Sequential Fractional Hybrid Inclusions: A Theoretical Study via Dhage’s Technique and Special Contractions

Abstract

The most important objective of the present research is to establish some theoretical existence results on a novel combined configuration of a Caputo sequential inclusion problem and the hybrid integro-differential one in which the boundary conditions are also formulated as the hybrid multi-order integro-differential conditions. In this respect, firstly, some inequalities are proven in relation to the corresponding integral equation. Then, we employ some newly defined theoretical techniques with the help of the product operators on a Banach algebra and also with the aid of some special functions including $\alpha$-$\psi$-contractions and $\alpha$-admissible mappings to extract the existence criteria corresponding to the given mixed sequential hybrid BVPs. Some important useful properties such as the approximate endpoint property, $\left(C_{\alpha }\right)$-property, and the compactness play a key role in this regard. The final part of the manuscript is devoted to formulating and computing two applicable examples to guarantee the correctness of the obtained results.

1. Introduction

Human beings need to recognize different interesting phenomena more than ever before. An appropriate approach to meet this demand is to employ the methods and techniques that are available in fractional calculus and, specifically, the fractional operators in modeling of events and processes. Many operators of this fractional type have appeared in recent years, and their consistency and flexibility are becoming known to mathematicians everyday. In such a way, it is convenient that we design complicated and general abstract mathematical models of processes in the format of applicable fractional BVPs. Several examples of the usability of such operators can be seen in branches of science including bio-mathematics, medical science, engineering, and so on; see [1,2,3,4]. All of these points imply that a wide range of scientists are attracted to work on various angles of applicability of such fractional BVPs along with some dynamical behaviors of solutions of these fractional systems. In the mentioned context, most mathematicians have turned toward studying advanced fractional modelings and relevant theoretical findings and graphical behaviors of solutions for these kinds of BVPs; see [5,6,7,8,9,10,11].

Miller with the help of Ross [12] introduced sequential structures of derivatives, which are illustrated by a product of the given derivatives. Later, new findings about different forms of these operators resulted in the publishing of several articles on sequential BVPs in fractional settings. Recently, Alsaedi et al. [13] addressed the primary version of the sequential fractional boundary problem of the Caputo type given by

$$\left\{\begin{array}{c}({}^C{\mathcal{D}}_{0^{+}}^{{\eta }^{*}}+{\mu }^{*}{}^C{\mathcal{D}}_{0^{+}}^{{\eta }^{*}-1})\hslash \left(z\right)={\breve{g}}_{*}(z,\hslash \left(z\right)),\hfill \\ \hslash \left(0\right)=0,{\hslash }^{\prime }\left(0\right)=0,\hslash \left(\sigma \right)=\gamma {}^{RL}{\mathcal{I}}_{0^{+}}^{{\theta }^{*}}\hslash \left(\delta \right)\hfill \end{array}\right.$$

so that ${\eta }^{*}\in (2,3]$, $\delta \in (0,1)$, $\sigma \in (\delta ,1)$ with ${\theta }^{*}>0$, $z\in [0,1]$, $\gamma ,{\mu }^{*}\in {\mathbb{R}}^{+}$, and ${\breve{g}}_{*}\in \mathcal{C}\left([0,1]\times \mathbb{R},\mathbb{R}\right)$. The authors in relation to the existence results used some usual notions of the nonlinear analysis theory [13]. For more details, check [14].

During the past few years, a new framework of problems called hybrid differential ones supplemented with abstract types of boundary conditions have been in the spotlight of most mathematicians; see, for example, [15,16,17,18,19,20]. In 2010, the study of this category of equations began with a brilliant paper by Lakshmikantham with the aid of Dhage [21]. Both of them regarded a newly designed hybrid differential equation and obtained its extremal possible solutions by proving several inequalities [21]. After this work, Zhao et al. [22] presented a fractional general extension of the aforementioned BVP in [21] and considered an FBVP consisting of fractional hybrid differential equations. Gradually, researchers have been attracted to this category of modern differential equations. For example, Ahmad et al. [23] carried out a pure mathematical analysis to find the necessary conditions for the existence results of the following Caputo nonlocal hybrid inclusion problem illustrated by

$${\displaystyle {}^C{\mathcal{D}}_{0^{+}}^{{\eta }^{*}}\left[\frac{\varpi \left(z\right)-{\sum }_{j=1}^k{}^{RL}{\mathcal{I}}_{0^{+}}^{{\theta }_j^{*}}{h}_j^{*}(z,\varpi \left(z\right))}{g^{*}(z,\varpi \left(z\right))}\right]\in \mathfrak{O}(z,\varpi \left(z\right)),}$$

equipped with conditions $\varpi \left(0\right)=\beta \left(z\right)$ and $\varpi \left(1\right)=\alpha \in \mathbb{R}$, where ${\eta }^{*}\in (1,2]$ and ${}^{RL}{\mathcal{I}}_{0^{+}}^{{\theta }^{*}}$ stands for the RL integral of order ${\theta }^{*}>0$ with ${\theta }^{*}\in \{{\theta }_1^{*},{\theta }_2^{*},\dots ,{\theta }_k^{*}\}$.

In 2020, Baleanu, Etemad, and Rezapour [24] developed a novel hybrid version of the thermostat fractional model, which describes its performance. Indeed, this model checks an amount of ambient heat and controls it according to the temperature assessed by active sensors. We can review the mentioned thermostat hybrid model as

$${\displaystyle {}^C{\mathcal{D}}_{0^{+}}^{{\eta }^{*}}\left[\frac{\mu \left(s\right)}{w^{*}(s,\mu \left(s\right))}\right]+\tilde{\mathsf{\Phi }}(s,\mu \left(s\right))=0,({\eta }^{*}\in (1,2],s\in [0,1])}$$

subject to the fractional hybrid boundary conditions

$$\left\{\begin{array}{c}{\displaystyle {}^C{\mathcal{D}}_{0^{+}}^1\left[\frac{\mu \left(s\right)}{w^{*}(s,\mu \left(s\right))}\right]{|}_{s=0}=0,}\hfill \\ {\displaystyle {\lambda }_{*}{}^C{\mathcal{D}}_{0^{+}}^{{\eta }^{*}-1}\left[\frac{\mu \left(s\right)}{w^{*}(s,\mu \left(s\right))}\right]{|}_{s=1}+\left[\frac{\mu \left(s\right)}{w^{*}(s,\mu \left(s\right))}\right]{|}_{s=\delta }=0,}\hfill \end{array}\right.$$

where ${\lambda }_{*}$ is a positive parameter, $\delta \in [0,1]$, and ${\eta }^{*}-1\in (0,1]$. Furthermore, ${}^C{\mathcal{D}}_{0^{+}}^1=\frac{\mathrm{d}}{\mathrm{d}z}$, ${}^C{\mathcal{D}}_{0^{+}}^{{\gamma }^{*}}$ illustrates the Caputo derivative of order ${\gamma }^{*}\in \{{\eta }^{*},{\eta }^{*}-1\}$ and $\tilde{\mathsf{\Phi }},w^{*}\in \mathcal{C}([0,1]\times \mathbb{R},\mathbb{R})$ with $w^{*}\ne 0$. The analytical techniques utilized by the authors rely on some fixed point concepts on self-maps and multifunctions.

In 2021, Abbas and Ragusa [25] dealt with a novel category of hybrid FDEs via proportional derivatives depending to a certain function as

$$\left\{\begin{array}{c}{\displaystyle {}^{\varrho }{\mathcal{D}}_{a^{+}}^{\delta ,\vartheta }\left[\frac{\mu \left(s\right)}{w^{*}(s,\mu \left(s\right))}\right]=\tilde{\mathsf{\Phi }}(s,\mu \left(s\right)),}\hfill \\ {\displaystyle {}^{\varrho }{\mathcal{I}}_{a^{+}}^{1-\delta ,\vartheta }\left[\frac{\mu \left(s\right)}{w^{*}(s,\mu \left(s\right))}\right]{|}_{s=a}=b\in \mathbb{R},}\hfill \end{array}\right.$$

where $\varrho ,\delta \in (0,1]$ and ${}^{\varrho }{\mathcal{D}}_{a^{+}}^{\delta ,\vartheta }$ displays the $\delta$th-proportional derivative depending on the increasing mapping $\vartheta$ with ${\vartheta }^{\prime }\left(s\right)>0$. They discussed the continuity of solutions of this BVP in terms of some inputs.

By considering and mixing the ideas of the above problems, we formulate a new combined structure of a Caputo sequential hybrid integro-differential inclusion as follows

$${\displaystyle {\mu }_1^{*}\left({}^C{\mathcal{D}}_{0^{+}}^{{\eta }^{*}}+{\mu }_2^{*}{}^C{\mathcal{D}}_{0^{+}}^{{\eta }^{*}-1}\right)\left[\frac{\varpi \left(z\right)}{{\tilde{S}}^{*}\left(z,\varpi \left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}\varpi \left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}\varpi \left(z\right)\right)}\right]\in \tilde{\mathfrak{O}}(z,\varpi \left(z\right))}$$

(1)

furnished with three-point hybrid multi-order integro-differential conditions:

$$\left\{\begin{array}{c}{\displaystyle \left[\frac{\varpi \left(z\right)}{{\tilde{S}}^{*}\left(z,\varpi \left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}\varpi \left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}\varpi \left(z\right)\right)}\right]{|}_{z=0}=0,}\hfill \\ {\displaystyle {}^C{\mathcal{D}}_{0^{+}}^1\left[\frac{\varpi \left(z\right)}{{\tilde{S}}^{*}\left(z,\varpi \left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}\varpi \left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}\varpi \left(z\right)\right)}\right]{|}_{z=0}=0,}\hfill \\ {\displaystyle {}^{RL}{\mathcal{I}}_{0^{+}}^{{\theta }_1^{*}}\left[\frac{\varpi \left(z\right)}{{\tilde{S}}^{*}\left(z,\varpi \left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}\varpi \left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}\varpi \left(z\right)\right)}\right]{|}_{z=1}}\hfill \\ {\displaystyle +{}^{RL}{\mathcal{I}}_{0^{+}}^{{\theta }_2^{*}}\left[\frac{\varpi \left(z\right)}{{\tilde{S}}^{*}\left(z,\varpi \left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}\varpi \left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}\varpi \left(z\right)\right)}\right]{|}_{z=\sigma }=0,}\hfill \end{array}\right.$$

(2)

where $z\in \mathfrak{T}:=[0,1]$, ${\eta }^{*}\in [2,3)$, ${\eta }^{*}-1\in [1,2)$, $\sigma \in (0,1)$, ${\mu }_1^{*},{\mu }_2^{*},{\theta }_1^{*},{\theta }_2^{*}>0$ with ${\theta }_1^{*}-1>0$ and ${\theta }_2^{*}-1>0$. Furthermore, for $k\in \mathbb{N}$, ${\delta }_1^{*},{\delta }_2^{*},\dots ,{\delta }_k^{*}>0$. Here, ${}^C{\mathcal{D}}_{0^{+}}^{(·)}$ and ${}^{RL}{\mathcal{I}}_{0^{+}}^{(·)}$ display the Caputo derivative and the RL integral. Further, notice that ${}^C{\mathcal{D}}_{0^{+}}^1=\frac{\mathrm{d}}{\mathrm{d}z}$. The continuous real-valued function ${\tilde{S}}^{*}\ne 0$ is defined on $\mathfrak{T}\times {\mathbb{R}}^{k+1}$, and the set-valued map $\tilde{\mathfrak{O}}$ is assumed to be considered on $\mathfrak{T}\times \mathbb{R}$ having some properties that are pointed out in the sequel. In the next stage, we turn to a special nonhybrid form of the above FBVP as follows:

$$\left\{\begin{array}{c}{\mu }_1^{*}\left({}^C{\mathcal{D}}_{0^{+}}^{{\eta }^{*}}+{\mu }_2^{*}{}^C{\mathcal{D}}_{0^{+}}^{{\eta }^{*}-1}\right)\varpi \left(z\right)\in \tilde{\mathfrak{O}}(z,\varpi \left(z\right)),\hfill \\ \varpi \left(0\right)=0,{\varpi }^{\prime }\left(0\right)=0,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\theta }_1^{*}}\varpi \left(1\right)+{}^{RL}{\mathcal{I}}_{0^{+}}^{{\theta }_2^{*}}\varpi \left(\sigma \right)=0,\hfill \end{array}\right.$$

(3)

where ${\eta }^{*}\in [2,3)$, ${\eta }^{*}-1\in [1,2)$, $z\in \mathfrak{T}:=[0,1]$, $0<\sigma <1$, ${\mu }_1^{*},{\mu }_2^{*}\in {\mathbb{R}}_{+}$, and ${}^{RL}{\mathcal{I}}_{0^{+}}^{(·)}$ illustrates the RL integral of both orders ${\theta }_1^{*},{\theta }_2^{*}>0$ with ${\theta }_1^{*}-1>0$ and ${\theta }_2^{*}-1>0$. As you see, it is notable that we derive the Caputo sequential inclusion problem (3) if we take ${\tilde{S}}^{*}\left(z,\varpi \left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}\varpi \left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}\varpi \left(z\right)\right)=1$ in the given problem (1) and (2). The readers should pay attention to this point that the combined construction of a sequential problem and a hybrid one is a novel and unique fractional modeling, and this kind of mixed sequential hybrid integro-differential inclusion problem has not been discussed in any literature so far. We organize the rest of the research as follows. In Section 2, we review some primitive notions briefly. In Section 3, we develop some newly defined theoretical methods to derive the required existence criteria corresponding to the given mixed BVPs by (1)–(3). Some existing important properties on operators and the existing space including the approximate endpoint property, the $\left(C_{\alpha }\right)$-property and the compactness play a key role in this regard. The last section of this manuscript is devoted to formulating and computing two simulation examples to validate the correctness of the results.

