\left(\eta \right)|x\left(\eta \right)|+{\theta }_5\left(\eta \right)|{}^c{\mathcal{D}}_{z_k,\eta }^{{\alpha }_1,\psi }x\left(\eta \right)|)\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3\left(z_i\right)+{\theta }_4\left(z_i\right)|x\left(z_i\right)|+{\theta }_5\left(z_i\right)|{}^c{\mathcal{D}}_{z_{i-1},z_i}^{{\alpha }_1,\psi }x\left(z_i\right)|)\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\times \frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}({\theta }_3\left(z_i\right)+{\theta }_4\left(z_i\right)|x\left(z_i\right)|+{\theta }_5\left(z_i\right)|{}^c{\mathcal{D}}_{z_{i-1},z_i}^{{\alpha }_1,\psi }x\left(z_i\right)|)\hfill \\ & & +A_1|x\left(z\right)|+N_1+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}(A_2|x\left(z\right)|+N_2)\left)\right)d\tau ](\psi \left(z\right)-\psi \left(z_k\right))\hfill \\ & & +\sum _{i=1}^p(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))+\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3\left(z_i\right)+{\theta }_4\left(z_i\right)|x\left(z_i\right)|+{\theta }_5\left(z_i\right)|{}^c{\mathcal{D}}_{z_{i-1},z_i}^{{\alpha }_1,\psi }x\left(z_i\right)|)\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\times \frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}({\theta }_3\left(z_i\right)+{\theta }_4\left(z_i\right)|x\left(z_i\right)|+{\theta }_5\left(z_i\right)|{}^c{\mathcal{D}}_{z_{i-1},z_i}^{{\alpha }_1,\psi }x\left(z_i\right)|)\hfill \\ & & +A_1|x\left(z\right)|+N_1+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}(A_2|x\left(z\right)|+N_2))\hfill \end{array}$$

From $\left(G_4\right)$ and (3), we have

$$\begin{array}{ccc}\hfill |{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x(z)|& \le & \frac{{\theta }_3^{*}+\frac{{(\psi (z)-\psi (z_k))}^{1-{\alpha }_1}}{\Gamma ({\alpha }_1)}{\theta }_0^{*}}{1-({\theta }_5^{*}+\frac{{(\psi (z)-\psi (z_k))}^{1-{\alpha }_1}}{\Gamma ({\alpha }_1)}{\theta }_2^{*})}\hfill \\ & & +\frac{({\theta }_4^{*}+\frac{{(\psi (z)-\psi (z_k))}^{1-{\alpha }_1}}{\Gamma ({\alpha }_1)}{\theta }_1^{*})}{1-({\theta }_5^{*}+\frac{{(\psi (z)-\psi (z_k))}^{1-{\alpha }_1}}{\Gamma ({\alpha }_1)}{\theta }_2^{*})}|x(z)|\hfill \end{array}$$

Let

$${\mathcal{A}}_1=\frac{{\theta }_3^{*}+\frac{{(\psi (z)-\psi (z_k))}^{1-{\alpha }_1}}{\Gamma ({\alpha }_1)}{\theta }_0^{*}}{1-({\theta }_5^{*}+\frac{{(\psi (z)-\psi (z_k))}^{1-{\alpha }_1}}{\Gamma ({\alpha }_1)}{\theta }_2^{*})}$$

and

$${\mathcal{A}}_2=\frac{({\theta }_4^{*}+\frac{{(\psi (z)-\psi (z_k))}^{1-{\alpha }_1}}{\Gamma ({\alpha }_1)}{\theta }_1^{*})}{1-({\theta }_5^{*}+\frac{{(\psi (z)-\psi (z_k))}^{1-{\alpha }_1}}{\Gamma ({\alpha }_1)}{\theta }_2^{*})}.$$

Then

$$\begin{array}{ccc}\hfill |{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x(z)|& \le & {\mathcal{A}}_1+{\mathcal{A}}_2|x(z)|.\hfill \end{array}$$

(8)

Let $|x(z)|\le r_1$, then we get

$$\begin{array}{ccc}\hfill |{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x(z)|& \le & {\mathcal{A}}_1+{\mathcal{A}}_2{r}_1.\hfill \end{array}$$

Using the above estimates, we get

$$\begin{array}{ccc}& \le & {\theta }_0^{*}+{\theta }_1^{*}{r}_1+{\theta }_2^{*}({\mathcal{A}}_1+{\mathcal{A}}_2{r}_1)+\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_4^{*}{r}_1+{\theta }_5^{*}({\mathcal{A}}_1+{\mathcal{A}}_2{r}_1))\hfill \\ & & +\frac{1}{|\Delta |}[{\theta }_0^{*}+{\theta }_1^{*}{r}_1+{\theta }_2^{*}({\mathcal{A}}_1+{\mathcal{A}}_2{r}_1)\hfill \\ & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_4^{*}{r}_1+{\theta }_5^{*}({\mathcal{A}}_1+{\mathcal{A}}_2{r}_1))\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_4^{*}{r}_1+{\theta }_5^{*}({\mathcal{A}}_1+{\mathcal{A}}_2{r}_1)\hfill \\ \hfill & & +\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\times \frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}({\theta }_3^{*}+{\theta }_4^{*}{r}_1+{\theta }_5^{*}({\mathcal{A}}_1+{\mathcal{A}}_2{r}_1))\hfill \\ & & +A_1{r}_1+N_1+\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}(A_2{r}_1+N_2))\hfill \\ & & +|a_1|{\int }_{z_k}^{{\delta }_{2k}}({\theta }_0^{*}+{\theta }_1^{*}{r}_1+{\theta }_2^{*}({\mathcal{A}}_1+{\mathcal{A}}_2{r}_1)\hfill \\ & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_4^{*}{r}_1+{\theta }_5^{*}({\mathcal{A}}_1+{\mathcal{A}}_2{r}_1))\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_4^{*}{r}_1+{\theta }_5^{*}({\mathcal{A}}_1+{\mathcal{A}}_2{r}_1))\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\times \frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}({\theta }_3^{*}+{\theta }_4^{*}{r}_1+{\theta }_5^{*}({\mathcal{A}}_1+{\mathcal{A}}_2{r}_1))\hfill \\ & & +A_1{r}_1+N_1+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}(A_2{r}_1+N_2)\left)\right)d\tau \hfill \end{array}$$

$$\begin{array}{ccc}& & +|a_2|{\int }_{{\delta }_{2k+1}}^{z_{k+1}}({\theta }_0^{*}+{\theta }_1^{*}{r}_1+{\theta }_2^{*}({\mathcal{A}}_1+{\mathcal{A}}_2{r}_1)\hfill \\ & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_4^{*}{r}_1+{\theta }_5^{*}({\mathcal{A}}_1+{\mathcal{A}}_2{r}_1))\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_4^{*}{r}_1+{\theta }_5^{*}({\mathcal{A}}_1+{\mathcal{A}}_2{r}_1))\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\times \frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}({\theta }_3^{*}+{\theta }_4^{*}{r}_1+{\theta }_5^{*}({\mathcal{A}}_1+{\mathcal{A}}_2{r}_1))\hfill \\ & & +A_1{r}_1+N_1+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}(A_2{r}_1+N_2)\left)\right)d\tau ](\psi \left(z\right)-\psi \left(z_k\right))\hfill \\ & & +\sum _{i=1}^p(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))+\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_4^{*}{r}_1+{\theta }_5^{*}({\mathcal{A}}_1+{\mathcal{A}}_2{r}_1))\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\times \frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}({\theta }_3^{*}+{\theta }_4^{*}{r}_1+{\theta }_5^{*}({\mathcal{A}}_1+{\mathcal{A}}_2{r}_1))\hfill \\ & & +A_1{r}_1+N_1+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}(A_2{r}_1+N_2)).\hfill \end{array}$$

Therefore,

$$\begin{array}{c}\hfill \parallel P_1{(x,y)\left(z\right)\parallel }_{{\mathbb{X}}^{{}^{\prime }}}\le {ϝ}_1.\end{array}$$

In the same way, we can prove that

$$\begin{array}{c}\hfill \parallel P_2{(x,y)\left(z\right)\parallel }_{{\mathbb{Y}}^{{}^{\prime }}}\le {ϝ}_2.\end{array}$$

Let $max\{{ϝ}_1,{ϝ}_2\}=ϝ$. Then we have

$$\begin{array}{c}\hfill {\parallel P(x,y)\left(z\right)\parallel }_{{\mathbb{U}}^{{}^{\prime }}\times {\mathbb{Y}}^{{}^{\prime }}}\le ϝ.\end{array}$$

The above inequality shows that the operator P is bounded. Now we need to show that the operator P is equicontinuous. For this let ${\omega }_1,{\omega }_2\in J_k$ such that ${\omega }_1<{\omega }_2$ where $k=0,1,2,\cdots ,p$.

Let $(x,y)\in Q_r$, and then we have

$$\begin{array}{ccc}& & |P_1(x,y)({\omega }_2)-P_1(x,y)({\omega }_1)|\hfill \\ & \le & |f_1({\omega }_2,x({\omega }_2),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x({\omega }_2))-f_1({\omega }_1,x({\omega }_1),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x({\omega }_1))|\hfill \\ & & +\frac{1}{\Gamma ({\alpha }_1)}|{\int }_{z_k}^{{\omega }_2}{\psi }^{{}^{\prime }}(s){(\psi ({\omega }_2)-\psi (s))}^{{\alpha }_1-1}{g}_1(s,x(s),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x(s))ds\hfill \\ & & -{\int }_{z_k}^{{\omega }_1}{\psi }^{{}^{\prime }}(s){(\psi ({\omega }_1)-\psi (s))}^{{\alpha }_1-1}{g}_1(s,x(s),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x(s))ds|\hfill \\ & & +\frac{1}{\Delta }[|f_1(\eta ,x(\eta ),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x(\eta ))+\frac{1}{\Gamma ({\alpha }_1)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}(s){(\psi (\eta )-\psi (s))}^{{\alpha }_1-1}{g}_1(s,x(s),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x(s))ds\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ & & +I_{i}x\left(z_i\right)+\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_{i}x\left(z_i\right))|\hfill \\ & & +|a_1|{\int }_{z_k}^{{\delta }_{2k}}||(f_1\left(\tau \right)+\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\tau \right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +I_{i}x\left(z_i\right)+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_{i}x\left(z_i\right)\left)\right)|d\tau \hfill \\ & & +|a_2|{\int }_{{\delta }_{2k+1}}^{z_{k+1}}|(f_1\left(\tau \right)+\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\tau \right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill \\ \hfill & & +I_{i}x\left(z_i\right)+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_{i}x\left(z_i\right)\left)\right)|d\tau ]||\left((\psi \left({\omega }_2\right)-\psi \left(z_k\right))-(\psi \left({\omega }_1\right)-\psi \left(z_k\right))\right)|\hfill \\ & & +\sum _{i=1}^p\frac{|\left((\psi \left({\omega }_2\right)-\psi \left(z_i\right))-(\psi \left({\omega }_1\right)-\psi \left(z_i\right))\right)|}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_i}|{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))|ds\hfill \\ & & +\sum _{i=1}^p\frac{|\left((\psi \left({\omega }_2\right)-\psi \left(z_i\right))-(\psi \left({\omega }_1\right)-\psi \left(z_i\right))\right)|}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_{i}x\left(z_i\right).\hfill \end{array}$$

From above inequality, if ${\omega }_1\to {\omega }_2$, we deduce that

$$|P_1(x,y)\left({\omega }_2\right)-P_1(x,y)\left({\omega }_1\right)|\to 0.$$

In the same way we can prove that

$$|P_2(x,y)\left({\omega }_2\right)-P_2(x,y)\left({\omega }_1\right)|\to 0.$$

Hence, by the Arzila-Ascoli theorem $P_1$ and $P_2$ are completely continuous. This shows that P is completely continuous.

Now let us define a set:

$$G=\{(x,y)\in {\mathbb{X}}^{{}^{\prime }}\times {\mathbb{Y}}^{{}^{\prime }};(x,y)=\lambda P(x,y);0<\lambda <1\}.$$

we prove that the set G is bounded.

For z∈J and $(x,y)\in G$ then $(x,y)=\lambda P(x,y)$ i.e., $x\left(z\right)=\lambda P_1(x,y)$ and $y\left(z\right)=\lambda P_2(x,y)$. Now

$$\begin{array}{ccc}& & |x\left(z\right)|=|\lambda P_1(x,y)|\hfill \\ & \le & \lambda \{|f_1(z,x\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(z\right))|\hfill \\ & & +\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^z{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z\right)-\psi \left(s\right))}^{{\alpha }_1-1}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))|ds\hfill \\ & & +\frac{1}{|\Delta |}[|f_1(\eta ,x\left(\eta \right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(\eta \right))|+\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\eta \right)-\psi \left(s\right))}^{{\alpha }_1-1}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))|ds\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))|ds\hfill \\ \hfill & & +\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))|ds\hfill \\ & & +I_{i}x\left(z_i\right)+\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_{i}x\left(z_i\right))\hfill \\ & & +|a_1|{\int }_{z_k}^{{\delta }_{2k}}(|f_1(\tau ,x\left(\tau \right),{}^c{\mathcal{D}}_{z_k,\tau }^{{\alpha }_1;\psi }x\left(\tau \right))|+\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\tau \right)-\psi \left(s\right))}^{{\alpha }_1-1}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))|ds\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))|ds\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))|ds\hfill \\ \hfill & & +I_{i}x\left(z_i\right)+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_{i}x\left(z_i\right)\left)\right)d\tau \hfill \\ & & +|a_2|{\int }_{{\delta }_{2k+1}}^{z_{k+1}}(|f_1(\tau ,x\left(\tau \right),{}^c{\mathcal{D}}_{z_k,\tau }^{{\alpha }_1;\psi }x\left(\tau \right))|+\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\tau \right)-\psi \left(s\right))}^{{\alpha }_1-1}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))|ds\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))|ds\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))|ds\hfill \\ \hfill & & +|I_{i}x(z_i|)+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}|J_{i}x(z_i|)\left)\right)d\tau ](\psi \left(z\right)-\psi \left(z_k\right))\hfill \\ & & +\sum _{i=1}^p(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))+\sum _{i=1}^p(\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_1}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}|g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(s\right))|ds\hfill \\ \hfill & & +\frac{\psi \left(z\right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_1}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}{g}_1\left(s\right)ds+|I_{i}x\left(z_i\right)|\hfill \\ & & +\frac{\psi \left(z\right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}|J_{i}x\left(z_i\right)|\left)\right\}\hfill \end{array}$$

$$\begin{array}{ccc}& \le & \lambda \{{\theta }_0\left(z\right)+{\theta }_1\left(z\right)|x\left(z\right)|+{\theta }_2\left(z\right)|{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1,\psi }x\left(z\right)|+\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3\left(z\right)+{\theta }_4\left(z\right)|x\left(z\right)|+{\theta }_5\left(z\right)|{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1,\psi }x\left(z\right)|)\hfill \\ & & +\frac{1}{|\Delta |}[{\theta }_0\left(\eta \right)+{\theta }_1\left(\eta \right)|x\left(\eta \right)|+{\theta }_2\left(\eta \right)|{}^c{\mathcal{D}}_{z_k,\eta }^{{\alpha }_1,\psi }x\left(\eta \right)|\hfill \\ & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3\left(\eta \right)+{\theta }_4\left(\eta \right)|x\left(\eta \right)|+{\theta }_5\left(\eta \right)|{}^c{\mathcal{D}}_{z_k,\eta }^{{\alpha }_1,\psi }x\left(\eta \right)|)\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3\left(z_i\right)+{\theta }_4\left(z_i\right)|x\left(z_i\right)|+{\theta }_5\left(z_i\right)|{}^c{\mathcal{D}}_{z_{i-1},z_i}^{{\alpha }_1,\psi }x\left(z_i\right)|)\hfill \\ \hfill & & +\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\times \frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}({\theta }_3\left(z_i\right)+{\theta }_4\left(z_i\right)|x\left(z_i\right)|+{\theta }_5\left(z_i\right)|{}^c{\mathcal{D}}_{z_{i-1},z_i}^{{\alpha }_1,\psi }x\left(z_i\right)|)\hfill \\ & & +A_1|x\left(z\right)|+N_1+\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}(A_2|x\left(z\right)|+N_2))\hfill \\ & & +|a_1|{\int }_{z_k}^{{\delta }_{2k}}({\theta }_0\left(\tau \right)+{\theta }_1\left(\tau \right)|x\left(\tau \right)|+{\theta }_2\left(\tau \right)|{}^c{\mathcal{D}}_{z_k,\tau }^{{\alpha }_1,\psi }x\left(\tau \right)|\hfill \\ & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3\left(\eta \right)+{\theta }_4\left(\eta \right)|x\left(\eta \right)|+{\theta }_5\left(\eta \right)|{}^c{\mathcal{D}}_{z_k,\eta }^{{\alpha }_1,\psi }x\left(\eta \right)|)\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3\left(z_i\right)+{\theta }_4\left(z_i\right)|x\left(z_i\right)|+{\theta }_5\left(z_i\right)|{}^c{\mathcal{D}}_{z_{i-1},z_i}^{{\alpha }_1,\psi }x\left(z_i\right)|)\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\times \frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}({\theta }_3\left(z_i\right)+{\theta }_4\left(z_i\right)|x\left(z_i\right)|+{\theta }_5\left(z_i\right)|{}^c{\mathcal{D}}_{z_{i-1},z_i}^{{\alpha }_1,\psi }x\left(z_i\right)|)\hfill \\ & & +A_1|x\left(z\right)|+N_1+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}(A_2|x\left(z\right)|+N_2)\left)\right)d\tau \hfill \\ & & +|a_2|{\int }_{{\delta }_{2k+1}}^{z_{k+1}}({\theta }_0\left(\tau \right)+{\theta }_1\left(\tau \right)|x\left(\eta \right)|+{\theta }_2\left(\tau \right)|{}^c{\mathcal{D}}_{z_k,\tau }^{{\alpha }_1,\psi }x\left(\tau \right)|\hfill \\ & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3\left(\eta \right)+{\theta }_4\left(\eta \right)|x\left(\eta \right)|+{\theta }_5\left(\eta \right)|{}^c{\mathcal{D}}_{z_k,\eta }^{{\alpha }_1,\psi }x\left(\eta \right)|)\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3\left(z_i\right)+{\theta }_4\left(z_i\right)|x\left(z_i\right)|+{\theta }_5\left(z_i\right)|{}^c{\mathcal{D}}_{z_{i-1},z_i}^{{\alpha }_1,\psi }x\left(z_i\right)|)\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\times \frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}({\theta }_3\left(z_i\right)+{\theta }_4\left(z_i\right)|x\left(z_i\right)|+{\theta }_5\left(z_i\right)|{}^c{\mathcal{D}}_{z_{i-1},z_i}^{{\alpha }_1,\psi }x\left(z_i\right)|)\hfill \\ & & +A_1|x\left(z\right)|+N_1+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}(A_2|x\left(z\right)|+N_2)\left)\right)d\tau ](\psi \left(z\right)-\psi \left(z_k\right))\hfill \\ & & +\sum _{i=1}^p(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))+\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3\left(z_i\right)+{\theta }_4\left(z_i\right)|x\left(z_i\right)|+{\theta }_5\left(z_i\right)|{}^c{\mathcal{D}}_{z_{i-1},z_i}^{{\alpha }_1,\psi }x\left(z_i\right)|)\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\times \frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}({\theta }_3\left(z_i\right)+{\theta }_4\left(z_i\right)|x\left(z_i\right)|+{\theta }_5\left(z_i\right)|{}^c{\mathcal{D}}_{z_{i-1},z_i}^{{\alpha }_1,\psi }x\left(z_i\right)|)\hfill \\ & & +A_1|x\left(z\right)|+N_1+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}(A_2|x\left(z\right)|+N_2)\left)\right\}\hfill \end{array}$$