2. Preliminaries

This part of the paper is devoted to recalling some basic notions and auxiliary concepts briefly. First of all, we assume that ${\eta }^{*}>0$. Then, one can recall the definition of the Riemann–Liouville integral for $\varpi :[0,+\infty )\to \mathbb{R}$ in the format of ${\displaystyle {}^{RL}{\mathcal{I}}_{0^{+}}^{{\eta }^{*}}\varpi \left(z\right)={\int }_0^z\frac{{(z-q)}^{{\eta }^{*}-1}}{\mathsf{\Gamma }\left({\eta }^{*}\right)}\varpi \left(q\right)\mathrm{d}q}$ such that there exists the R.H.S. value finitely [26,27]. Hereafter, we regard ${\eta }^{*}\in (n-1,n)$. For $\varpi \in {\mathcal{AC}}_{\mathbb{R}}^{\left(n\right)}\left([0,+\infty )\right)$, the derivative in the Caputo sense can be defined by the following framework:

$${\displaystyle {}^C{\mathcal{D}}_{0^{+}}^{{\eta }^{*}}\varpi \left(z\right)={\int }_0^z\frac{{(z-q)}^{n-{\eta }^{*}-1}}{\mathsf{\Gamma }(n-{\eta }^{*})}{\varpi }^{\left(n\right)}\left(q\right)\mathrm{d}q}$$

such that the R.H.S integral exists [26,27]. In addition, for an arbitrary sufficiently smooth map $\varpi :[0,+\infty )\to \mathbb{R}$, one can define the sequential fractional derivative as follows:

$${\mathcal{D}}_{0^{+}}^{{\eta }^{*}}\varpi \left(z\right)=({\mathcal{D}}_{0^{+}}^{{\eta }_1^{*}}{\mathcal{D}}_{0^{+}}^{{\eta }_2^{*}}\dots {\mathcal{D}}_{0^{+}}^{{\eta }_n^{*}})\varpi \left(z\right),$$

where ${\eta }^{*}=({\eta }_1^{*},{\eta }_2^{*},\dots ,{\eta }_n^{*})$ is a multi-index [12]. It is notable that the sequential derivative operator ${\mathcal{D}}_{0^{+}}^{{\eta }^{*}}$ can be considered one of the Caputo, RL, Hadamard, Caputo–Hadamard, etc. In the present research, we shall utilize the sequential derivative in the Caputo setting, which is illustrated by the following. For $n=1+\left[{\eta }^{*}\right]$, the Caputo sequential derivative of the given map $\varpi :[0,+\infty )\to \mathbb{R}$ is arranged as

$${}^C{\mathcal{D}}_{0^{+}}^{{\eta }^{*}}\varpi \left(z\right)={\mathcal{D}}_{0^{+}}^{-(n-{\eta }^{*})}{\left(\frac{\mathrm{d}}{\mathrm{d}z}\right)}^n\varpi \left(z\right),$$

where ${\mathcal{D}}_{0^{+}}^{-(n-{\eta }^{*})}\varpi \left(z\right)={}^{RL}{\mathcal{I}}_{0^{+}}^{(n-{\eta }^{*})}\varpi \left(z\right)$ stands for the RL integral of order $n-{\eta }^{*}$ [26]. In a monograph presented by Miller and Ross [12], it was demonstrated that the general solution of ${}^C{\mathcal{D}}_{0^{+}}^{{\eta }^{*}}\varpi \left(z\right)=0$ is illustrated by $\varpi \left(z\right)=c_0+c_{1}z+c_2{z}^2+\dots +c_{n-1}{z}^{n-1}$, and we obtain

$${}^{RL}{\mathcal{I}}_{0^{+}}^{{\eta }^{*}}\left({}^C{\mathcal{D}}_{0^{+}}^{{\eta }^{*}}\varpi \left(z\right)\right)=\varpi \left(z\right)+\sum _{k=0}^{n-1}{c}_k{z}^k=\varpi \left(z\right)+c_0+c_{1}z+c_2{z}^2+\dots +c_{n-1}{z}^{n-1},$$

in which $c_i$ are real numbers with $n=\left[{\eta }^{*}\right]+1$. Assuming the normed space $(\mathfrak{W},\parallel ·{\parallel }_{\mathfrak{W}})$, we introduce the notations $\mathcal{P}\left(\mathfrak{W}\right)$, ${\mathcal{P}}_{cls}\left(\mathfrak{W}\right)$, ${\mathcal{P}}_{bnd}\left(\mathfrak{W}\right)$, ${\mathcal{P}}_{cmp}\left(\mathfrak{W}\right)$, and ${\mathcal{P}}_{cvx}\left(\mathfrak{W}\right)$ to illustrate the set of all nonempty, closed, bounded, compact, and convex subsets of $\mathfrak{W}$, respectively. A metric ${\mathbb{PH}}_{d_{\mathfrak{W}}}:\mathcal{P}\left(\mathfrak{W}\right)\times \mathcal{P}\left(\mathfrak{W}\right)\to {\mathbb{R}}^{*}$ is named as the Pompeiu–Hausdorff metric if

$${\mathbb{PH}}_{d_{\mathfrak{W}}}({\mathcal{E}}_1,{\mathcal{E}}_2)=max\{\underset{e_1\in {\mathcal{E}}_1}{sup}{d}_{\mathfrak{W}}(e_1,{\mathcal{E}}_2),\underset{e_2\in {\mathcal{E}}_2}{sup}{d}_{\mathfrak{W}}({\mathcal{E}}_1,e_2)\}$$

so that $d_{\mathfrak{W}}({\mathcal{E}}_1,e_2)={inf}_{e_1\in {\mathcal{E}}_1}{d}_{\mathfrak{W}}(e_1,e_2)$ and $d_{\mathfrak{W}}(e_1,{\mathcal{E}}_2)={inf}_{e_2\in {\mathcal{E}}_2}{d}_{\mathfrak{W}}(e_1,e_2)$ [28]. We represent all selections of $\tilde{\mathfrak{O}}$ at point $\varpi \in {\mathcal{C}}_{\mathbb{R}}\left(\mathfrak{T}\right)$ by

$${\left(\mathbb{SEL}\right)}_{\tilde{\mathfrak{O}},\varpi }:=\{\widehat{\kappa }\in {\mathcal{L}}_{\mathbb{R}}^1\left(\mathfrak{T}\right):\widehat{\kappa }\left(z\right)\in \tilde{\mathfrak{O}}(z,\varpi \left(z\right))\}$$

for a.e. $z\in \mathfrak{T}:=[0,1]$ [28,29]. Further, ${\left(\mathbb{SEL}\right)}_{\tilde{\mathfrak{O}},\varpi }\ne \varnothing$ if $dim\left(\mathfrak{W}\right)<\infty$ [28].

A novel category of nonnegative nondecreasing mappings as $\psi :[0,\infty )\to [0,\infty )$ was constructed by Samet et al. [30] in which ${\sum }_{n=1}^{\infty }{\psi }^n\left(z\right)<\infty$. This category of functions is illustrated by the notation $\mathsf{\Psi }$. The most important property of these functions is that $\psi \left(z\right)<z$ for each $z>0$ [30]. After this work, Mohammadi et al. introduced a generalized framework for set-valued maps as follows [31]. A multifunction $\tilde{\mathfrak{O}}:\mathfrak{W}\to {\mathcal{P}}_{bnd,cls}\left(\mathfrak{W}\right)$ is named an $\alpha$-$\psi$-contraction whenever for each $\varpi ,{\varpi }^{\prime }\in \mathfrak{W}$, we have

$$\alpha (\varpi ,{\varpi }^{\prime }){\mathbb{PH}}_{d_{\mathfrak{W}}}(\tilde{\mathfrak{O}}\varpi ,\tilde{\mathfrak{O}}{\varpi }^{\prime })\le \psi \left(d_{\mathfrak{W}}(\varpi ,{\varpi }^{\prime })\right).$$

We also say that the normed space $\mathfrak{W}$ possesses the $\left(C_{\alpha }\right)$-property if for each convergent sequence $\left\{{\varpi }_n\right\}\subseteq \mathfrak{W}$ with ${\varpi }_n\to \varpi$ and $\alpha ({\varpi }_n,{\varpi }_{n+1})\ge 1$ for any $n\in \mathbb{N}$, there exists a subsequence $\left\{{\varpi }_{n_j}\right\}$ of $\left\{{\varpi }_n\right\}$ provided that $\alpha ({\varpi }_{n_j},\varpi )\ge 1$ for each $j\in \mathbb{N}$. In the same direction, we say that $\tilde{\mathfrak{O}}$ is $\alpha$-admissible if for every $\varpi \in \mathfrak{W}$ and ${\varpi }^{\prime }\in \tilde{\mathfrak{O}}\left(\varpi \right)$ with $\alpha (\varpi ,{\varpi }^{\prime })\ge 1$, we have $\alpha ({\varpi }^{\prime },{\varpi }^{\prime \prime })\ge 1$ for all ${\varpi }^{\prime \prime }\in \tilde{\mathfrak{O}}\left({\varpi }^{\prime }\right)$ [31]. Eventually, an element $\varpi \in \mathfrak{W}$ is said to be the endpoint of $\tilde{\mathfrak{O}}:\mathfrak{W}\to \mathcal{P}\left(\mathfrak{W}\right)$ if the equality $\tilde{\mathfrak{O}}\left(\varpi \right)=\left\{\varpi \right\}$ holds [32]. In addition, $\tilde{\mathfrak{O}}$ is a set-valued map having an approximate endpoint property (APPX-endpoint property) if ${inf}_{\varpi \in \mathfrak{W}}{sup}_{\varrho \in \tilde{\mathfrak{O}}\varpi }{d}_{\mathfrak{W}}(\varpi ,\varrho )=0$ [32]. We need the following lemmas to establish theoretical results about the existence criteria of solutions in this research.