Using (4.3), we have

$$\begin{array}{ccc}\hfill |x(z)|& \le & \lambda \{{\theta }_0^{*}+{\theta }_1^{*}|x\left(z\right)|+{\theta }_2^{*}({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(z\right)|)+\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_4^{*}{r}_1+{\theta }_5^{*}({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(z\right)|\left)\right)\hfill \\ & & +\frac{1}{|\Delta |}[{\theta }_0^{*}+{\theta }_1^{*}|x\left(\eta \right)|+{\theta }_2^{*}({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(\eta \right))|\hfill \\ & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_4^{*}{r}_1+{\theta }_5^{*}\left(\right({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(\eta \right)|\left)\right))\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_4^{*}|x\left(\eta \right)||+{\theta }_5^{*}\left(\right({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(\eta \right)|\left)\right))\hfill \\ \hfill & & +\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\times \frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}({\theta }_3^{*}+{\theta }_4^{*}x\left(\eta \right)+{\theta }_5^{*}\left(\right({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(\eta \right)|\left)\right))\hfill \\ & & +A_{1}x\left(z\right)+N_1+\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}(A_{2}x\left(z\right)+N_2))\hfill \\ & & +|a_1|{\int }_{z_k}^{{\delta }_{2k}}({\theta }_0^{*}+{\theta }_1^{*}|x\left(\tau \right)|+{\theta }_2^{*}({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(\tau \right)|)\hfill \\ & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_4^{*}|x\left(\eta \right)|+{\theta }_5^{*}\left(\right({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(\eta \right)|\left)\right))\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_4^{*}|x\left(\eta \right)|+{\theta }_5^{*}\left(\right({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(\eta \right)|\left)\right))\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\times \frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}({\theta }_3^{*}+{\theta }_4^{*}|x\left(\eta \right)|+{\theta }_5^{*}\left(\right({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(\eta \right)|\left)\right))\hfill \\ & & +A_1|x\left(z\right)|+N_1+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}(A_2|x\left(z\right)|+N_2)\left)\right)d\tau \hfill \\ & & +|a_2|{\int }_{{\delta }_{2k+1}}^{z_{k+1}}({\theta }_0^{*}+{\theta }_1^{*}|x\left(\tau \right)|+{\theta }_2^{*}({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(\tau \right)|)\hfill \\ & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_4^{*}|x\left(\eta \right)|+{\theta }_5^{*}\left(\right({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(\eta \right)|\left)\right))\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_4^{*}|x\left(\eta \right)|+{\theta }_5^{*}\left(\right({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(\eta \right)|\left)\right))\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\times \frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}({\theta }_3^{*}+{\theta }_4^{*}|x\left(\eta \right)|+{\theta }_5^{*}\left(\right({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(\eta \right)|\left)\right))\hfill \\ & & +A_1|x\left(z\right)|+N_1+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}(A_2|x\left(z\right)|+N_2)\left)\right)d\tau ](\psi \left(z\right)-\psi \left(z_k\right))\hfill \\ & & +\sum _{i=1}^p(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))+\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_4^{*}|x\left(\eta \right)|+{\theta }_5^{*}\left(\right({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(\eta \right)|)\left)\right)\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\times \frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}({\theta }_3^{*}+{\theta }_4^{*}|x\left(\eta \right)|+{\theta }_5^{*}\left(\right({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(\eta \right)|\left)\right))\hfill \\ & & +A_1|x\left(z\right)|+N_1+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}(A_2|x\left(z\right)|+N_2)\left)\right\}.\hfill \end{array}$$

By further simplification, we get that

$$\begin{array}{ccc}\hfill |x(z)|& \le & \lambda \{{\theta }_0^{*}+{\theta }_1^{*}|x\left(z\right)|+{\theta }_2^{*}({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(z\right)|)+\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_4^{*}{r}_1+{\theta }_5^{*}({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(z\right)|\left)\right)\hfill \\ & & +\frac{1}{|\Delta |}[{\theta }_0^{*}+{\theta }_1^{*}|x\left(\eta \right)|+{\theta }_2^{*}\left(\right({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(\eta \right)|\left)\right)\hfill \\ & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_4^{*}{r}_1+{\theta }_5^{*}({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(\eta \right)|\left)\right)\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_4^{*}|x\left(\eta \right)||+{\theta }_5^{*}({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(\eta \right)|\left)\right)\hfill \\ \hfill & & +\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\times \frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}({\theta }_3^{*}+{\theta }_4^{*}x\left(\eta \right)+{\theta }_5^{*}({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(\eta \right)|\left)\right)\hfill \\ & & +A_{1}x\left(z\right)+N_1+\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}(A_2|x\left(z\right)|+N_2))\hfill \\ & & +|a_1|{\int }_{z_k}^{{\delta }_{2k}}({\theta }_0^{*}+{\theta }_1^{*}|x\left(\tau \right)|+{\theta }_2^{*}{\mathcal{A}}_1+{\mathcal{A}}_2|x\left(\tau \right)|\hfill \\ & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_4^{*}|x\left(\eta \right)|+{\theta }_5^{*}({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(\eta \right)|\left)\right)\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_4^{*}|x\left(\eta \right)|+{\theta }_5^{*}({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(\eta \right)|\left)\right)\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\times \frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}({\theta }_3^{*}+{\theta }_4^{*}|x\left(\eta \right)|+{\theta }_5^{*}({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(\eta \right)|\left)\right)\hfill \\ & & +A_1|x\left(z\right)|+N_1+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}(A_2|x\left(z\right)|+N_2)\left)\right)d\tau \hfill \\ & & +|a_2|{\int }_{{\delta }_{2k+1}}^{z_{k+1}}({\theta }_0^{*}+{\theta }_1^{*}|x\left(\tau \right)|+{\theta }_2^{*}{\mathcal{A}}_1+{\mathcal{A}}_2|x\left(\tau \right)|\hfill \\ & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_4^{*}|x\left(\eta \right)|+{\theta }_5^{*}({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(\eta \right)|\left)\right)\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_4^{*}|x\left(\eta \right)|+{\theta }_5^{*}({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(\eta \right)|\left)\right)\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\times \frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}({\theta }_3^{*}+{\theta }_4^{*}|x\left(\eta \right)|+{\theta }_5^{*}({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(\eta \right)|\left)\right)\hfill \\ & & +A_1|x\left(z\right)|+N_1+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}(A_2|x\left(z\right)|+N_2)\left)\right)d\tau ](\psi \left(z\right)-\psi \left(z_k\right))\hfill \\ & & +\sum _{i=1}^p(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))+\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_4^{*}|x\left(\eta \right)|+{\theta }_5^{*}({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(\eta \right)|))\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\times \frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}({\theta }_3^{*}+{\theta }_4^{*}|x\left(\eta \right)|+{\theta }_5^{*}({\mathcal{A}}_1+{\mathcal{A}}_2|x\left(\eta \right)|\left)\right)\hfill \\ & & +A_1|x\left(z\right)|+N_1+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}(A_2|x\left(z\right)|+N_2)\left)\right\}.\hfill \end{array}$$

Let us assume that ${\mathcal{M}}^{*}<1$, where

$$\begin{array}{ccc}\hfill {\mathcal{M}}^{*}& =& \lambda [{\theta }_1^{*}+{\theta }_2^{*}{\mathcal{A}}_2+\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_4^{*}+{\theta }_5^{*}{\mathcal{A}}_2)\hfill \\ & & +\frac{1}{|\Delta |}[{\theta }_1^{*}+{\theta }_2^{*}{\mathcal{A}}_2+\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_4^{*}+{\theta }_5^{*}{\mathcal{A}}_2)\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_4^{*}+{\theta }_5^{*}{\mathcal{A}}_2)\hfill \\ \hfill & & +\frac{(\psi \left(\eta \right)-\psi \left(z_i\right)){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)}({\theta }_4^{*}+{\theta }_5^{*}{\mathcal{A}}_2)\hfill \\ & & +A_1+\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\left(A_2\right))\hfill \\ & & +|a_1|{\int }_{z_k}^{{\delta }_{2k}}({\theta }_1^{*}|x\left(\tau \right)|+{\theta }_2^{*}{\mathcal{A}}_2|x\left(\tau \right)|\hfill \\ & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_4^{*}+{\theta }_5^{*}{\mathcal{A}}_2)\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_4^{*}+{\theta }_5^{*}{\mathcal{A}}_2)\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\times \frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}({\theta }_4^{*}+{\theta }_5^{*}{\mathcal{A}}_2)\hfill \\ & & +A_1+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{A}_2\left)\right)({\delta }_{2k}-z_k)\hfill \\ & & +|a_2|{\int }_{{\delta }_{2k+1}}^{z_{k+1}}({\theta }_1^{*}|x\left(\tau \right)|+{\theta }_2^{*}{\mathcal{A}}_2|x\left(\tau \right)|\hfill \\ & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_4^{*}+{\theta }_5^{*}{\mathcal{A}}_2)\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_4^{*}+{\theta }_5^{*}{\mathcal{A}}_2)\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\times \frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}({\theta }_4^{*}+{\theta }_5^{*}{\mathcal{A}}_2)\hfill \\ & & +A_1+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\left(A_2\right)\left)\right)(z_{k+1}-{\delta }_{2k+1})](\psi \left(z\right)-\psi \left(z_k\right))\hfill \\ & & +\sum _{i=1}^p(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))+\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_4^{*}+{\theta }_5^{*}{\mathcal{A}}_2)\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\times \frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}({\theta }_4^{*}+{\theta }_5^{*}{\mathcal{A}}_2)\hfill \\ & & +A_1+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{A}_2\left)\right].\hfill \end{array}$$

Then we have

$$\begin{array}{ccc}\hfill \parallel x\left(z\right)\parallel & \le & \frac{\lambda }{1-{\mathcal{M}}^{*}}\{{\theta }_0^{*}+{\theta }_2^{*}{\mathcal{A}}_1+\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_5^{*}{\mathcal{A}}_1)\hfill \\ & & +\frac{1}{|\Delta |}[{\theta }_0^{*}+{\theta }_2^{*}\left({\mathcal{A}}_1\right)\hfill \\ & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_5^{*}{\mathcal{A}}_1\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_5^{*}{\mathcal{A}}_1)\hfill \\ \hfill & & +\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\times \frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}({\theta }_3^{*}+{\theta }_4^{*}x\left(\eta \right)+{\theta }_5^{*}{\mathcal{A}}_1)\hfill \\ & & +N_1+\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{N}_2)\hfill \\ & & +|a_1|({\theta }_0^{*}+{\theta }_2^{*}{\mathcal{A}}_1\hfill \\ & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_5^{*}\left({\mathcal{A}}_1\right))\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_5^{*}{\mathcal{A}}_1)\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\times \frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}({\theta }_3^{*}+{\theta }_5^{*}{\mathcal{A}}_1)\hfill \\ & & +N_1+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{N}_2\left)\right)\left({\delta }_{2k}-z_k\right)\hfill \\ & & +|a_2|({\theta }_0^{*}+{\theta }_2^{*}{\mathcal{A}}_1\hfill \\ & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_5^{*}{\mathcal{A}}_1)\hfill \\ \hfill & & +\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_5^{*}{\mathcal{A}}_1)\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\times \frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}({\theta }_3^{*}+{\theta }_5^{*}{\mathcal{A}}_1)\hfill \\ & & +N_1+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{N}_2\left)\right)\left(z_{k+1}-{\delta }_{2k+1}\right)](\psi \left(z\right)-\psi \left(z_k\right))\hfill \\ & & +\sum _{i=1}^p(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))+\sum _{i=1}^p(\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}({\theta }_3^{*}+{\theta }_5^{*}{\mathcal{A}}_1)\hfill \\ \hfill & & +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\times \frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}({\theta }_3^{*}+{\theta }_5^{*}{\mathcal{A}}_1)\hfill \\ & & +N_1+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{N}_2\left)\right\}.\hfill \end{array}$$

Thus there exists a positive constant ${ϝ}_1$, such that

$$\begin{array}{c}\hfill {\parallel x\parallel }_{{\mathbb{X}}^{{}^{\prime }}}\le {ϝ}_1.\end{array}$$

In the same way, we can prove that there exists ${ϝ}_1$, such that

$$\begin{array}{c}\hfill {\parallel y\parallel }_{{\mathbb{Y}}^{{}^{\prime }}}\le {ϝ}_2.\end{array}$$

Let $max\{{ϝ}_1,{ϝ}_2\}=ϝ$. Then we have

$$\begin{array}{c}\hfill {\parallel (x,y)\parallel }_{{\mathbb{U}}^{{}^{\prime }}\times {\mathbb{Y}}^{{}^{\prime }}}\le ϝ.\end{array}$$

Thus the set G is bounded, and by the Schaefer’s fixed point theorem, the operator P has at least one fixed point, i.e., the problem (1) has at least one solution. □

5. Ulam’s Stability Results

Using Definition 4, in this section we give the Ulam-Hyers stability of the problem (3).

Theorem 3.

If assumptions $\left(G_1\right)$, $\left(G_2\right)$ are satisfied, then the coupled system (1) is Ulam-Hyers stable.

Proof. 

Let $(x,y)\in {\mathbb{X}}^{{}^{\prime }}\times {\mathbb{Y}}^{{}^{\prime }}$ be an approximate solution of the inequality:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}|{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }[x\left(z\right)-f_1(z,x\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(z\right))]-g_1(z,x\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(z\right))|<{ϵ}_1, z\in (z_k,z_{k+1}],\hfill & \hfill \\ |\Delta x\left(z_k\right)-I_k\left(x\left(z_k\right)\right)|<{ϵ}_1, k=1,2,\cdots ,p,\hfill & \hfill \\ |\Delta x^{{}^{\prime }}\left(z_k\right)-J_k\left(x\left(z_k\right)\right)|<{ϵ}_1.\hfill & \hfill \\ |{}^c{\mathcal{D}}_{z_k,z}^{{\beta }_1;\psi }[y\left(z\right)-f_2(z,y\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\beta }_1;\psi }y\left(z\right))]-g_2(z,y\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\beta }_1;\psi }y\left(z\right))|<{ϵ}_2,\hfill & \hfill \\ |\Delta y\left(z_k\right)-I_k^{*}\left(y\left(z_k\right)\right)|<{ϵ}_2, \hfill & \hfill \\ |\Delta y^{{}^{\prime }}\left(z_k\right)-J_k^{*}\left(y\left(z_k\right)\right)|<{ϵ}_2, k=1,2,\cdots ,p.\hfill \end{array}\right.\end{array}$$

(9)

From the inequality (9), we have

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }[x\left(z\right)-f_1(z,x\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(z\right))]=g_1(z,x\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(z\right))+\phi \left(z\right),\hfill & \hfill \\ z\in (z_k,z_{k+1}], k=0,1,2,\cdots ,p.\hfill & \hfill \\ \Delta x\left(z_k\right)=I_k\left(x\left(z_k\right)\right)+{\phi }_k\left(z\right), k=1,2,\cdots ,p,\hfill & \hfill \\ \Delta x^{{}^{\prime }}\left(z_k\right)=J_k\left(x\left(z_k\right)\right)+{\phi }_k\left(z\right).\hfill & \hfill \\ {}^c{\mathcal{D}}_{z_k,z}^{{\beta }_1;\psi }[y\left(z\right)-f_2(z,y\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\beta }_1;\psi }y\left(z\right))]=g_2(z,y\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\beta }_1;\psi }y\left(z\right))+\varphi \left(z\right),\hfill & \hfill \\ z\in (z_k,z_{k+1}], k=0,1,2,\cdots ,p,\hfill & \hfill \\ \Delta y\left(z_k\right)=I_k^{*}\left(y\left(z_k\right)\right)+{\varphi }_k\left(z\right), k=1,2,\cdots ,p,\hfill & \hfill \\ \Delta y^{{}^{\prime }}\left(z_k\right)=J_k^{*}\left(y\left(z_k\right)\right)+{\varphi }_k\left(z\right).\hfill \end{array}\right.\end{array}$$

(10)

The x and y parts of the solution $(x,y)$ of problem (10) are equivalent to

$$\begin{array}{c}\hfill x\left(z\right)=\left\{\begin{array}{cc}{f}_1(z,x\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(z\right))+\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^z{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,s}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill & \hfill \\ +\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^z{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z\right)-\psi \left(s\right))}^{{\alpha }_1-1}\phi \left(s\right)ds\hfill & \hfill \\ +\frac{1}{\Delta }[f_1(\eta ,x\left(\eta \right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(\eta \right))+\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\eta \right)-\psi \left(s\right))}^{{\alpha }_1-1}\hfill & \hfill \\ \times g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,s}^{{\alpha }_1;\psi }x\left(s\right))ds+\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\eta \right)-\psi \left(s\right))}^{{\alpha }_1-1}\phi \left(s\right)ds\hfill & \hfill \\ +\sum _{i=1}^p(\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,s}^{{\alpha }_1;\psi }x\left(s\right))ds+\frac{1}{\Gamma \left({\alpha }_1\right)}\hfill & \hfill \\ \times {\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}\phi \left(s\right)ds+\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}\hfill & \hfill \\ \times g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,s}^{{\alpha }_1;\psi }x\left(s\right))ds+\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}\phi \left(s\right)ds\hfill & \hfill \\ +I_{i}x\left(z_i\right)+{\phi }_i\left(z\left(i\right)\right)+\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_{i}x\left(z_i\right)+\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\phi \left(z_i\right))\hfill & \hfill \\ -a_1{\int }_{z_k}^{{\delta }_{2k}}(f_1(\tau ,x\left(\tau \right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(\tau \right))+\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^{\tau }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\tau \right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,s}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill & \hfill \\ +\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^{\tau }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\tau \right)-\psi \left(s\right))}^{{\alpha }_1-1}\phi \left(s\right)ds+\sum _{i=1}^p(\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}\hfill & \hfill \\ g_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,s}^{{\alpha }_1;\psi }x\left(s\right))ds+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,s}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill & \hfill \\ +I_{i}x\left(z_i\right)+{\phi }_i\left(z\left(i\right)\right)+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_{i}x\left(z_i\right)+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\phi \left(z_i\right)\left)\right)d\tau \hfill & \hfill \\ -a_2{\int }_{{\delta }_{2k+1}}^{z_{k+1}}(f_1(\tau ,x\left(\tau \right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(\tau \right))+\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\tau \right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,s}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill & \hfill \\ +\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\tau \right)-\psi \left(s\right))}^{{\alpha }_1-1}\phi \left(s\right)ds\hfill & \hfill \\ +\sum _{i=1}^p(\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,s}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill & \hfill \\ +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,s}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill & \hfill \\ +I_{i}x\left(z_i\right)+{\phi }_i\left(z\left(i\right)\right)+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_{i}x\left(z_i\right)+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\phi \left(z_i\right)\left)\right)d\tau ](\psi \left(z\right)-\psi \left(z_k\right))\hfill & \hfill \\ +\sum _{i=1}^p(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))+\sum _{i=1}^p\frac{1}{\Gamma \left({\alpha }_1\right)}{\int }_{z_{i-1}}^{z_1}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-1}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,s}^{{\alpha }_1;\psi }x\left(s\right))ds\hfill & \hfill \\ +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\alpha }_1-1)}{\int }_{z_{i-1}}^{z_1}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\alpha }_1-2}{g}_1(s,x\left(s\right),{}^c{\mathcal{D}}_{z_k,s}^{{\alpha }_1;\psi }x\left(s\right))ds+\sum _{i=1}^p{I}_{i}x\left(z_i\right)\hfill & \hfill \\ +\sum _{i=1}^p{\phi }_i\left(z\left(i\right)\right)+\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_{i}x\left(z_i\right)+\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{\phi }_i\left(z\left(i\right)\right),\hfill \end{array}\right.\end{array}$$