Lemma 1

([33]). The Banach space $\mathfrak{W}$ is assumed to be separable. Suppose that the multi-function $\tilde{\mathfrak{O}}:[0,1]\times \mathfrak{W}\to {\mathcal{P}}_{cmp,cvx}\left(\mathfrak{W}\right)$ is ${\mathcal{L}}^1$-Carathéodory and the linear mapping ${\tilde{\mathbb{P}}}^{*}:{\mathcal{L}}_{\mathfrak{W}}^1\left([0,1]\right)\to {\mathcal{C}}_{\mathfrak{W}}\left([0,1]\right)$ is continuous. Then, ${\tilde{\mathbb{P}}}^{*}\circ {\left(\mathbb{SEL}\right)}_{\tilde{\mathfrak{O}}}:{\mathcal{C}}_{\mathfrak{W}}\left([0,1]\right)\to {\mathcal{P}}_{cmp,cvx}\left({\mathcal{C}}_{\mathfrak{W}}\left([0,1]\right)\right)$ defines an operator in ${\mathcal{C}}_{\mathfrak{W}}\left([0,1]\right)\times {\mathcal{C}}_{\mathfrak{W}}\left([0,1]\right)$ via action

$$\varpi \mapsto ({\tilde{\mathbb{P}}}^{*}\circ {\left(\mathbb{SEL}\right)}_{\tilde{\mathfrak{O}}})\left(\varpi \right)={\tilde{\mathbb{P}}}^{*}\left({\left(\mathbb{SEL}\right)}_{\tilde{\mathfrak{O}},\varpi }\right)$$

which has a closed graph.

Lemma 2

([34]). Let $\mathfrak{W}$ be a Banach algebra and some items be valid for ${\mathbb{A}}_1^{*}:\mathfrak{W}\to \mathfrak{W}$ and ${\mathbb{A}}_2^{*}:\mathfrak{W}\to {\mathcal{P}}_{cmp,cvx}\left(\mathfrak{W}\right)$ including:

(1S)

${\mathbb{A}}_1^{*}$ is Lipschitz with $l^{*}>0$;

(2S)

${\mathbb{A}}_2^{*}$ has the upper semi-continuity and the compactness properties;

(3S)

$2l^{*}\widehat{\mathbb{O}}<1$, provided that $\widehat{\mathbb{O}}=\parallel {\mathbb{A}}_2^{*}\left(\mathfrak{W}\right)\parallel$.

In that case, either $\left(a\right)$${\mathsf{\Sigma }}^{*}=\{v^{*}\in \mathfrak{W}|{\alpha }_0{v}^{*}\in \left({\mathbb{A}}_1^{*}{v}^{*}\right)\left({\mathbb{A}}_2^{*}{v}^{*}\right),{\alpha }_0>1\}$ is unbounded or $\left(b\right)$ a member exists in $\mathfrak{W}$ with $\varpi \in \left({\mathbb{A}}_1^{*}\varpi \right)\left({\mathbb{A}}_2^{*}\varpi \right)$.

Lemma 3

([31]). Let the metric space $(\mathfrak{W},d_{\mathfrak{W}})$ be complete and α be a nonnegative mapping defined on ${\mathfrak{W}}^2$, $\psi \in \mathsf{\Psi }$ be a map that increases strictly, and $\tilde{\mathfrak{O}}:\mathfrak{W}\to {\mathcal{P}}_{cls,bnd}\left(\mathfrak{W}\right)$ be α-admissible and an α-ψ-contraction via $\alpha (\varpi ,{\varpi }^{\prime })\ge 1$ for some $\varpi \in \mathfrak{W}$ and ${\varpi }^{\prime }\in \tilde{\mathfrak{O}}\left(\varpi \right)$. In this case, $\tilde{\mathfrak{O}}$ includes a fixed point if $\mathfrak{W}$ contains the $\left(C_{\alpha }\right)$-property.

Lemma 4

([32]). Let the metric space $(\mathfrak{W},d_{\mathfrak{W}})$ be complete and $\psi :[0,\infty )\to [0,\infty )$ involve the upper semi-continuity specification via $\psi \left(z\right)<z$ and ${lim\; inf}_{z\to \infty }(z-\psi \left(z\right))>0$ for all $z>0$. Besides, we assume that $\tilde{\mathfrak{O}}:\mathfrak{W}\to {\mathcal{P}}_{cls,bnd}\left(\mathfrak{W}\right)$ is such that ${\mathbb{PH}}_{d_{\mathfrak{W}}}(\tilde{\mathfrak{O}}\varpi ,\tilde{\mathfrak{O}}{\varpi }^{\prime })\le \psi \left(d_{\mathfrak{W}}(\varpi ,{\varpi }^{\prime })\right)$ for each $\varpi ,{\varpi }^{\prime }\in \mathfrak{W}$. Then, $\tilde{\mathfrak{O}}$ involves an endpoint uniquely iff $\tilde{\mathfrak{O}}$ possesses the APPX-endpoint property.

3. Main Results

In two previous sections, we assembled some auxiliary and useful notions to achieve our main goals. Now, in the following, we establish other required lemmas to derive the main existence items. To do this, we first regard the sup norm given by ${\parallel \varpi \parallel }_{\mathfrak{W}}={sup}_{z\in \mathfrak{T}}\left|\varpi \left(z\right)\right|$ on the space $\mathfrak{W}=\{\varpi \left(z\right):\varpi \left(z\right)\in {\mathcal{C}}_{\mathbb{R}}\left(\mathfrak{T}\right)\}$. In this case, the Banach space $(\mathfrak{W},\parallel ·{\parallel }_{\mathfrak{W}})$ along with the multiplication action defined as $(\varpi ·{\varpi }^{\prime })\left(z\right)=\varpi \left(z\right){\varpi }^{\prime }\left(z\right)$ is a Banach algebra for all $\varpi ,{\varpi }^{\prime }\in \mathfrak{W}$. In addition, we specify the following constant for the sake of simplicity in computation:

$$\begin{array}{cc}\hfill {\widehat{\mathsf{\Omega }}}_{*}& :=\frac{{\mu }_2^{*}}{\mathsf{\Gamma }({\theta }_1^{*}+2)}-\frac{1}{\mathsf{\Gamma }({\theta }_1^{*}+1)}+\frac{{\mu }_2^{*}{\sigma }^{{\theta }_2^{*}+1}}{\mathsf{\Gamma }({\theta }_2^{*}+2)}-\frac{{\sigma }^{{\theta }_2^{*}}}{\mathsf{\Gamma }({\theta }_2^{*}+1)}\hfill \\ \hfill & +{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{e}^{-{\mu }_2^{*}q}\mathrm{d}q+{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{e}^{-{\mu }_2^{*}q}\mathrm{d}q\ne 0.\hfill \end{array}$$

(4)

On this basis, we explore an integral framework for the possible solution of the proposed Caputo sequential hybrid boundary problem of the inclusion version (1) and (2).

Lemma 5.

Let ${\tilde{h}}_{*}\in \mathfrak{W}$, ${\eta }^{*}\in [2,3)$, ${\eta }^{*}-1\in [1,2)$, $\sigma \in (0,1)$, ${\mu }_1^{*},{\mu }_2^{*},{\theta }_1^{*},{\theta }_2^{*}>0$ with ${\theta }_1^{*}-1>0$ and ${\theta }_2^{*}-1>0$. Furthermore, for $k\in \mathbb{N}$, ${\delta }_1^{*},{\delta }_2^{*},\dots ,{\delta }_k^{*}>0$. Then, a solution for the sequential hybrid FDE:

$${\displaystyle {\mu }_1^{*}\left({}^C{\mathcal{D}}_{0^{+}}^{{\eta }^{*}}+{\mu }_2^{*}{}^C{\mathcal{D}}_{0^{+}}^{{\eta }^{*}-1}\right)\left[\frac{\varpi \left(z\right)}{{\tilde{S}}^{*}\left(z,\varpi \left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}\varpi \left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}\varpi \left(z\right)\right)}\right]={\tilde{h}}_{*}\left(z\right)}$$

(5)

furnished with hybrid multi-order integro-differential conditions:

$$\left\{\begin{array}{c}{\displaystyle \left[\frac{\varpi \left(z\right)}{{\tilde{S}}^{*}\left(z,\varpi \left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}\varpi \left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}\varpi \left(z\right)\right)}\right]{|}_{z=0}=0,}\hfill \\ {\displaystyle {}^C{\mathcal{D}}_{0^{+}}^1\left[\frac{\varpi \left(z\right)}{{\tilde{S}}^{*}\left(z,\varpi \left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}\varpi \left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}\varpi \left(z\right)\right)}\right]{|}_{z=0}=0,}\hfill \\ {\displaystyle {}^{RL}{\mathcal{I}}_{0^{+}}^{{\theta }_1^{*}}\left[\frac{\varpi \left(z\right)}{{\tilde{S}}^{*}\left(z,\varpi \left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}\varpi \left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}\varpi \left(z\right)\right)}\right]{|}_{z=1}}\hfill \\ {\displaystyle +{}^{RL}{\mathcal{I}}_{0^{+}}^{{\theta }_2^{*}}\left[\frac{\varpi \left(z\right)}{{\tilde{S}}^{*}\left(z,\varpi \left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}\varpi \left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}\varpi \left(z\right)\right)}\right]{|}_{z=\sigma }=0,}\hfill \end{array}\right.$$

(6)

is displayed as $\varpi \left(z\right)$ iff

$$\begin{array}{cc}\hfill \varpi \left(z\right)& ={\tilde{S}}^{*}\left(z,\varpi \left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}\varpi \left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}\varpi \left(z\right)\right)(\frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\tilde{h}}_{*}\left(r\right)\mathrm{d}r\mathrm{d}q\hfill \\ \hfill & +\frac{1-e^{-{\mu }_2^{*}z}-{\mu }_2^{*}z}{{\mu }_1^{*}{\widehat{\mathsf{\Omega }}}_{*}}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\tilde{h}}_{*}\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\tilde{h}}_{*}\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\left]\right),\hfill \end{array}$$

(7)

where ${\widehat{\mathsf{\Omega }}}_{*}$ is given in (4).

Proof. 

We suppose that the function ${\tilde{\varpi }}_0^{*}$ satisfies the given Caputo sequential hybrid Equation (5). Then,

$${\displaystyle {\mu }_1^{*}{}^C{\mathcal{D}}_{0^{+}}^{{\eta }^{*}}\left(1+{\mu }_2^{*}{}^C{\mathcal{D}}_{0^{+}}^{-1}\right)\left[\frac{{\tilde{\varpi }}_0^{*}\left(z\right)}{{\tilde{S}}^{*}\left(z,{\tilde{\varpi }}_0^{*}\left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}{\tilde{\varpi }}_0^{*}\left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}{\tilde{\varpi }}_0^{*}\left(z\right)\right)}\right]={\tilde{h}}_{*}\left(z\right).}$$

In the next step, by taking the ${\eta }^{*}$-th order integral of the Riemann–Liouville type on the above equality, the following non-homogeneous integro-differential equation results:

$${\displaystyle {\mu }_1^{*}\left(1+{\mu }_2^{*}{}^C{\mathcal{D}}_{0^{+}}^{-1}\right)\left[\frac{{\tilde{\varpi }}_0^{*}\left(z\right)}{{\tilde{S}}^{*}\left(z,{\tilde{\varpi }}_0^{*}\left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}{\tilde{\varpi }}_0^{*}\left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}{\tilde{\varpi }}_0^{*}\left(z\right)\right)}\right]={}^{RL}{\mathcal{I}}_{0^{+}}^{{\eta }^{*}}{\tilde{h}}_{*}\left(z\right)+c_0+c_{1}z+c_2{z}^2.}$$

We try to find these unknowns $c_0,c_1,c_2\in \mathbb{R}$ provided

$$\begin{array}{cc}\hfill {\mu }_1^{*}[& {\displaystyle \frac{{\tilde{\varpi }}_0^{*}\left(z\right)}{{\tilde{S}}^{*}\left(z,{\tilde{\varpi }}_0^{*}\left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}{\tilde{\varpi }}_0^{*}\left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}{\tilde{\varpi }}_0^{*}\left(z\right)\right)}]={}^{RL}{\mathcal{I}}_{0^{+}}^{{\eta }^{*}}{\tilde{h}}_{*}\left(z\right)+c_0+c_{1}z+c_2{z}^2}\hfill \\ \hfill & {\displaystyle -{\mu }_1^{*}{\mu }_2^{*}{\int }_0^z\left[\frac{{\tilde{\varpi }}_0^{*}\left(q\right)}{{\tilde{S}}^{*}\left(q,{\tilde{\varpi }}_0^{*}\left(q\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}{\tilde{\varpi }}_0^{*}\left(q\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}{\tilde{\varpi }}_0^{*}\left(q\right)\right)}\right]\mathrm{d}q.}\hfill \end{array}$$

(8)