(11)

$$\begin{array}{c}\hfill y\left(z\right)=\left\{\begin{array}{cc}{f}_2(z,x\left(z\right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(z\right))+\frac{1}{\Gamma \left({\beta }_1\right)}{\int }_{z_k}^z{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z\right)-\psi \left(s\right))}^{{\beta }_1-1}{g}_2(s,y\left(s\right),{}^c{\mathcal{D}}_{z_k,s}^{{\beta }_1;\psi }y\left(s\right))ds\hfill & \hfill \\ +\frac{1}{\Gamma \left({\beta }_1\right)}{\int }_{z_k}^z{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z\right)-\psi \left(s\right))}^{{\beta }_1-1}\varphi \left(s\right)ds\hfill & \hfill \\ +\frac{1}{\Delta }[f_2(\eta ,x\left(\eta \right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(\eta \right))+\frac{1}{\Gamma \left({\beta }_1\right)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\eta \right)-\psi \left(s\right))}^{{\beta }_1-1}{g}_2(s,y\left(s\right),{}^c{\mathcal{D}}_{z_k,s}^{{\beta }_1;\psi }y\left(s\right))ds\hfill & \hfill \\ +\frac{1}{\Gamma \left({\beta }_1\right)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\eta \right)-\psi \left(s\right))}^{{\beta }_1-1}\varphi \left(s\right)ds\hfill & \hfill \\ +\sum _{i=1}^p(\frac{1}{\Gamma \left({\beta }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\beta }_1-1}{g}_2(s,y\left(s\right),{}^c{\mathcal{D}}_{z_k,s}^{{\beta }_1;\psi }y\left(s\right))ds\hfill & \hfill \\ +\frac{1}{\Gamma \left({\beta }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\beta }_1-1}\varphi \left(s\right)ds\hfill & \hfill \\ +\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\beta }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\beta }_1-2}{g}_2(s,y\left(s\right),{}^c{\mathcal{D}}_{z_k,s}^{{\beta }_1;\psi }y\left(s\right))ds\hfill & \hfill \\ +\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\beta }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\beta }_1-2}\varphi \left(s\right)ds\hfill & \hfill \\ +I_i^{*}y\left(z_i\right)+{\varphi }_i\left(z_{(}i\right))+\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_i^{*}y\left(z_i\right)+\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\varphi \left(z_i\right))\hfill & \hfill \\ -a_1{\int }_{z_k}^{{\delta }_{2k}}(f_2(\tau ,x\left(\tau \right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(\tau \right))+\frac{1}{\Gamma \left({\beta }_1\right)}{\int }_{z_k}^{\tau }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\tau \right)-\psi \left(s\right))}^{{\beta }_1-1}{g}_2(s,y\left(s\right),{}^c{\mathcal{D}}_{z_k,s}^{{\beta }_1;\psi }y\left(s\right))ds\hfill & \hfill \\ +\frac{1}{\Gamma \left({\beta }_1\right)}{\int }_{z_k}^{\tau }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\tau \right)-\psi \left(s\right))}^{{\beta }_1-1}\varphi \left(s\right)ds\hfill & \hfill \\ +\sum _{i=1}^p(\frac{1}{\Gamma \left({\beta }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\beta }_1-1}{g}_2(s,y\left(s\right),{}^c{\mathcal{D}}_{z_k,s}^{{\beta }_1;\psi }y\left(s\right))ds\hfill & \hfill \\ +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\beta }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\beta }_1-2}{g}_2(s,y\left(s\right),{}^c{\mathcal{D}}_{z_k,s}^{{\beta }_1;\psi }y\left(s\right))ds\hfill & \hfill \\ +I_i^{*}y\left(z_i\right)+{\varphi }_i\left(z_{(}i\right))+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_i^{*}y\left(z_i\right)+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\varphi \left(z_i\right)\left)\right)d\tau -a_2{\int }_{{\delta }_{2k+1}}^{z_{k+1}}(f_2(\tau ,x\left(\tau \right),{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(\tau \right))\hfill & \hfill \\ +\frac{1}{\Gamma \left({\beta }_1\right)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\tau \right)-\psi \left(s\right))}^{{\beta }_1-1}{g}_2(s,y\left(s\right),{}^c{\mathcal{D}}_{z_k,s}^{{\beta }_1;\psi }y\left(s\right))ds\hfill & \hfill \\ \frac{1}{\Gamma \left({\beta }_1\right)}{\int }_{z_k}^{\eta }{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(\tau \right)-\psi \left(s\right))}^{{\beta }_1-1}\varphi \left(s\right)ds\hfill & \hfill \\ +\sum _{i=1}^p(\frac{1}{\Gamma \left({\beta }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\beta }_1-1}{g}_2(s,y\left(s\right),{}^c{\mathcal{D}}_{z_k,s}^{{\beta }_1;\psi }y\left(s\right))ds\hfill & \hfill \\ +\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\beta }_1-1)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\beta }_1-2}{g}_2(s,y\left(s\right),{}^c{\mathcal{D}}_{z_k,s}^{{\beta }_1;\psi }y\left(s\right))ds\hfill & \hfill \\ +I_i^{*}y\left(z_i\right)+{\varphi }_i\left(z_i\right)+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_i^{*}y\left(z_i\right)+\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\varphi \left(z_i\right)\left)\right)d\tau ](\psi \left(z\right)-\psi \left(z_k\right))\hfill & \hfill \\ +\sum _{i=1}^p(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))+\sum _{i=1}^p\frac{1}{\Gamma \left({\beta }_1\right)}{\int }_{z_{i-1}}^{z_i}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\beta }_1-1}{g}_2(s,y\left(s\right),{}^c{\mathcal{D}}_{z_k,s}^{{\beta }_1;\psi }y\left(s\right))ds\hfill & \hfill \\ +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma ({\beta }_1-1)}{\int }_{z_{i-1}}^{z_1}{\psi }^{{}^{\prime }}\left(s\right){(\psi \left(z_i\right)-\psi \left(s\right))}^{{\beta }_1-2}{g}_2(s,y\left(s\right),{}^c{\mathcal{D}}_{z_k,s}^{{\beta }_1;\psi }y\left(s\right))ds+\sum _{i=1}^p{I}_i^{*}y\left(z_i\right)+\sum _{i=1}^p{\varphi }_i\left(z_{(}i\right))\hfill & \hfill \\ +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}{J}_i^{*}y\left(z_i\right)+\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\varphi \left(z_i\right),\hfill \end{array}\right.\end{array}$$

(12)

Now let $(x,y)$ be the solution of (3) and $(x_1,y_1)$ be the solution of (5.2), then

$$\begin{array}{ccc}\hfill |x\left(z\right)-\overline{x}\left(z\right)|& \le & M_{f_1}\parallel x-\overline{x}\parallel +N_{f_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel +\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}{L}_{g_1}\parallel x-\overline{x}\parallel \hfill \\ \hfill & & +\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}{K}_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel \hfill \\ \hfill & & +\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}|\phi \left(s\right)|\hfill \\ \hfill & & +(\frac{M_{f_1}}{|\Delta |}\parallel x-\overline{x}\parallel +\frac{N_{f_1}}{|\Delta |}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel \hfill \\ \hfill & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}{L}_{g_1}\parallel x-\overline{x}\parallel +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}{K}_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel \hfill \\ \hfill & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{|\Delta |\Gamma ({\alpha }_1+1)}|\phi \left(s\right)|\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}{L}_{g_1}\parallel x-\overline{x}\parallel \hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}{K}_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{|\Delta |\Gamma ({\alpha }_1+1)}|\phi \left(s\right)|\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}{L}_{g_1}\parallel x-\overline{x}\parallel \hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}{K}_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel \hfill \\ \hfill & & \sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}|\varphi \left(z\right)|\hfill \\ \hfill & & +\sum _{i=1}^p\frac{A_1}{|\Delta |}|x\left(z_i\right)-\overline{x}\left(z_i\right)|+\sum _{i=1}^p\frac{1}{|\Delta |}|\varphi \left(z\right)|+\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}{A}_2|x\left(z_i\right)-\overline{x}\left(z_i\right)|\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}|\varphi \left(z\right)|\hfill \\ \hfill & & +\frac{|a_1|}{|\Delta |}({\delta }_{2k}-z_k)M_{f_1}\parallel x-\overline{x}\parallel +\frac{|a_1|}{|\Delta |}({\delta }_{2k}-z_k)N_{f_1}\hfill \\ \hfill & & +|{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel +\frac{|a_1|}{|\Delta |}({\delta }_{2k}-z_k)|\varphi \left(z\right)|\hfill \\ \hfill & & +\frac{{(\psi \left(\tau \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)L_{g_1}\parallel x-\overline{x}\parallel \hfill \end{array}$$

$$\begin{array}{ccc}& & +\frac{{(\psi \left(\tau \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)K_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel \hfill \\ \hfill & & +\frac{{(\psi \left(\tau \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)|\varphi \left(z\right)|\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)L_{g_1}\parallel x-\overline{x}\parallel \hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)K_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel \hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)|\varphi \left(z\right)|\hfill \\ \hfill & & +\sum _{i=1}^p\frac{(\psi \left(\tau \right)-\psi \left(z_i\right)){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)L_{g_1}\parallel x-\overline{x}\parallel \hfill \\ \hfill & & +\sum _{i=1}^p\frac{(\psi \left(\tau \right)-\psi \left(z_i\right)){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)K_{g_1}\hfill \\ \hfill & & \times \parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel \hfill \\ \hfill & & +\sum _{i=1}^p\frac{(\psi \left(\tau \right)-\psi \left(z_i\right)){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)|\varphi \left(z\right)|\hfill \\ \hfill & & +\sum _{i=1}^p\frac{A_1}{|\Delta |}|a_1|({\delta }_{2k}-z_k)|x\left(z_i\right)-\overline{x}\left(z_i\right)|+\sum _{i=1}^p\frac{1}{|\Delta |}|a_1|({\delta }_{2k}-z_k)|\varphi \left(z\right)|\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)A_2|x\left(z_i\right)-\overline{x}\left(z_i\right)|\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)|\varphi \left(z\right)|\hfill \\ & & +\frac{|a_2|}{|\Delta |}(z_{k+1}-{\delta }_{2k+1})M_{f_1}\parallel x-\overline{x}\parallel \hfill \\ \hfill & & +\frac{|a_2|(z_{k+1}-{\delta }_{2k+1})}{|\Delta |}{N}_{f_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel +\frac{|a_2|}{|\Delta |}(z_{k+1}-{\delta }_{2k+1})|\varphi \left(z\right)|\hfill \\ \hfill & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma \left({\alpha }_1\right)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})L_{g_1}\parallel x-\overline{x}\parallel \hfill \end{array}$$

$$\begin{array}{ccc}\hfill & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma \left({\alpha }_1\right)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})K_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel \hfill \\ \hfill & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma \left({\alpha }_1\right)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})|\varphi \left(z\right)|\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})L_{g_1}\parallel x-\overline{x}\parallel \hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})K_{g_1}\hfill \\ \hfill & & \times \parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel \hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})|\varphi \left(z\right)|\hfill \\ \hfill & & +\sum _{i=1}^p\frac{A_1}{|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})|x\left(z_i\right)-\overline{x}\left(z_i\right)|+\sum _{i=1}^p\frac{1}{|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})|\varphi \left(z_i\right)|\hfill \\ \hfill & & +\sum _{i=1}^p{A}_2\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})\parallel x-\overline{x}\parallel \hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})|\varphi \left(z_i\right)|)\left(\psi \left(z\right)-\psi \left(z_k\right)\right)\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{\alpha }}{\Gamma ({\alpha }_1+1)}{L}_{g_1}\parallel x-\overline{x}\parallel \hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}{K}_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel \hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}|\varphi \left(z\right)|\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)}{L}_{g_1}\parallel x-\overline{x}\parallel \hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)}{K}_{g_1}\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel \hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)}|\varphi \left(z\right)|\hfill \\ \hfill & & +p{A}_1\parallel x-\overline{x}\parallel +p|\varphi \left(z\right)|+\sum _{i=1}^p{A}_2\frac{\psi \left(z\right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}\parallel x-\overline{x}\parallel +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}|\varphi \left(z\right)|.\hfill \end{array}$$

$$\begin{array}{ccc}\hfill |x\left(z\right)-\overline{x}\left(z\right)|& \le & (M_{f_1}+\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}{L}_{g_1}\hfill \\ \hfill & & +(\frac{M_{f_1}}{|\Delta |}+\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}{L}_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}{L}_{g_1}+\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}{L}_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{A_1}{|\Delta |}+\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}{A}_2\hfill \\ \hfill & & +\frac{|a_1|}{|\Delta |}({\delta }_{2k}-z_k)M_{f_1}\hfill \\ \hfill & & +\frac{{(\psi \left(\tau \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)L_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)L_{g_1}\hfill \\ \hfill & & +\frac{(\psi \left(\tau \right)-\psi \left(z_i\right)){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)L_{g_1}+p\frac{A_1}{|\Delta |}|a_1|({\delta }_{2k}-z_k)\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)A_2+\frac{|a_2|}{|\Delta |}(z_{k+1}-{\delta }_{2k+1})M_{f_1}\hfill \\ \hfill & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma \left({\alpha }_1\right)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})L_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})L_{g_1}\hfill \\ \hfill & & +p\frac{A_1}{|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})\hfill \\ \hfill & & +\frac{A_2}{|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)})\left(\psi \left(z\right)-\psi \left(z_k\right)\right)\hfill \\ \hfill & & \sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{\alpha }}{\Gamma ({\alpha }_1+1)}{L}_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)}{L}_{g_1}\hfill \\ \hfill & & +p{A}_1+A_2\sum _{i=1}^p\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)})\parallel x-\overline{x}\parallel \hfill \end{array}$$

$$\begin{array}{ccc}& & +(N_{f_1}+\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}{K}_{g_1}\hfill \\ \hfill & & +(\frac{N_{f_1}}{|\Delta |}+\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}{K}_{g_1}+\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}{K}_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}{K}_{g_1}+\frac{|a_1|}{|\Delta |}({\delta }_{2k}-z_k)N_{f_1}\hfill \\ \hfill & & +\frac{{(\psi \left(\tau \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)K_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)K_{g_1}\hfill \\ \hfill & & +\frac{(\psi \left(\tau \right)-\psi \left(z_i\right)){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)K_{g_1}+\frac{|a_2|(z_{k+1}-{\delta }_{2k+1})}{|\Delta |}{N}_{f_1}\hfill \\ \hfill & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma \left({\alpha }_1\right)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})K_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})K_{g_1})\left(\psi \left(z\right)-\psi \left(z_k\right)\right)\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}{K}_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)}{K}_{g_1})\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\parallel \hfill \\ \hfill & & +(\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}+\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{|\Delta |\Gamma ({\alpha }_1+1)}+\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{|\Delta |\Gamma ({\alpha }_1+1)}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}+\frac{|a_1|}{|\Delta |}({\delta }_{2k}-z_k)\hfill \\ \hfill & & +\frac{{(\psi \left(\tau \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)\hfill \\ \hfill & & +\sum _{i=1}^p\frac{(\psi \left(\tau \right)-\psi \left(z_i\right)){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)+\frac{|a_2|}{|\Delta |}(z_{k+1}-{\delta }_{2k+1})\hfill \\ \hfill & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma \left({\alpha }_1\right)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})+\sum _{i=1}^p\frac{1}{|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1}))\left(\psi \left(z\right)-\psi \left(z_k\right)\right)\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)}+\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}.){ϵ}_1\hfill \end{array}$$

Assume that,

$$\begin{array}{ccc}\hfill {\Omega }_5& =& M_{f_1}+\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}{L}_{g_1}\hfill \\ \hfill & & +(\frac{M_{f_1}}{|\Delta |}+\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}{L}_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}{L}_{g_1}+\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}{L}_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{A_1}{|\Delta |}+\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}{A}_2\hfill \\ \hfill & & +\frac{|a_1|}{|\Delta |}({\delta }_{2k}-z_k)M_{f_1}\hfill \\ \hfill & & +\frac{{(\psi \left(\tau \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)L_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)L_{g_1}\hfill \\ \hfill & & +\frac{(\psi \left(\tau \right)-\psi \left(z_i\right)){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)L_{g_1}+p\frac{A_1}{|\Delta |}|a_1|({\delta }_{2k}-z_k)\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)A_2+\frac{|a_2|}{|\Delta |}(z_{k+1}-{\delta }_{2k+1})M_{f_1}\hfill \\ \hfill & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma \left({\alpha }_1\right)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})L_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})L_{g_1}\hfill \\ \hfill & & +p\frac{A_1}{|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})\hfill \\ \hfill & & +\frac{A_2}{|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})\sum _{i=1}^p\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)})\left(\psi \left(z\right)-\psi \left(z_k\right)\right)\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{\alpha }}{\Gamma ({\alpha }_1+1)}{L}_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)}{L}_{g_1}\hfill \\ \hfill & & +p{A}_1+A_2\sum _{i=1}^p\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}.\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\Omega }_6& =& N_{f_1}+\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}{K}_{g_1}\hfill \\ \hfill & & +(\frac{N_{f_1}}{|\Delta |}+\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}{K}_{g_1}+\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}{K}_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}{K}_{g_1}+\frac{|a_1|}{|\Delta |}({\delta }_{2k}-z_k)N_{f_1}\hfill \\ \hfill & & +\frac{{(\psi \left(\tau \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)K_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)K_{g_1}\hfill \\ \hfill & & +\frac{(\psi \left(\tau \right)-\psi \left(z_i\right)){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)K_{g_1}+\frac{|a_2|(z_{k+1}-{\delta }_{2k+1})}{|\Delta |}{N}_{f_1}\hfill \\ \hfill & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma \left({\alpha }_1\right)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})K_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})K_{g_1})\left(\psi \left(z\right)-\psi \left(z_k\right)\right)\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}{K}_{g_1}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)}{K}_{g_1}.\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\Omega }_7& =& \frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}+\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{|\Delta |\Gamma ({\alpha }_1+1)}+\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{|\Delta |\Gamma ({\alpha }_1+1)}\hfill \\ \hfill & & \sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}+\frac{|a_1|}{|\Delta |}({\delta }_{2k}-z_k)\hfill \\ \hfill & & +\frac{{(\psi \left(\tau \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)\hfill \\ \hfill & & +\sum _{i=1}^p\frac{(\psi \left(\tau \right)-\psi \left(z_i\right)){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)+\frac{|a_2|}{|\Delta |}(z_{k+1}-{\delta }_{2k+1})\hfill \\ \hfill & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\alpha }_1}}{\Gamma \left({\alpha }_1\right)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})+\sum _{i=1}^p\frac{1}{|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})\hfill \\ \hfill & & \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1}))\left(\psi \left(z\right)-\psi \left(z_k\right)\right)\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\alpha }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\alpha }_1\right)}+\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}.\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\Omega }_8& =& M_{f_2}+\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{{\beta }_1}}{\Gamma ({\beta }_1+1)}{L}_{g_2}\hfill \\ \hfill & & +(\frac{M_{f_2}}{|\Delta |}+\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\beta }_1}}{\Gamma ({\beta }_1+1)|\Delta |}{L}_{g_2}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_1}}{\Gamma ({\beta }_1+1)|\Delta |}{L}_{g_2}+\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\beta }_1\right)|\Delta |}{L}_{g_2}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{A_1}{|\Delta |}+\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}{A}_2\hfill \\ \hfill & & +\frac{|a_1|}{|\Delta |}({\delta }_{2k}-z_k)M_{f_2}\hfill \\ \hfill & & +\frac{{(\psi \left(\tau \right)-\psi \left(z_k\right))}^{{\beta }_1}}{\Gamma ({\beta }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)L_{g_2}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_1}}{\Gamma ({\beta }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)L_{g_2}\hfill \\ \hfill & & +\frac{(\psi \left(\tau \right)-\psi \left(z_i\right)){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\beta }_1\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)L_{g_2}+p\frac{A_1}{|\Delta |}|a_1|({\delta }_{2k}-z_k)\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)A_2+\frac{|a_2|}{|\Delta |}(z_{k+1}-{\delta }_{2k+1})M_{f_2}\hfill \\ \hfill & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\beta }_1}}{\Gamma \left({\beta }_1\right)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})L_{g_2}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_1}}{\Gamma ({\beta }_1+1)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})L_{g_2}\hfill \\ \hfill & & +p\frac{A_1}{|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})\hfill \\ \hfill & & +\frac{A_2}{|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})\sum _{i=1}^p\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)})\left(\psi \left(z\right)-\psi \left(z_k\right)\right)\hfill \\ \hfill & & \sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{\alpha }}{\Gamma ({\beta }_1+1)}{L}_{g_2}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\beta }_1\right)}{L}_{g_2}\hfill \\ \hfill & & +p{A}_1+A_2\sum _{i=1}^p\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}.\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\Omega }_9& =& N_{f_2}+\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{{\beta }_1}}{\Gamma ({\beta }_1+1)}{K}_{g_2}\hfill \\ \hfill & & +(\frac{N_{f_2}}{|\Delta |}+\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\beta }_1}}{\Gamma ({\beta }_1+1)|\Delta |}{K}_{g_2}+\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_1}}{\Gamma ({\beta }_1+1)|\Delta |}{K}_{g_2}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\beta }_1\right)|\Delta |}{K}_{g_2}+\frac{|a_1|}{|\Delta |}({\delta }_{2k}-z_k)N_{f_2}\hfill \\ \hfill & & +\frac{{(\psi \left(\tau \right)-\psi \left(z_k\right))}^{{\beta }_1}}{\Gamma ({\beta }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)K_{g_2}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_1}}{\Gamma ({\beta }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)K_{g_2}\hfill \\ \hfill & & +\frac{(\psi \left(\tau \right)-\psi \left(z_i\right)){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\beta }_1\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)K_{g_2}+\frac{|a_2|(z_{k+1}-{\delta }_{2k+1})}{|\Delta |}{N}_{f_2}\hfill \\ \hfill & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\beta }_1}}{\Gamma \left({\beta }_1\right)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})K_{g_2}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_1}}{\Gamma ({\beta }_1+1)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})K_{g_2})\left(\psi \left(z\right)-\psi \left(z_k\right)\right)\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_1}}{\Gamma ({\beta }_1+1)}{K}_{g_2}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\beta }_1\right)}{K}_{g_2}.\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\Omega }_{10}& =& \frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{{\beta }_1}}{\Gamma ({\beta }_1+1)}+\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\beta }_1}}{|\Delta |\Gamma ({\beta }_1+1)}+\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_1}}{|\Delta |\Gamma ({\beta }_1+1)}\hfill \\ \hfill & & \sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\beta }_1\right)|\Delta |}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\eta \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}+\frac{|a_1|}{|\Delta |}({\delta }_{2k}-z_k)\hfill \\ \hfill & & +\frac{{(\psi \left(\tau \right)-\psi \left(z_k\right))}^{{\beta }_1}}{\Gamma ({\beta }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_1}}{\Gamma ({\beta }_1+1)|\Delta |}|a_1|({\delta }_{2k}-z_k)\hfill \\ \hfill & & +\sum _{i=1}^p\frac{(\psi \left(\tau \right)-\psi \left(z_i\right)){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\beta }_1\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}|a_1|({\delta }_{2k}-z_k)+\frac{|a_2|}{|\Delta |}(z_{k+1}-{\delta }_{2k+1})\hfill \\ \hfill & & +\frac{{(\psi \left(\eta \right)-\psi \left(z_k\right))}^{{\beta }_1}}{\Gamma \left({\beta }_1\right)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_1}}{\Gamma ({\beta }_1+1)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})+\sum _{i=1}^p\frac{1}{|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1})\hfill \\ \hfill & & \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(\tau \right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)|\Delta |}|a_2|(z_{k+1}-{\delta }_{2k+1}))\left(\psi \left(z\right)-\psi \left(z_k\right)\right)\hfill \\ \hfill & & +\sum _{i=1}^p\frac{{(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_1}}{\Gamma ({\beta }_1+1)}\hfill \\ \hfill & & +\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right){(\psi \left(z_i\right)-\psi \left(z_{i-1}\right))}^{{\beta }_1-1}}{{\psi }^{{}^{\prime }}\left(z_i\right)\Gamma \left({\beta }_1\right)}+\sum _{i=1}^p\frac{\psi \left(z\right)-\psi \left(z_i\right)}{{\psi }^{{}^{\prime }}\left(z_i\right)}.\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \parallel x-\overline{x}\parallel & \le & {\Omega }_5\parallel x-\overline{x}\parallel +{\Omega }_6\parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\left(z\right)\parallel +{\Omega }_7{ϵ}_1.\hfill \end{array}$$