The latter equality implies that ${\mu }_1^{*}\left[\frac{{\tilde{\varpi }}_0^{*}\left(0\right)}{{\tilde{S}}^{*}\left(0,{\tilde{\varpi }}_0^{*}\left(0\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}{\tilde{\varpi }}_0^{*}\left(0\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}{\tilde{\varpi }}_0^{*}\left(0\right)\right)}\right]=c_0$. In addition, by taking the Caputo derivative of Equation (8) of the first order with respect to z, we obtain

$$\begin{array}{cc}\hfill {\mu }_1^{*}{}^C{\mathcal{D}}_{0^{+}}^1& {\displaystyle \left[\frac{{\tilde{\varpi }}_0^{*}\left(z\right)}{{\tilde{S}}^{*}\left(z,{\tilde{\varpi }}_0^{*}\left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}{\tilde{\varpi }}_0^{*}\left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}{\tilde{\varpi }}_0^{*}\left(z\right)\right)}\right]={}^{RL}{\mathcal{I}}_{0^{+}}^{{\eta }^{*}-1}{\tilde{h}}_{*}\left(z\right)+c_1+2c_{2}z}\hfill \\ \hfill & {\displaystyle -{\mu }_1^{*}{\mu }_2^{*}\left[\frac{{\tilde{\varpi }}_0^{*}\left(z\right)}{{\tilde{S}}^{*}\left(z,{\tilde{\varpi }}_0^{*}\left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}{\tilde{\varpi }}_0^{*}\left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}{\tilde{\varpi }}_0^{*}\left(z\right)\right)}\right].}\hfill \end{array}$$

By multiplying both sides of the above equality by $e^{{\mu }_2^{*}z}$, one can write

$$\begin{array}{cc}\hfill {\mu }_1^{*}{}^C{\mathcal{D}}_{0^{+}}^1& {\displaystyle \left[\frac{{\tilde{\varpi }}_0^{*}\left(z\right)}{{\tilde{S}}^{*}\left(z,{\tilde{\varpi }}_0^{*}\left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}{\tilde{\varpi }}_0^{*}\left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}{\tilde{\varpi }}_0^{*}\left(z\right)\right)}\right]e^{{\mu }_2^{*}z}=e^{{\mu }_2^{*}z}{}^{RL}{\mathcal{I}}_{0^{+}}^{{\eta }^{*}-1}{\tilde{h}}_{*}\left(z\right)+c_1{e}^{{\mu }_2^{*}z}}\hfill \\ \hfill & {\displaystyle +2c_{2}z{e}^{{\mu }_2^{*}z}-{\mu }_1^{*}{\mu }_2^{*}\left[\frac{{\tilde{\varpi }}_0^{*}\left(z\right)}{{\tilde{S}}^{*}\left(z,{\tilde{\varpi }}_0^{*}\left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}{\tilde{\varpi }}_0^{*}\left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}{\tilde{\varpi }}_0^{*}\left(z\right)\right)}\right]e^{{\mu }_2^{*}z}.}\hfill \end{array}$$

After performing some direct computations and necessary simplifications, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\tilde{\varpi }}_0^{*}\left(z\right)}{{\tilde{S}}^{*}\left(z,{\tilde{\varpi }}_0^{*}\left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}{\tilde{\varpi }}_0^{*}\left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}{\tilde{\varpi }}_0^{*}\left(z\right)\right)}=\frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\tilde{h}}_{*}\left(r\right)\mathrm{d}r\mathrm{d}q}\hfill \\ \hfill & +\frac{c_0}{{\mu }_1^{*}}{e}^{-{\mu }_2^{*}z}+\frac{c_1}{{\mu }_1^{*}{\mu }_2^{*}}(1-e^{-{\mu }_2^{*}z})+\frac{2c_2}{{\mu }_1^{*}{\mu }_2^{{*}^2}}({\mu }_2^{*}z-1+e^{-{\mu }_2^{*}z}).\hfill \end{array}$$

(9)

Here, in light of the hybrid multi-order integro-differential conditions (6), we obtain $c_0=c_1=0$ and

$$\begin{array}{cc}\hfill c_2& =-\frac{{\mu }_2^{{*}^2}}{2{\widehat{\mathsf{\Omega }}}_{*}}{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\tilde{h}}_{*}\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & -\frac{{\mu }_2^{{*}^2}}{2{\widehat{\mathsf{\Omega }}}_{*}}{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\tilde{h}}_{*}\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q,\hfill \end{array}$$

where we have mentioned before the nonzero constant ${\widehat{\mathsf{\Omega }}}_{*}$ in (4). Eventually, we insert all three values $c_0$, $c_1$, and $c_2$ into (9) and (9) becomes

$$\begin{array}{cc}\hfill {\tilde{\varpi }}_0^{*}\left(z\right)& ={\tilde{S}}^{*}\left(z,{\tilde{\varpi }}_0^{*}\left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}{\tilde{\varpi }}_0^{*}\left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}{\tilde{\varpi }}_0^{*}\left(z\right)\right)(\frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\tilde{h}}_{*}\left(r\right)\mathrm{d}r\mathrm{d}q\hfill \\ \hfill & +\frac{1-e^{-{\mu }_2^{*}z}-{\mu }_2^{*}z}{{\mu }_1^{*}{\widehat{\mathsf{\Omega }}}_{*}}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\tilde{h}}_{*}\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\tilde{h}}_{*}\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\left]\right).\hfill \end{array}$$

(10)

This indicates that the function ${\tilde{\varpi }}_0^{*}$ can be regarded as a solution for (7), and the argument is finished. The converse is evident. □

Next, we present the following inequalities, which are useful in the sequel.

Lemma 6.

Let ${\tilde{h}}_{*}:\mathfrak{T}\to \mathbb{R}$ be continuous via $\parallel {\tilde{h}}_{*}\parallel ={sup}_{z\in \mathfrak{T}}\left|{\tilde{h}}_{*}\left(z\right)\right|$. Then, the following inequalities are valid:

(E1)

${\displaystyle \left|{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\tilde{h}}_{*}\left(r\right)\mathrm{d}r\mathrm{d}q\right|\le \frac{1-e^{-{\mu }_2^{*}}}{{\mu }_2^{*}\mathsf{\Gamma }\left({\eta }^{*}\right)}\parallel {\tilde{h}}_{*}\parallel }$;

(E2)

${\displaystyle \left|{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\tilde{h}}_{*}\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\right|\le \frac{{\mu }_2^{*}+e^{-{\mu }_2^{*}}-1}{{\mu }_2^{{*}^2}\mathsf{\Gamma }\left({\theta }_1^{*}\right)\mathsf{\Gamma }\left({\eta }^{*}\right)}\parallel {\tilde{h}}_{*}\parallel }$;

(E3)

${\displaystyle \left|{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\tilde{h}}_{*}\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\right|\le }$${\displaystyle \frac{{\sigma }^{{\theta }_2^{*}+{\eta }^{*}-1}({\mu }_2^{*}\sigma +e^{-{\mu }_2^{*}\sigma }-1)}{{\mu }_2^{{*}^2}\mathsf{\Gamma }\left({\theta }_2^{*}\right)\mathsf{\Gamma }\left({\eta }^{*}\right)}\parallel {\tilde{h}}_{*}\parallel }$.

Proof. 

$\left(\mathrm{E}1\right)$ In the first stage, a simple computation yields

$${\displaystyle {\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\mathrm{d}r=-\frac{{(q-r)}^{{\eta }^{*}-1}}{\mathsf{\Gamma }\left({\eta }^{*}\right)}{|}_0^q=\frac{q^{{\eta }^{*}-1}}{\mathsf{\Gamma }\left({\eta }^{*}\right)}.}$$

In the next step, one can write

$$\begin{array}{cc}\hfill {\displaystyle {\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}}& {\displaystyle {\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\mathrm{d}r\mathrm{d}q={\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}\frac{q^{{\eta }^{*}-1}}{\mathsf{\Gamma }\left({\eta }^{*}\right)}\mathrm{d}q\le \frac{z^{{\eta }^{*}-1}}{\mathsf{\Gamma }\left({\eta }^{*}\right)}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}\mathrm{d}q}\hfill \\ \hfill & \le \frac{1}{\mathsf{\Gamma }\left({\eta }^{*}\right)}\left(-\frac{1}{{\mu }_2^{*}}{e}^{-{\mu }_2^{*}(z-q)}\right){|}_0^z=-\frac{1}{{\mu }_2^{*}\mathsf{\Gamma }\left({\eta }^{*}\right)}\left(1-e^{-{\mu }_2^{*}z}\right).\hfill \end{array}$$

Thus, we obtain

$${\displaystyle \left|{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\tilde{h}}_{*}\left(r\right)\mathrm{d}r\mathrm{d}q\right|\le \frac{1-e^{-{\mu }_2^{*}}}{{\mu }_2^{*}\mathsf{\Gamma }\left({\eta }^{*}\right)}\parallel {\tilde{h}}_{*}\parallel .}$$

$\left(\mathrm{E}2\right)$ By some easy calculations, we have

$${\displaystyle {\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\mathrm{d}r=-\frac{{(m-r)}^{{\eta }^{*}-1}}{\mathsf{\Gamma }\left({\eta }^{*}\right)}{|}_0^m=\frac{m^{{\eta }^{*}-1}}{\mathsf{\Gamma }\left({\eta }^{*}\right)}.}$$

(11)

In addition, one may write

$$\begin{array}{cc}\hfill {\displaystyle {\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}}& {\displaystyle {\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\mathrm{d}r\mathrm{d}m={\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}\frac{m^{{\eta }^{*}-1}}{\mathsf{\Gamma }\left({\eta }^{*}\right)}\mathrm{d}m\le \frac{q^{{\eta }^{*}-1}}{\mathsf{\Gamma }\left({\eta }^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}\mathrm{d}m}\hfill \\ \hfill & =\frac{q^{{\eta }^{*}-1}}{\mathsf{\Gamma }\left({\eta }^{*}\right)}\left(-\frac{1}{{\mu }_2^{*}}{e}^{-{\mu }_2^{*}(q-m)}\right){|}_0^q=-\frac{q^{{\eta }^{*}-1}}{{\mu }_2^{*}\mathsf{\Gamma }\left({\eta }^{*}\right)}\left(1-e^{-{\mu }_2^{*}q}\right).\hfill \end{array}$$

(12)

Then, by mixing the above results, we obtain

$$\begin{array}{cc}\hfill {\displaystyle |{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}}& {\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\tilde{h}}_{*}\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q|\hfill \\ \hfill & {\displaystyle \le \parallel {\tilde{h}}_{*}\parallel {\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}\frac{q^{{\eta }^{*}-1}(1-e^{-{\mu }_2^{*}q})}{{\mu }_2^{*}\mathsf{\Gamma }\left({\eta }^{*}\right)}\mathrm{d}q}\hfill \\ \hfill & \le \parallel {\tilde{h}}_{*}\parallel \frac{1}{{\mu }_2^{*}\mathsf{\Gamma }\left({\theta }_1^{*}\right)\mathsf{\Gamma }\left({\eta }^{*}\right)}{\int }_0^1(1-e^{-{\mu }_2^{*}q})\mathrm{d}q=\parallel {\tilde{h}}_{*}\parallel \frac{1}{{\mu }_2^{*}\mathsf{\Gamma }\left({\theta }_1^{*}\right)\mathsf{\Gamma }\left({\eta }^{*}\right)}\left(1+\frac{1}{{\mu }_2^{*}}{e}^{-{\mu }_2^{*}}-\frac{1}{{\mu }_2^{*}}\right)\hfill \\ \hfill & =\parallel {\tilde{h}}_{*}\parallel \frac{1}{{\mu }_2^{{*}^2}\mathsf{\Gamma }\left({\theta }_1^{*}\right)\mathsf{\Gamma }\left({\eta }^{*}\right)}\left({\mu }_2^{*}+e^{-{\mu }_2^{*}}-1\right).\hfill \end{array}$$

$\left(\mathrm{E}3\right)$ By similar computations and according to (11) and (12), we have

$${\displaystyle {\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\mathrm{d}r\mathrm{d}m\le -\frac{q^{{\eta }^{*}-1}}{{\mu }_2^{*}\mathsf{\Gamma }\left({\eta }^{*}\right)}\left(1-e^{-{\mu }_2^{*}q}\right).}$$