Since

$$\begin{array}{ccc}\hfill \parallel {}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }x\left(z\right)-{}^c{\mathcal{D}}_{z_k,z}^{{\alpha }_1;\psi }\overline{x}\left(z\right)\parallel & \le & \frac{\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\alpha }_1}}{\Gamma \left({\alpha }_1\right)}\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}{M}_{f_1}+L_{g_1}}{(1-\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\alpha }_1}}{\Gamma \left({\alpha }_1\right)}\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}{N}_{f_1}-K_{g_1})}\parallel x-\overline{x}\parallel .\hfill \end{array}$$

So we have,

$$\begin{array}{ccc}\hfill \parallel x-\overline{x}\parallel & \le & {\Omega }_5\parallel x-\overline{x}\parallel +{\Omega }_6(\frac{\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\alpha }_1}}{\Gamma \left({\alpha }_1\right)}\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}{M}_{f_1}+L_{g_1}}{(1-\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\alpha }_1}}{\Gamma \left({\alpha }_1\right)}\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}{N}_{f_1}-K_{g_1})}\hfill \\ & & \times \parallel x-\overline{x}\parallel +{\Omega }_7{ϵ}_1,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \parallel x-\overline{x}\parallel & \le & \frac{{\Omega }_7}{1-({\Omega }_5+{\Omega }_6\frac{\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\alpha }_1}}{\Gamma \left({\alpha }_1\right)}\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}{M}_{f_1}+L_{g_1}}{(1-\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\alpha }_1}}{\Gamma \left({\alpha }_1\right)}\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}{N}_{f_1}-K_{g_1})})}{ϵ}_1.\hfill \end{array}$$

Where we assumed that

$$\begin{array}{c}\hfill {\Omega }_5+{\Omega }_6\frac{\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\alpha }_1}}{\Gamma \left({\alpha }_1\right)}\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}{M}_{f_1}+L_{g_1}}{(1-\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\alpha }_1}}{\Gamma \left({\alpha }_1\right)}\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}{N}_{f_1}-K_{g_1})}<1.\end{array}$$

Similarly we have,

$$\begin{array}{ccc}\hfill \parallel y-\overline{y}\parallel & \le & \frac{{\Omega }_{10}}{1-({\Omega }_8+{\Omega }_9\frac{\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\beta }_1}}{\Gamma \left({\beta }_1\right)}\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}{M}_{f_2}+L_{g_2}}{(1-\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\beta }_1}}{\Gamma \left({\beta }_1\right)}\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}{N}_{f_2}-K_{g_2})})}{ϵ}_2\hfill \end{array}$$

where we assumed that

$$\begin{array}{c}\hfill {\Omega }_8+{\Omega }_9\frac{\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\beta }_1}}{\Gamma \left({\beta }_1\right)}\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}{M}_{f_2}+L_{g_2}}{(1-\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\beta }_1}}{\Gamma \left({\beta }_1\right)}\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}{N}_{f_2}-K_{g_2})}<1.\end{array}$$

Let

$$max\{{ϵ}_1,{ϵ}_2\}=ϵ,$$

then

$$\begin{array}{ccc}\hfill \parallel x-\overline{x}\parallel +\parallel y-\overline{y}\parallel & \le & \frac{{\Omega }_7}{1-{\Omega }_5-{\Omega }_6\frac{\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\alpha }_1}}{\Gamma \left({\alpha }_1\right)}\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}{M}_{f_1}+L_{g_1}}{(1-\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\alpha }_1}}{\Gamma \left({\alpha }_1\right)}\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}{N}_{f_1}-K_{g_1})}}ϵ\hfill \\ & & +\frac{{\Omega }_{10}}{1-{\Omega }_8-{\Omega }_9\frac{\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\beta }_1}}{\Gamma \left({\beta }_1\right)}\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}{M}_{f_2}+L_{g_2}}{(1-\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\beta }_1}}{\Gamma \left({\beta }_1\right)}\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}{N}_{f_2}-K_{g_2})}}ϵ.\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \parallel x-\overline{x}\parallel +\parallel y-\overline{y}\parallel & \le & [\frac{{\Omega }_7}{1-{\Omega }_5-{\Omega }_6\frac{\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\alpha }_1}}{\Gamma \left({\alpha }_1\right)}\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}{M}_{f_1}+L_{g_1}}{(1-\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\alpha }_1}}{\Gamma \left({\alpha }_1\right)}\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}{N}_{f_1}-K_{g_1})}}\hfill \\ & & +\frac{{\Omega }_{10}}{1-{\Omega }_8-{\Omega }_9\frac{\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\beta }_1}}{\Gamma \left({\beta }_1\right)}\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}{M}_{f_2}+L_{g_2}}{(1-\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\beta }_1}}{\Gamma \left({\beta }_1\right)}\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}{N}_{f_2}-K_{g_2})}}]ϵ.\hfill \end{array}$$

Let

$$\begin{array}{ccc}\hfill \Theta & =& [\frac{{\Omega }_7}{1-{\Omega }_5-{\Omega }_6\frac{\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\alpha }_1}}{\Gamma \left({\alpha }_1\right)}\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}{M}_{f_1}+L_{g_1}}{(1-\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\alpha }_1}}{\Gamma \left({\alpha }_1\right)}\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}{N}_{f_1}-K_{g_1})}}\hfill \\ & & +\frac{{\Omega }_{10}}{1-{\Omega }_8-{\Omega }_9\frac{\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\beta }_1}}{\Gamma \left({\beta }_1\right)}\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}{M}_{f_2}+L_{g_2}}{(1-\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\beta }_1}}{\Gamma \left({\beta }_1\right)}\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}{N}_{f_2}-K_{g_2})}}].\hfill \end{array}$$

Hence, we have

$$\parallel (x,y)-(\overline{x},\overline{y}){\parallel }_{{\mathbb{X}}^{{}^{\prime }}\times {\mathbb{Y}}^{{}^{\prime }}}\le \Theta ϵ.$$

Thus system (1) is Ulam-Hyers stable. □

6. Example

In this portion, we discuss an example related to our main results.