Thus,

$$\begin{array}{cc}\hfill {\displaystyle |{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}}& {\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\tilde{h}}_{*}\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q|\hfill \\ \hfill & {\displaystyle \le \parallel {\tilde{h}}_{*}\parallel {\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}\frac{q^{{\eta }^{*}-1}(1-e^{-{\mu }_2^{*}q})}{{\mu }_2^{*}\mathsf{\Gamma }\left({\eta }^{*}\right)}\mathrm{d}q}\hfill \\ \hfill & \le \parallel {\tilde{h}}_{*}\parallel \frac{1}{{\mu }_2^{*}\mathsf{\Gamma }\left({\theta }_2^{*}\right)\mathsf{\Gamma }\left({\eta }^{*}\right)}{\int }_0^{\sigma }{\sigma }^{{\theta }_2^{*}-1}{\sigma }^{{\eta }^{*}-1}(1-e^{-{\mu }_2^{*}q})\mathrm{d}q\hfill \\ \hfill & \le \parallel {\tilde{h}}_{*}\parallel \frac{{\sigma }^{{\theta }_2^{*}+{\eta }^{*}-1}}{{\mu }_2^{*}\mathsf{\Gamma }\left({\theta }_2^{*}\right)\mathsf{\Gamma }\left({\eta }^{*}\right)}{\int }_0^{\sigma }(1-e^{-{\mu }_2^{*}q})\mathrm{d}q=\parallel {\tilde{h}}_{*}\parallel \frac{{\sigma }^{{\theta }_2^{*}+{\eta }^{*}-1}}{{\mu }_2^{*}\mathsf{\Gamma }\left({\theta }_2^{*}\right)\mathsf{\Gamma }\left({\eta }^{*}\right)}\left(\sigma +\frac{1}{{\mu }_2^{*}}{e}^{-{\mu }_2^{*}\sigma }-\frac{1}{{\mu }_2^{*}}\right)\hfill \\ \hfill & =\parallel {\tilde{h}}_{*}\parallel \frac{{\sigma }^{{\theta }_2^{*}+{\eta }^{*}-1}}{{\mu }_2^{{*}^2}\mathsf{\Gamma }\left({\theta }_2^{*}\right)\mathsf{\Gamma }\left({\eta }^{*}\right)}\left({\mu }_2^{*}\sigma +e^{-{\mu }_2^{*}\sigma }-1\right).\hfill \end{array}$$

□

Definition 1.

A function $\varpi \in {\mathcal{AC}}_{\mathbb{R}}\left(\mathfrak{T}\right)$ is a solution of the suggested sequential multi-order hybrid inclusion boundary problem (1) and (2) if some function $\widehat{\kappa }\in {\mathcal{L}}_{\mathbb{R}}^1\left(\mathfrak{T}\right)$ exists with $\widehat{\kappa }\left(z\right)\in \tilde{\mathfrak{O}}\left(z,\varpi \left(z\right)\right)$ for almost all $z\in \mathfrak{T}$ that satisfies the following hybrid multi-order integro-differential boundary conditions:

$$\left\{\begin{array}{c}{\displaystyle \left[\frac{\varpi \left(z\right)}{{\tilde{S}}^{*}\left(z,\varpi \left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}\varpi \left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}\varpi \left(z\right)\right)}\right]{|}_{z=0}=0,}\hfill \\ {\displaystyle {}^C{\mathcal{D}}_{0^{+}}^1\left[\frac{\varpi \left(z\right)}{{\tilde{S}}^{*}\left(z,\varpi \left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}\varpi \left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}\varpi \left(z\right)\right)}\right]{|}_{z=0}=0,}\hfill \\ {\displaystyle {}^{RL}{\mathcal{I}}_{0^{+}}^{{\theta }_1^{*}}\left[\frac{\varpi \left(z\right)}{{\tilde{S}}^{*}\left(z,\varpi \left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}\varpi \left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}\varpi \left(z\right)\right)}\right]{|}_{z=1}}\hfill \\ {\displaystyle +{}^{RL}{\mathcal{I}}_{0^{+}}^{{\theta }_2^{*}}\left[\frac{\varpi \left(z\right)}{{\tilde{S}}^{*}\left(z,\varpi \left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}\varpi \left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}\varpi \left(z\right)\right)}\right]{|}_{z=\sigma }=0,}\hfill \end{array}\right.$$

and

$$\begin{array}{cc}\hfill \varpi \left(z\right)& ={\tilde{S}}^{*}\left(z,\varpi \left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}\varpi \left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}\varpi \left(z\right)\right)(\frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}q\hfill \\ \hfill & +\frac{1-e^{-{\mu }_2^{*}z}-{\mu }_2^{*}z}{{\mu }_1^{*}{\widehat{\mathsf{\Omega }}}_{*}}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\left]\right)\hfill \end{array}$$

for any $z\in \mathfrak{T}$.

For now, by taking into account Lemmas 5 and 6, we provide some existence theorems for two Caputo sequential multi-order BVPs (1)–(3).

Theorem 1.

Assume that a continuous real mapping ${\tilde{S}}^{*}\ne 0$ is defined on product space $\mathfrak{T}\times {\mathfrak{W}}^{k+1}$ and $\tilde{\mathfrak{O}}:\mathfrak{T}\times \mathfrak{W}\to {\mathcal{P}}_{cvx,cmp}\left(\mathfrak{W}\right)$. Along with these, the following statements are valid:

(HPS1)

There exists a bounded positive-valued mapping $\tilde{\upsilon }:\mathfrak{T}\to {\mathbb{R}}^{+}$ so that for each member ${\varpi }_1,\dots ,{\varpi }_{k+1},{\varpi }_1^{\prime },\dots ,{\varpi }_{k+1}^{\prime }\in \mathfrak{W}$ and $z\in \mathfrak{T}$, we have

$${\displaystyle |{\tilde{S}}^{*}\left(z,{\varpi }_1\left(z\right),\dots ,{\varpi }_{k+1}\left(z\right)\right)-{\tilde{S}}^{*}\left(z,{\varpi }_1^{\prime }\left(z\right),\dots ,{\varpi }_{k+1}^{\prime }\left(z\right)\right)|\le \tilde{\upsilon }\left(z\right)\sum _{i=1}^{k+1}|{\varpi }_i\left(z\right)-{\varpi }_i^{\prime }\left(z\right)|;}$$

(HPS2)

$\tilde{\mathfrak{O}}:\mathfrak{T}\times \mathfrak{W}\to {\mathcal{P}}_{cvx,cmp}\left(\mathfrak{W}\right)$ is supposed to be ${\mathcal{L}}^1$-Caratheodory;

(HPS3)

There is a positive function $x^{*}\in {\mathcal{L}}_{{\mathbb{R}}^{+}}^1\left(\mathfrak{T}\right)$ such that for any $\varpi \in \mathfrak{W}$ and for almost all $z\in \mathfrak{T}$, we have

$$\parallel \tilde{\mathfrak{O}}(z,\varpi )\parallel =sup\left\{\right|\widehat{\kappa }|:\widehat{\kappa }\in \tilde{\mathfrak{O}}(z,\varpi \left(z\right))\}\le x^{*}\left(z\right);$$

(HPS4)

There is a number $r_{*}\in {\mathbb{R}}^{+}$ provided that

$${\displaystyle r_{*}>\frac{{\tilde{\mathbb{S}}}^{*}{\mathbb{M}}_{*}\parallel x^{*}\parallel }{{\displaystyle 1-{\tilde{\upsilon }}^{*}\left(1+\frac{1}{\mathsf{\Gamma }({\delta }_1^{*}+1)}+\dots +\frac{1}{\mathsf{\Gamma }({\delta }_k^{*}+1)}\right){\mathbb{M}}_{*}\parallel x^{*}\parallel }},}$$

(13)

where $\parallel x^{*}\parallel ={sup}_{z\in \mathfrak{T}}\left\{\right|x\left(z\right)\left|\right\}$, ${\displaystyle {\tilde{\mathbb{S}}}^{*}=\underset{z\in \mathfrak{T}}{sup}\left|{\tilde{S}}^{*}(z,0,\dots ,0)\right|}$, ${\displaystyle {\tilde{\upsilon }}^{*}=\underset{z\in \mathfrak{T}}{sup}\left|\tilde{\upsilon }\left(z\right)\right|}$ and

$${\displaystyle {\mathbb{M}}_{*}=\frac{|1-e^{-{\mu }_2^{*}}|}{{\mu }_1^{*}{\mu }_2^{*}\mathsf{\Gamma }\left({\eta }^{*}\right)}+\frac{|1-e^{-{\mu }_2^{*}}|+{\mu }_2^{*}}{{\mu }_1^{*}\left|{\widehat{\mathsf{\Omega }}}_{*}\right|}\left(\frac{|{\mu }_2^{*}+e^{-{\mu }_2^{*}}-1|}{{\mu }_2^{{*}^2}\mathsf{\Gamma }\left({\theta }_1^{*}\right)\mathsf{\Gamma }\left({\eta }^{*}\right)}+\frac{{\sigma }^{{\theta }_2^{*}+{\eta }^{*}-1}|{\mu }_2^{*}\sigma +e^{-{\mu }_2^{*}\sigma }-1|}{{\mu }_2^{{*}^2}\mathsf{\Gamma }\left({\theta }_2^{*}\right)\mathsf{\Gamma }\left({\eta }^{*}\right)}\right).}$$

(14)

Then, the Caputo sequential hybrid multi-order integro-differential inclusion BVP (1) and (2) has a solution if an inequality ${\displaystyle {\tilde{\upsilon }}^{*}\left(1+\frac{1}{\mathsf{\Gamma }({\delta }_1^{*}+1)}+\dots +\frac{1}{\mathsf{\Gamma }({\delta }_k^{*}+1)}\right){\mathbb{M}}_{*}\parallel x^{*}\parallel <\frac{1}{2}}$ holds strictly.

Proof. 

First of all, for every $\varpi \in \mathfrak{W}$, we construct the following collection of selections for $\tilde{\mathfrak{O}}$ as follows:

$${\left(\mathbb{SEL}\right)}_{\tilde{\mathfrak{O}},\varpi }=\left\{\widehat{\kappa }\in {\mathcal{L}}^1\left(\mathfrak{T}\right):\widehat{\kappa }\left(z\right)\in \tilde{\mathfrak{O}}\left(z,\varpi \left(z\right)\right)\right\}$$

for almost all $z\in \mathfrak{T}$. In addition, we consider a set-valued map ${\mathcal{G}}^{*}:\mathfrak{W}\to \mathcal{P}\left(\mathfrak{W}\right)$ by ${\mathcal{G}}^{*}\left(\varpi \right)=\{g^{*}\in \mathfrak{W}:g^{*}\left(z\right)={\rho }^{*}\left(z\right)\mathrm{for}z\in \mathfrak{T}\}$, where

$$\begin{array}{cc}\hfill {\rho }^{*}\left(z\right)& ={\tilde{S}}^{*}\left(z,\varpi \left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}\varpi \left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}\varpi \left(z\right)\right)(\frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}q\hfill \\ \hfill & +\frac{1-e^{-{\mu }_2^{*}z}-{\mu }_2^{*}z}{{\mu }_1^{*}{\widehat{\mathsf{\Omega }}}_{*}}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\left]\right)\hfill \end{array}$$

for some $\widehat{\kappa }\in {\left(\mathbb{SEL}\right)}_{\tilde{\mathfrak{O}},\varpi }$. It is evident that $g_0^{*}\in \mathfrak{W}$ is considered as a solution for the Caputo sequential hybrid multi-order inclusion BVP (1) and (2) iff $g_0^{*}$ is a fixed point of ${\mathcal{G}}^{*}$. To begin the main proof, with due attention to Lemma 5, we formulate two different structural mappings ${\mathbb{A}}_1^{*}:\mathfrak{W}\to \mathfrak{W}$ by $\left({\mathbb{A}}_1^{*}\varpi \right)\left(z\right)={\tilde{S}}^{*}\left(z,\varpi \left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}\varpi \left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}\varpi \left(z\right)\right)$ and ${\mathbb{A}}_2^{*}:\mathfrak{W}\to \mathcal{P}\left(\mathfrak{W}\right)$ by

$$\left({\mathbb{A}}_2^{*}\varpi \right)\left(z\right)=\{v\in \mathfrak{W}:v\left(z\right)=\phi \left(z\right)\mathrm{for}z\in \mathfrak{T}\},$$

where

$$\begin{array}{cc}\hfill \phi \left(z\right)& =\frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}q\hfill \\ \hfill & +\frac{1-e^{-{\mu }_2^{*}z}-{\mu }_2^{*}z}{{\mu }_1^{*}{\widehat{\mathsf{\Omega }}}_{*}}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q]\hfill \end{array}$$

for some $\widehat{\kappa }\in {\left(\mathbb{SEL}\right)}_{\tilde{\mathfrak{O}},\varpi }$. Then, one can represent the product operator equation:

$${\mathcal{G}}^{*}\left(\varpi \right)={\mathbb{A}}_1^{*}\varpi {\mathbb{A}}_2^{*}\varpi .$$

The basic aim at present is to verify that ${\mathbb{A}}_1^{*}$ and ${\mathbb{A}}_2^{*}$ settle all hypotheses of Lemma 2. We in the first place intend to confirm that ${\mathbb{A}}_1^{*}$ is Lipschitz on space $\mathfrak{W}$. Let ${\varpi }_1,{\varpi }_2\in \mathfrak{W}$ be arbitrary. By proceeding under the hypothesis ($\mathrm{HPS}1$), it is deduced that

$$\begin{array}{cc}\hfill |\left({\mathbb{A}}_1^{*}{\varpi }_1\right)\left(z\right)& -\left({\mathbb{A}}_1^{*}{\varpi }_2\right)\left(z\right)|=|{\tilde{S}}^{*}\left(z,{\varpi }_1\left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}{\varpi }_1\left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}{\varpi }_1\left(z\right)\right)\hfill \\ \hfill & -{\tilde{S}}^{*}\left(z,{\varpi }_2\left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}{\varpi }_2\left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}{\varpi }_2\left(z\right)\right)|\hfill \\ \hfill & \le \tilde{\upsilon }\left(z\right)\left(|{\varpi }_1\left(z\right)-{\varpi }_2\left(z\right)|+\frac{1}{\mathsf{\Gamma }({\delta }_1^{*}+1)}|{\varpi }_1\left(z\right)-{\varpi }_2\left(z\right)|+\dots +\frac{1}{\mathsf{\Gamma }({\delta }_k^{*}+1)}|{\varpi }_1\left(z\right)-{\varpi }_2\left(z\right)|\right)\hfill \\ \hfill & =\tilde{\upsilon }\left(z\right)(1+\frac{1}{\mathsf{\Gamma }({\delta }_1^{*}+1)}+\dots +\frac{1}{\mathsf{\Gamma }({\delta }_k^{*}+1)})|{\varpi }_1\left(z\right)-{\varpi }_2\left(z\right)|\hfill \end{array}$$

for all $z\in \mathfrak{T}:=[0,1]$. Hence, we obtain

$${\displaystyle \parallel {\mathbb{A}}_1^{*}{\varpi }_1-{\mathbb{A}}_1^{*}{\varpi }_2{\parallel }_{\mathfrak{W}}\le {\tilde{\upsilon }}^{*}\left(1+\frac{1}{\mathsf{\Gamma }({\delta }_1^{*}+1)}+\dots +\frac{1}{\mathsf{\Gamma }({\delta }_k^{*}+1)}\right){\parallel {\varpi }_1-{\varpi }_2\parallel }_{\mathfrak{W}}}$$

for all ${\varpi }_1,{\varpi }_2\in \mathfrak{W}$. It follows that the single-valued mapping ${\mathbb{A}}_1^{*}$ is Lipschitz via constant ${\displaystyle {\tilde{\upsilon }}^{*}\left(1+\frac{1}{\mathsf{\Gamma }({\delta }_1^{*}+1)}+\dots +\frac{1}{\mathsf{\Gamma }({\delta }_k^{*}+1)}\right)}$. In the subsequent stage, we proceed to determine that ${\mathbb{A}}_2^{*}$ involves convex values. To emphasize the correctness of such a claim, let ${\varpi }_1,{\varpi }_2\in {\mathbb{A}}_2^{*}\varpi$ be arbitrary. Choose functions $\widehat{{\kappa }_1},\widehat{{\kappa }_2}\in {\left(\mathbb{SEL}\right)}_{\tilde{\mathfrak{O}},\varpi }$ such that

$$\begin{array}{cc}\hfill {\varpi }_l\left(z\right)& =\frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\widehat{\kappa }}_l\left(r\right)\mathrm{d}r\mathrm{d}q\hfill \\ \hfill & +\frac{1-e^{-{\mu }_2^{*}z}-{\mu }_2^{*}z}{{\mu }_1^{*}{\widehat{\mathsf{\Omega }}}_{*}}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\widehat{\kappa }}_l\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\widehat{\kappa }}_l\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q]\hfill \end{array}$$

for all $z\in \mathfrak{T}$ (a.e.) and for $l=1,2$. Further, let $\lambda \in (0,1)$. In this case, one may write

$$\begin{array}{cc}\hfill \lambda {\varpi }_1& \left(z\right)+(1-\lambda ){\varpi }_2\left(z\right)=\frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\left[\lambda {\widehat{\kappa }}_1\left(r\right)+(1-\lambda ){\widehat{\kappa }}_2\left(r\right)\right]\mathrm{d}r\mathrm{d}q\hfill \\ \hfill & +\frac{1-e^{-{\mu }_2^{*}z}-{\mu }_2^{*}z}{{\mu }_1^{*}{\widehat{\mathsf{\Omega }}}_{*}}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\left[\lambda {\widehat{\kappa }}_1\left(r\right)+(1-\lambda ){\widehat{\kappa }}_2\left(r\right)\right]\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\left[\lambda {\widehat{\kappa }}_1\left(r\right)+(1-\lambda ){\widehat{\kappa }}_2\left(r\right)\right]\mathrm{d}r\mathrm{d}m\mathrm{d}q]\hfill \end{array}$$

for all $z\in \mathfrak{T}$ (a.e.). By assumption, we know that $\tilde{\mathfrak{O}}$ is convex-valued; thus, we immediately realize that ${\left(\mathbb{SEL}\right)}_{\tilde{\mathfrak{O}},\varpi }$ is a convex set. Therefore, we obtain that $\lambda {\widehat{\kappa }}_1\left(z\right)+(1-\lambda ){\widehat{\kappa }}_2\left(z\right)\in {\left(\mathbb{SEL}\right)}_{\tilde{\mathfrak{O}},\varpi }$ for each $z\in \mathfrak{T}$, and so, ${\mathbb{A}}_2^{*}w$ belongs to collection ${\mathcal{P}}_{cvx}\left(\mathfrak{W}\right)$ for each $\varpi \in \mathfrak{W}$.

By continuing the proof process, we are going to investigate the complete continuity of ${\mathbb{A}}_2^{*}$ on $\mathfrak{W}$. To reach this aim, we have to prove two notions for the set ${\mathbb{A}}_2^{*}\left(\mathfrak{W}\right)$ including equi-continuity and uniform boundedness. We first check that ${\mathbb{A}}_2^{*}$ relatesevery bounded set to a bounded one in $\mathfrak{W}$. For an arbitrary number ${\tilde{r}}_{*}\in {\mathbb{R}}^{+}$, we build a bounded ball ${\mathbb{V}}_{{\tilde{r}}_{*}}={\{\varpi \in \mathfrak{W}:\parallel \varpi \parallel }_{\mathfrak{W}}\le {\tilde{r}}_{*}\}$. In that phase, for every $\varpi \in {\mathbb{V}}_{{\tilde{r}}_{*}}$ and $v\in {\mathbb{A}}_2^{*}\varpi$, there is a function $\widehat{\kappa }\in {\left(\mathbb{SEL}\right)}_{\tilde{\mathfrak{O}},\varpi }$ provided that

$$\begin{array}{cc}\hfill v\left(z\right)& =\frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}q\hfill \\ \hfill & +\frac{1-e^{-{\mu }_2^{*}z}-{\mu }_2^{*}z}{{\mu }_1^{*}{\widehat{\mathsf{\Omega }}}_{*}}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q]\hfill \end{array}$$

for each $z\in \mathfrak{T}$. Then, the estimate of v is implemented as follows:

$$\begin{array}{cc}\hfill \left|v\right(z\left)\right|& \le \frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\left|\widehat{\kappa }\left(r\right)\right|\mathrm{d}r\mathrm{d}q\hfill \\ \hfill & +\frac{|1-e^{-{\mu }_2^{*}z}|+{\mu }_2^{*}z}{{\mu }_1^{*}\left|{\widehat{\mathsf{\Omega }}}_{*}\right|}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\left|\widehat{\kappa }\left(r\right)\right|\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\left|\widehat{\kappa }\left(r\right)\right|\mathrm{d}r\mathrm{d}m\mathrm{d}q]\hfill \\ \hfill & \le \frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{x}^{*}\left(r\right)\mathrm{d}r\mathrm{d}q\hfill \\ \hfill & +\frac{|1-e^{-{\mu }_2^{*}z}|+{\mu }_2^{*}z}{{\mu }_1^{*}\left|{\widehat{\mathsf{\Omega }}}_{*}\right|}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{x}^{*}\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{x}^{*}\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q]\hfill \\ \hfill & {\displaystyle \le \left[\frac{1-e^{-{\mu }_2^{*}}}{{\mu }_1^{*}{\mu }_2^{*}\mathsf{\Gamma }\left({\eta }^{*}\right)}+\frac{|1-e^{-{\mu }_2^{*}}|+{\mu }_2^{*}}{{\mu }_1^{*}\left|{\widehat{\mathsf{\Omega }}}_{*}\right|}\left(\frac{{\mu }_2^{*}+e^{-{\mu }_2^{*}}-1}{{\mu }_2^{{*}^2}\mathsf{\Gamma }\left({\theta }_1^{*}\right)\mathsf{\Gamma }\left({\eta }^{*}\right)}+\frac{{\sigma }^{{\theta }_2^{*}+{\eta }^{*}-1}({\mu }_2^{*}\sigma +e^{-{\mu }_2^{*}\sigma }-1)}{{\mu }_2^{{*}^2}\mathsf{\Gamma }\left({\theta }_2^{*}\right)\mathsf{\Gamma }\left({\eta }^{*}\right)}\right)\right]\parallel x^{*}\parallel }\hfill \\ \hfill & ={\mathbb{M}}_{*}\parallel x^{*}\parallel ,\hfill \end{array}$$

so that ${\mathbb{M}}_{*}$ is illustrated by (14). Therefore, $\parallel v\parallel \le {\mathbb{M}}_{*}\parallel x^{*}\parallel$, which confirms the uniform boundedness of ${\mathbb{A}}_2^{*}\left(\mathfrak{W}\right)$. In the subsequent step, we verify that ${\mathbb{A}}_2^{*}$ maps every bounded set to an equi-continuous subset. Let $\varpi \in {\mathbb{V}}_{{\tilde{r}}_{*}}$ and $v\in {\mathbb{A}}_2^{*}\varpi$. We select a function $\widehat{\kappa }\in {\left(\mathbb{SEL}\right)}_{\tilde{\mathfrak{O}},\varpi }$ so that

$$\begin{array}{cc}\hfill v\left(z\right)& =\frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}q\hfill \\ \hfill & +\frac{1-e^{-{\mu }_2^{*}z}-{\mu }_2^{*}z}{{\mu }_1^{*}{\widehat{\mathsf{\Omega }}}_{*}}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q]\hfill \end{array}$$

for all $z\in \mathfrak{T}$. Let us suppose that $z_1,z_2\in \mathfrak{T}$ along with $z_1<z_2$. Then, we can write

$$\begin{array}{cc}\hfill \left|v\right(z_2)& -v\left(z_1\right)|\le |\frac{1}{{\mu }_1^{*}}{\int }_0^{z_2}{e}^{-{\mu }_2^{*}(z_2-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\tilde{r}}_{*}\mathrm{d}r\mathrm{d}q\hfill \\ \hfill & -\frac{1}{{\mu }_1^{*}}{\int }_0^{z_1}{e}^{-{\mu }_2^{*}(z_1-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\tilde{r}}_{*}\mathrm{d}r\mathrm{d}q|\hfill \\ \hfill & +\frac{(e^{-{\mu }_2^{*}{z}_1}-e^{-{\mu }_2^{*}{z}_2})+{\mu }_2^{*}(z_2-z_1)}{{\mu }_1^{*}\left|{\widehat{\mathsf{\Omega }}}_{*}\right|}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\tilde{r}}_{*}\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\tilde{r}}_{*}\mathrm{d}r\mathrm{d}m\mathrm{d}q].\hfill \end{array}$$