Example 1.

$$\begin{array}{ccc}\hfill {}^c{\mathcal{D}}^{\frac{5}{3};x}[x\left(z\right)-(\frac{cosz|x(z)|}{450(1+|x\left(z\right)|)}+\frac{|{}^c{\mathcal{D}}^{\frac{5}{3};x}x\left(z\right)|}{460(1+|{}^c{\mathcal{D}}^{\frac{5}{3};x}x\left(z\right)|)})]& =& \frac{3}{7}+\frac{4+|x(z)|}{1600(1+|x\left(z\right)|)}\hfill \\ & & +\frac{|{}^c{\mathcal{D}}^{\frac{5}{3};x}x\left(z\right)|}{1650+|{}^c{\mathcal{D}}^{\frac{5}{3};x}x\left(z\right)|},\hfill \\ & & z\in [0,1], z\ne \frac{4}{5}.\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {}^c{\mathcal{D}}^{\frac{5}{3};y}[y\left(z\right)-(\frac{cosz|y(z)|}{450(1+|y\left(z\right)|)}+\frac{|{}^c{\mathcal{D}}^{\frac{5}{3};y}y\left(z\right)|}{460(1+|{}^c{\mathcal{D}}^{\frac{5}{3};y}y\left(z\right)|)})]& =& \frac{5}{7}+\frac{|siny\left(z\right)|}{1200(1+|siny\left(z\right)|)}\hfill \\ & & +\frac{1}{1250}|cos{}^c{\mathcal{D}}^{\frac{5}{3};y}y\left(z\right)|.\hfill \\ & & z\in [0,1],z\ne \frac{5}{6}.\hfill \end{array}$$

$$x\left(0\right)=0, x\left(\frac{1}{4}\right)={\int }_0^{\frac{1}{4}}x\left(\tau \right)d\tau +{\int }_{\frac{1}{3}}^{\frac{4}{5}}x\left(\tau \right)d\tau .$$

$$y\left(0\right)=0, y\left(\frac{1}{4}\right)={\int }_0^{\frac{1}{4}}y\left(\tau \right)d\tau +{\int }_{\frac{1}{3}}^{\frac{5}{6}}y\left(\tau \right)d\tau .$$

$$I_{1}x\left(\frac{4}{5}\right)=\frac{1}{230+|x(z)|}, J_{1}x\left(\frac{4}{5}\right)=\frac{1}{260+|{}^c{\mathcal{D}}^{\frac{5}{3};x}x\left(z\right)|}.$$

$$I_1^{*}y\left(\frac{5}{6}\right)=\frac{1}{250+|y(z)|}, J_1^{*}y\left(\frac{5}{6}\right)=\frac{1}{300+|{}^c{\mathcal{D}}^{\frac{5}{3};y}y\left(z\right)|}.$$

We see in the proposed problem, that ${\alpha }_1={\beta }_1=\frac{5}{3}$, and $z_j\ne \frac{4}{5}$, where j = 1, 2, ..., 60.

For $z\in [0,1]$ and $x_1\left(z\right),x_2\left(z\right),y_1\left(z\right),y_2\left(z\right)\in \mathbb{R},$ we have,

$$\begin{array}{c}\hfill |f_1(z,x_1\left(z\right),{}^c{\mathcal{D}}^{\frac{5}{3};x}{x}_1\left(z\right))-f_1(z,x_2\left(z\right),{}^c{\mathcal{D}}^{\frac{5}{3};x}{x}_2\left(z\right))|\le \frac{1}{450}|x_1\left(z\right)-x_2\left(z\right)|+\frac{1}{460}|{}^c{\mathcal{D}}^{\frac{5}{3};x}{x}_1\left(z\right)-{}^c{\mathcal{D}}^{\frac{5}{3};x}{x}_2\left(z\right)|.\end{array}$$

$$\begin{array}{c}\hfill |g_1(z,x_1\left(z\right),{}^c{\mathcal{D}}^{\frac{5}{3};x}{x}_1\left(z\right))-g_1(z,x_2\left(z\right),{}^c{\mathcal{D}}^{\frac{5}{3};x}{x}_2\left(z\right))|\le \frac{1}{1600}|x_1\left(z\right)-x_2\left(z\right)|+\frac{1}{1650}|{}^c{\mathcal{D}}^{\frac{5}{3};x}{x}_1\left(z\right)-{}^c{\mathcal{D}}^{\frac{5}{3};x}{x}_2\left(z\right)|.\end{array}$$

$$\begin{array}{c}\hfill |f_2(z,y_1\left(z\right),{}^c{\mathcal{D}}^{\frac{5}{3};y}{y}_1(z))-f_2(z,y_2\left(z\right),{}^c{\mathcal{D}}^{\frac{5}{3};y}{y}_2\left(z\right))|\le \frac{1}{450}|y_1\left(z\right)-y_2\left(z\right)|+\frac{1}{460}|{}^c{\mathcal{D}}^{\frac{5}{3};y}{y}_1\left(z\right)-{}^c{\mathcal{D}}^{\frac{5}{3};y}{y}_2\left(z\right)|.\end{array}$$

$$\begin{array}{c}\hfill |g_2(z,y_1\left(z\right),{}^c{\mathcal{D}}^{\frac{5}{3};y}{y}_1\left(z\right))-g_2(z,y_2\left(z\right),{}^c{\mathcal{D}}^{\frac{5}{3};y}{y}_2\left(z\right))|\le \frac{1}{1200}|y_1\left(z\right)-y_2\left(z\right)|+\frac{1}{1250}|{}^c{\mathcal{D}}^{\frac{5}{3};y}{y}_1\left(z\right)-{}^c{\mathcal{D}}^{\frac{5}{3};y}{y}_2\left(z\right)|.\end{array}$$

$$\begin{array}{c}\hfill |I{x}_1\left(z\right)\left(z_i\right)-I{x}_2\left(z\right)\left(z_i\right)|\le \frac{1}{130}|x_1\left(z\right)-x_2\left(z\right)|, |J{x}_1\left(z\right)\left(z_i\right)-J{x}_2\left(z\right)\left(z_i\right)|\le \frac{1}{160}|x_1\left(z\right)-x_2\left(z\right)|.\end{array}$$

$$\begin{array}{c}\hfill |I^{*}{y}_1\left(z\right)\left(z_j\right)-I^{*}{y}_1\left(z\right)\left(z_j\right)|\le \frac{1}{150}|y_1\left(z\right)-y_2\left(z\right)|, |J^{*}{y}_1\left(z\right)\left(z_j\right)-J^{*}{y}_2\left(z\right)\left(z_j\right)|\le \frac{1}{200}|y_1\left(z\right)-y_2\left(z\right)|.\end{array}$$

From above inequalities, we obtain that $M_{f_1}=\frac{1}{450},N_{f_1}=\frac{1}{460},L_{g_1}=\frac{1}{1600},K_{g_1}=\frac{1}{1650},M_{f_2}=\frac{1}{450},N_{f_2}=\frac{1}{460},L_{g_2}=\frac{1}{1200},K_{g_2}=\frac{1}{1250},{\mu }_1=\frac{1}{130},{\mu }_2=\frac{1}{160},{\mu }_3=\frac{1}{150},{\mu }_4=\frac{1}{200}.$

Since

$$\begin{array}{ccc}\hfill Z_1& =& ({\Omega }_0{M}_{f_1}+{\Omega }_1{L}_{g_1}+{\Omega }_2)\hfill \\ & & +({\Omega }_0{N}_{f_1}+{\Omega }_1{K}_{g_1}+{\Omega }_2)\left(\frac{\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\alpha }_1}}{\Gamma \left({\alpha }_1\right)}\left(\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}\right)M_{f_1}+L_{g_1}}{1-(\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\alpha }_1}}{\Gamma \left({\alpha }_1\right)}\left(\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}\right)N_{f_1}+K_{g_1})}\right),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill Z_2& =& ({\Omega }_0{M}_{f_2}+{\Omega }_3{L}_{g_2}+{\Omega }_4)\hfill \\ & & +({\Omega }_0{N}_{f_2}+{\Omega }_3{K}_{g_2}+{\Omega }_4)\left(\frac{\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\beta }_1}}{\Gamma \left({\beta }_1\right)}\left(\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}\right)M_{f_2}+L_{g_2}}{1-(\frac{{(\psi \left(z\right)-\psi \left(z_k\right))}^{1-{\beta }_1}}{\Gamma \left({\beta }_1\right)}\left(\frac{1}{{\psi }^{{}^{\prime }}\left(z\right)}\frac{d}{dz}\right)N_{f_2}+K_{g_2})}\right).\hfill \end{array}$$

On calculating $Z_1$ and $Z_2$ we have $Z_1=0.40370658619<1$ and $Z_2=0.51479638<1$. Then $max\{Z_1,Z_2\}<1$, and the coupled system (1) has unique solution.

Also on calculating $\Theta =8.92827998857$, and $ϵ=0.00214$, we get $\Theta ϵ=0.01910651917>0$. Therefore, the coupled system (1) is Ulam-Hyers stable.

7. Conclusions

In this article, we studied the existence and uniqueness property of the coupled system of impulsive hybrid fractional differential equations with slit-strips integral boundary conditions. We utilized the Schaefer’s fixed point theorem for the existence of at least one solution of the problem (1). For the uniqueness of the solution of problem (1) we used Banach contraction principle. Also we studied the Ulam-Hyers stability of the proposed problem (1). Finally the we provide an example for the support of the results. Our obtained results can be utilized in the impulsive problems involving the scattering by slits silicon strips detectors for scanned multi-slit X-ray imaging, the acoustic impedance of baffled strips radiators, diffraction from an elastic knife-edge adjacent to a strip, sound fields of infinitely long strips, dielectric-loaded multiple slits in a conducting plane, lattice engineering. The Ulam-Hyers stability means that for any approximation in specific region we will get to the exact distinction, so the obtained results can be utilized in numerical analysis and approximation theory of the related impulsive problems.