In light of the above arguments, we realize that the limit value of the right-hand expressions equals zero without depending on $\varpi \in {\mathbb{V}}_{{\tilde{r}}_{*}}$ whenever $z_1\to z_2$. Consequently, we are in a position that one can refer to as the Arzela–Ascoli theorem to deduce that the operator ${\mathbb{A}}_2^{*}:{\mathcal{C}}_{\mathbb{R}}\left(\mathfrak{T}\right)\to \mathcal{P}({\mathcal{C}}_{\mathbb{R}}\left(\mathfrak{T}\right)$ is completely continuous. In the sequel, we intend to prove that ${\mathbb{A}}_2^{*}$ is an operator having a closed graph, which implies its upper semi-continuity property. Let ${\varpi }_n\in {\mathbb{V}}_{{\tilde{r}}_{*}}$ and $v_n\in {\mathbb{A}}_2^{*}{\varpi }_n$ so that ${\varpi }_n\to {\varpi }^{*}$ and $v_n\to v^{*}$. We claim that the inclusion $v^{*}\in {\mathbb{A}}_2^{*}{\varpi }^{*}$ is valid. For each $n\ge 1$ and $v_n\in {\mathbb{A}}_2^{*}{\varpi }_n$, choose ${\widehat{\kappa }}_n\in {\left(\mathbb{SEL}\right)}_{\tilde{\mathfrak{O}},{\varpi }_n}$ so that

$$\begin{array}{cc}\hfill v_n\left(z\right)& =\frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\widehat{\kappa }}_n\left(r\right)\mathrm{d}r\mathrm{d}q\hfill \\ \hfill & +\frac{1-e^{-{\mu }_2^{*}z}-{\mu }_2^{*}z}{{\mu }_1^{*}{\widehat{\mathsf{\Omega }}}_{*}}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\widehat{\kappa }}_n\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\widehat{\kappa }}_n\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q]\hfill \end{array}$$

for any $z\in \mathfrak{T}$. We have to check the existence of a function ${\widehat{\kappa }}^{*}\in {\left(\mathbb{SEL}\right)}_{\tilde{\mathfrak{O}},{\varpi }^{*}}$ provided that

$$\begin{array}{cc}\hfill v^{*}\left(z\right)& =\frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\widehat{\kappa }}^{*}\left(r\right)\mathrm{d}r\mathrm{d}q\hfill \\ \hfill & +\frac{1-e^{-{\mu }_2^{*}z}-{\mu }_2^{*}z}{{\mu }_1^{*}{\widehat{\mathsf{\Omega }}}_{*}}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\widehat{\kappa }}^{*}\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\widehat{\kappa }}^{*}\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q]\hfill \end{array}$$

for each $z\in \mathfrak{T}$. To achieve this aim, we construct the continuous linear map ${\tilde{\mathbb{P}}}^{*}:{\mathcal{L}}_{\mathbb{R}}^1\left(\mathfrak{T}\right)\to \mathfrak{W}={\mathcal{C}}_{\mathbb{R}}\left(\mathfrak{T}\right)$ as follows:

$$\begin{array}{cc}\hfill {\tilde{\mathbb{P}}}^{*}\left(\widehat{\kappa }\right)\left(z\right)=\varpi \left(z\right)& =\frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}q\hfill \\ \hfill & +\frac{1-e^{-{\mu }_2^{*}z}-{\mu }_2^{*}z}{{\mu }_1^{*}{\widehat{\mathsf{\Omega }}}_{*}}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q]\hfill \end{array}$$

for any $z\in \mathfrak{T}$. One can directly confirm

$$\begin{array}{cc}\hfill \parallel v_n\left(z\right)& -v^{*}\left(z\right)\parallel =\parallel \frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\left({\widehat{\kappa }}_n\left(r\right)-{\widehat{\kappa }}^{*}\left(r\right)\right)\mathrm{d}r\mathrm{d}q\hfill \\ \hfill & +\frac{1-e^{-{\mu }_2^{*}z}-{\mu }_2^{*}z}{{\mu }_1^{*}{\widehat{\mathsf{\Omega }}}_{*}}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\left({\widehat{\kappa }}_n\left(r\right)-{\widehat{\kappa }}^{*}\left(r\right)\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\left({\widehat{\kappa }}_n\left(r\right)-{\widehat{\kappa }}^{*}\left(r\right)\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q]\parallel \to 0.\hfill \end{array}$$

Hence, by the assumptions of Lemma 2, it is realized that ${\tilde{\mathbb{P}}}^{*}\circ {\left(\mathbb{SEL}\right)}_{\tilde{\mathfrak{O}}}$ has a closed graph. For the sake of holding the inclusion $v_n\in {\tilde{\mathbb{P}}}^{*}\left({\left(\mathbb{SEL}\right)}_{\tilde{\mathfrak{O}},{\varpi }_n}\right)$ and also ${\varpi }_n\to {\varpi }^{*}$, there exists a function ${\widehat{\kappa }}^{*}\in {\left(\mathbb{SEL}\right)}_{\tilde{\mathfrak{O}},{\varpi }^{*}}$ provided that

$$\begin{array}{cc}\hfill v^{*}\left(z\right)& =\frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\widehat{\kappa }}^{*}\left(r\right)\mathrm{d}r\mathrm{d}q\hfill \\ \hfill & +\frac{1-e^{-{\mu }_2^{*}z}-{\mu }_2^{*}z}{{\mu }_1^{*}{\widehat{\mathsf{\Omega }}}_{*}}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\widehat{\kappa }}^{*}\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\widehat{\kappa }}^{*}\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q]\hfill \end{array}$$

for all $z\in \mathfrak{T}$. Hence, $v^{*}\in {\mathbb{A}}_2^{*}{\varpi }^{*}$ and so ${\mathbb{A}}_2^{*}$ is an operator having a closed graph. From this point, we find that ${\mathbb{A}}_2^{*}$ is upper semi-continuous. Meanwhile, notice that by assumption, we know that ${\mathbb{A}}_2^{*}$ is compact-valued. In the following, by taking into account the hypothesis ($\mathrm{HPS}3$) and continuing a similar argument, we have

$$\begin{array}{cc}\hfill \widehat{\mathbb{O}}& =\parallel {\mathbb{A}}_2^{*}\left(\mathfrak{W}\right)\parallel =\underset{z\in \mathfrak{T}}{sup}\left\{\right|{\mathbb{A}}_2^{*}\varpi |:\varpi \in \mathfrak{W}\}\hfill \\ \hfill & {\displaystyle \le \left[\frac{1-e^{-{\mu }_2^{*}}}{{\mu }_1^{*}{\mu }_2^{*}\mathsf{\Gamma }\left({\eta }^{*}\right)}+\frac{|1-e^{-{\mu }_2^{*}}|+{\mu }_2^{*}}{{\mu }_1^{*}\left|{\widehat{\mathsf{\Omega }}}_{*}\right|}\left(\frac{{\mu }_2^{*}+e^{-{\mu }_2^{*}}-1}{{\mu }_2^{{*}^2}\mathsf{\Gamma }\left({\theta }_1^{*}\right)\mathsf{\Gamma }\left({\eta }^{*}\right)}+\frac{{\sigma }^{{\theta }_2^{*}+{\eta }^{*}-1}({\mu }_2^{*}\sigma +e^{-{\mu }_2^{*}\sigma }-1)}{{\mu }_2^{{*}^2}\mathsf{\Gamma }\left({\theta }_2^{*}\right)\mathsf{\Gamma }\left({\eta }^{*}\right)}\right)\right]\parallel x^{*}\parallel }\hfill \\ \hfill & ={\mathbb{M}}_{*}\parallel x^{*}\parallel .\hfill \end{array}$$

As we can observe, an inequality $\widehat{\mathbb{O}}\le {\mathbb{M}}_{*}\parallel x^{*}\parallel$ holds. Therefore, one can write

$${\displaystyle {\tilde{\upsilon }}^{*}\left(1+\frac{1}{\mathsf{\Gamma }({\delta }_1^{*}+1)}+\dots +\frac{1}{\mathsf{\Gamma }({\delta }_k^{*}+1)})\widehat{\mathbb{O}}\le {\tilde{\upsilon }}^{*}(1+\frac{1}{\mathsf{\Gamma }({\delta }_1^{*}+1)}+\dots +\frac{1}{\mathsf{\Gamma }({\delta }_k^{*}+1)}\right){\mathbb{M}}_{*}\parallel x^{*}\parallel <\frac{1}{2}.}$$

By setting ${\displaystyle l^{*}={\tilde{\upsilon }}^{*}\left(1+\frac{1}{\mathsf{\Gamma }({\delta }_1^{*}+1)}+\dots +\frac{1}{\mathsf{\Gamma }({\delta }_k^{*}+1)}\right)}$, clearly ${\displaystyle l^{*}\widehat{\mathbb{O}}<\frac{1}{2}}$. Until now, it is seen that Lemma 2 is settled on ${\mathbb{A}}_1^{*}$ and ${\mathbb{A}}_2^{*}$. In this position, it is sufficient to check that one of (a) or (b) holds. Our claim is that (b) is not valid. To confirm this claim, with due attention to Lemma 2 and hypothesis ($\mathrm{HPS}4$), one can suppose that $\varpi$ is an arbitrary member belonging to ${\mathsf{\Sigma }}^{*}$ with $\parallel \varpi \parallel =r_{*}$. Obviously, ${\alpha }_0\varpi \left(z\right)\in {\mathbb{A}}_1^{*}\varpi \left(z\right){\mathbb{A}}_2^{*}\varpi \left(z\right)$ for any ${\alpha }_0>1$. By selecting a suitable function $\widehat{\kappa }\in {\left(\mathbb{SEL}\right)}_{\tilde{\mathfrak{O}},\varpi }$, one can write

$$\begin{array}{cc}\hfill \varpi \left(z\right)& {\displaystyle =\frac{1}{{\alpha }_0}{\tilde{S}}^{*}\left(z,\varpi \left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}\varpi \left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}\varpi \left(z\right)\right)(\frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}q}\hfill \\ \hfill & +\frac{1-e^{-{\mu }_2^{*}z}-{\mu }_2^{*}z}{{\mu }_1^{*}{\widehat{\mathsf{\Omega }}}_{*}}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\left]\right)\hfill \end{array}$$

for each ${\alpha }_0>1$ and for any $z\in \mathfrak{T}$. Therefore, we obtain:

$$\begin{array}{cc}\hfill \left|\varpi \right(z\left)\right|& {\displaystyle =\frac{1}{{\alpha }_0}\left|{\tilde{S}}^{*}\left(z,\varpi \left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}\varpi \left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}\varpi \left(z\right)\right)\right|(\frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\left|\widehat{\kappa }\left(r\right)\right|\mathrm{d}r\mathrm{d}q}\hfill \\ \hfill & +\frac{|1-e^{-{\mu }_2^{*}z}|+{\mu }_2^{*}z}{{\mu }_1^{*}\left|{\widehat{\mathsf{\Omega }}}_{*}\right|}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\left|\widehat{\kappa }\left(r\right)\right|\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\left|\widehat{\kappa }\left(r\right)\right|\mathrm{d}r\mathrm{d}m\mathrm{d}q\left]\right)\hfill \\ \hfill & =\left[|{\tilde{S}}^{*}\left(z,\varpi \left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}\varpi \left(z\right),\dots ,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_k^{*}}\varpi \left(z\right)\right)-{\tilde{S}}^{*}\left(z,0,0,\dots ,0\right)|+|{\tilde{S}}^{*}\left(z,0,0,\dots ,0\right)|\right]\hfill \\ \hfill & \times (\frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\left|\widehat{\kappa }\left(r\right)\right|\mathrm{d}r\mathrm{d}q\hfill \\ \hfill & +\frac{|1-e^{-{\mu }_2^{*}z}|+{\mu }_2^{*}z}{{\mu }_1^{*}\left|{\widehat{\mathsf{\Omega }}}_{*}\right|}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\left|\widehat{\kappa }\left(r\right)\right|\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\left|\widehat{\kappa }\left(r\right)\right|\mathrm{d}r\mathrm{d}m\mathrm{d}q\left]\right)\hfill \\ \hfill & {\displaystyle \le [{\tilde{\upsilon }}^{*}(1+\frac{1}{\mathsf{\Gamma }({\delta }_1^{*}+1)}+\dots +\frac{1}{\mathsf{\Gamma }({\delta }_k^{*}+1)})\parallel \varpi \parallel +{\tilde{\mathbb{S}}}^{*}\left]\right(\frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{x}^{*}\left(r\right)\mathrm{d}r\mathrm{d}q}\hfill \\ \hfill & +\frac{|1-e^{-{\mu }_2^{*}z}|+{\mu }_2^{*}z}{{\mu }_1^{*}\left|{\widehat{\mathsf{\Omega }}}_{*}\right|}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{x}^{*}\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{x}^{*}\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\left]\right)\hfill \\ \hfill & {\displaystyle \le \left[{\tilde{\upsilon }}^{*}\left(1+\frac{1}{\mathsf{\Gamma }({\delta }_1^{*}+1)}+\dots +\frac{1}{\mathsf{\Gamma }({\delta }_k^{*}+1)}\right)\parallel \varpi \parallel +{\tilde{\mathbb{S}}}^{*}\right]{\mathbb{M}}_{*}\parallel x^{*}\parallel }\hfill \end{array}$$

for all $z\in \mathfrak{T}$. After some simplifications, we reach the following inequality

$${\displaystyle r_{*}\le \frac{{\tilde{\mathbb{S}}}^{*}{\mathbb{M}}_{*}\parallel x^{*}\parallel }{{\displaystyle 1-{\tilde{\upsilon }}^{*}\left(1+\frac{1}{\mathsf{\Gamma }({\delta }_1^{*}+1)}+\dots +\frac{1}{\mathsf{\Gamma }({\delta }_k^{*}+1)}\right){\mathbb{M}}_{*}\parallel x^{*}\parallel }}.}$$

In view of Inequality (13), the impossibility of Condition (b) of Lemma 2 is deduced. Hence, we have $\varpi \in {\mathbb{A}}_1^{*}\varpi {\mathbb{A}}_2^{*}\varpi$. Eventually, we manage to verify that there exists a fixed point for ${\mathcal{G}}^{*}$ and the sequential hybrid multi-order integro-differential inclusion BVP (1) and (2) involves a solution. □

In what follows, we carry out our procedure to derive other existence criteria for the Caputo sequential nonhybrid multi-order inclusion BVP (3) by applying two novel pure analytical theorems.

Definition 2.

The function $\varpi \in {\mathcal{AC}}_{\mathbb{R}}\left(\mathfrak{T}\right)$ is defined to be a solution of the given Caputo sequential nonhybrid multi-order inclusion BVP (3) if $\widehat{\kappa }\in {\mathcal{L}}_{\mathbb{R}}^1\left(\mathfrak{T}\right)$ exists via $\widehat{\kappa }\in \tilde{\mathfrak{O}}\left(z,\varpi \left(z\right)\right)$ for all $z\in \mathfrak{T}$ (a.e.), which satisfies the multi-order integro-derivative boundary conditions:

$$\varpi \left(0\right)=0,{\varpi }^{\prime }\left(0\right)=0,{}^{RL}{\mathcal{I}}_{0^{+}}^{{\theta }_1^{*}}\varpi \left(1\right)+{}^{RL}{\mathcal{I}}_{0^{+}}^{{\theta }_2^{*}}\varpi \left(\sigma \right)=0,$$

and

$$\begin{array}{cc}\hfill \varpi \left(z\right)& =\frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}q\hfill \\ \hfill & +\frac{1-e^{-{\mu }_2^{*}z}-{\mu }_2^{*}z}{{\mu }_1^{*}{\widehat{\mathsf{\Omega }}}_{*}}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q]\hfill \end{array}$$

for any $z\in \mathfrak{T}$.

In addition, for each $\varpi \in \mathfrak{W}$, we build the following collection of selections for $\tilde{\mathfrak{O}}$:

$${\left(\mathbb{SEL}\right)}_{\tilde{\mathfrak{O}},\varpi }=\left\{\widehat{\kappa }\in L^1\left(\mathfrak{T}\right):\widehat{\kappa }\left(z\right)\in \tilde{\mathfrak{O}}\left(z,\varpi \left(z\right)\right)\right\}$$

for almost all $z\in \mathfrak{T}$. From here onwards, we consider $\mathfrak{P}:\mathfrak{W}\to \mathcal{P}\left(\mathfrak{W}\right)$ by

$$\mathfrak{P}\left(\varpi \right)=\left\{\mathsf{\Phi }\in \mathfrak{W}:\mathrm{there}\mathrm{is}\widehat{\kappa }\in {\left(\mathbb{SEL}\right)}_{\tilde{\mathfrak{O}},\varpi }\mathrm{so}\mathrm{that}\mathsf{\Phi }\left(z\right)=b\left(z\right)\mathrm{for}\mathrm{any}z\in \mathfrak{T}\right\},$$

(15)

where

$$\begin{array}{cc}\hfill b\left(z\right)& =\frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}q\hfill \\ \hfill & +\frac{1-e^{-{\mu }_2^{*}z}-{\mu }_2^{*}z}{{\mu }_1^{*}{\widehat{\mathsf{\Omega }}}_{*}}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q].\hfill \end{array}$$

Theorem 2.

Let $\tilde{\mathfrak{O}}:\mathfrak{T}\times \mathfrak{W}\to {\mathcal{P}}_{cmp}\left(\mathfrak{W}\right)$ be compact. Suppose that all six hypotheses hold:

(HPS5)

An integrable operator $\tilde{\mathfrak{O}}$ is bounded, and also, $\tilde{\mathfrak{O}}(·,\varpi ):\mathfrak{T}\to {\mathcal{P}}_{cmp}\left(\mathfrak{W}\right)$ is a measurable set for every $\varpi \in \mathfrak{W}$;

(HPS6)

There are $\psi \in \mathsf{\Psi }$ and $\beta \in {\mathcal{C}}_{{\mathbb{R}}^{\ge 0}}\left(\mathfrak{T}\right)$ provided that for any $z\in \mathfrak{T}$ and $\varpi ,{\varpi }^{\prime }\in \mathfrak{W}$, we have

$${\mathbb{PH}}_{d_{\mathfrak{W}}}\left(\tilde{\mathfrak{O}}(z,\varpi ),\tilde{\mathfrak{O}}(z,{\varpi }^{\prime })\right)\le \psi \left(|\varpi -{\varpi }^{\prime }|\right)\frac{\beta \left(z\right)}{{\mathbb{M}}_{*}\parallel \beta \parallel },$$

(16)

where ${sup}_{z\in \mathfrak{T}}\left|\beta \left(z\right)\right|=\parallel \beta \parallel$ and ${\mathbb{M}}_{*}$ is displayed in (14);

(HPS7)

$\tilde{\epsilon }:\mathfrak{W}\times \mathfrak{W}\to \mathbb{R}$ exists so that $\tilde{\epsilon }\left(\varpi ,{\varpi }^{\prime }\right)\ge 0$ for each $\varpi ,{\varpi }^{\prime }\in \mathfrak{W}$;

(HPS8)

Let ${\left\{{\varpi }_n\right\}}_{n\ge 1}\subset \mathfrak{W}$ go to ϖ and $\tilde{\epsilon }\left({\varpi }_n\left(z\right),{\varpi }_{n+1}\left(z\right)\right)\ge 0$ for any $z\in \mathfrak{T}$. Then, a subsequence ${\left\{{\varpi }_{n_l}\right\}}_{l\ge 1}$ of $\left\{{\varpi }_n\right\}$ exists such that $\tilde{\epsilon }\left({\varpi }_{n_l}\left(z\right),\varpi \left(z\right)\right)\ge 0$ for any $z\in \mathfrak{T}$ and $l\ge 1$;

(HPS9)

There exist two elements ${\varpi }_0\in \mathfrak{W}$ and $\mathsf{\Phi }\in \mathfrak{P}\left({\varpi }_0\right)$ such that $\tilde{\epsilon }\left({\varpi }_0\left(z\right),\mathsf{\Phi }\left(z\right)\right)\ge 0$ for all $z\in \mathfrak{T}$, in which $\mathfrak{P}:\mathfrak{W}\to \mathcal{P}\left(\mathfrak{W}\right)$ is the same operator illustrated by (15);

(HPS10)

For every $\varpi \in \mathfrak{W}$ and $\mathsf{\Phi }\in \mathfrak{P}\left(\varpi \right)$ along with $\tilde{\epsilon }\left(\varpi \left(z\right),\mathsf{\Phi }\left(z\right)\right)\ge 0$, there exists $b\in \mathfrak{P}\left(\varpi \right)$ provided that $\tilde{\epsilon }\left(\mathsf{\Phi }\left(z\right),b\left(z\right)\right)\ge 0$ for each $z\in \mathfrak{T}$.

Then, the Caputo sequential nonhybrid multi-order inclusion BVP (3) has a solution.

Proof. 

For the same reason as before, it is explicit that a solution of the given Caputo sequential nonhybrid multi-order FBVP (3) is a fixed point of $\mathfrak{P}:\mathfrak{W}\to \mathcal{P}\left(\mathfrak{W}\right)$ defined by (15). By $\left(\mathrm{HPS}5\right)$, the measurability of $z\mapsto \tilde{\mathfrak{O}}\left(z,\varpi \left(z\right)\right)$ is an obvious fact, and so, it is closed-valued for any $\varpi \in \mathfrak{W}$. Therefore, we realize that $\tilde{\mathfrak{O}}$ involves a measurable selection and ${\left(\mathbb{SEL}\right)}_{\tilde{\mathfrak{O}},\varpi }$ is nonempty. Subsequently, we attempt to verify that $\mathfrak{P}\left(\varpi \right)$ is closed in $\mathfrak{W}$ for each $\varpi \in \mathfrak{W}$. To implement the process, we regard a sequence ${\left\{{\varpi }_n\right\}}_{n\ge 1}$ of $\mathfrak{P}\left(\varpi \right)$ having the property ${\varpi }_n\to \varpi$. For each n, we choose ${\widehat{\kappa }}_n\in {\left(\mathbb{SEL}\right)}_{\tilde{\mathfrak{O}},\varpi }$ so that

$$\begin{array}{cc}\hfill {\varpi }_n\left(z\right)& =\frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\widehat{\kappa }}_n\left(r\right)\mathrm{d}r\mathrm{d}q\hfill \\ \hfill & +\frac{1-e^{-{\mu }_2^{*}z}-{\mu }_2^{*}z}{{\mu }_1^{*}{\widehat{\mathsf{\Omega }}}_{*}}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\widehat{\kappa }}_n\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\widehat{\kappa }}_n\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q]\hfill \end{array}$$

for all $z\in \mathfrak{T}$ (a.e.). In light of the compactness of $\tilde{\mathfrak{O}}$, we can pass it into a subsequence to reach ${\left\{{\widehat{\kappa }}_n\right\}}_{n\ge 1}$, which approaches some $\widehat{\kappa }\in {\mathcal{L}}^1\left(\mathfrak{T}\right)$. Hence, there exists $\widehat{\kappa }\in {\left(\mathbb{SEL}\right)}_{\tilde{\mathfrak{O}},\varpi }$, and so,

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}{\varpi }_n\left(z\right)& =\frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}q\hfill \\ \hfill & +\frac{1-e^{-{\mu }_2^{*}z}-{\mu }_2^{*}z}{{\mu }_1^{*}{\widehat{\mathsf{\Omega }}}_{*}}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q]\hfill \\ \hfill & =\varpi \left(z\right)\hfill \end{array}$$

for any $z\in \mathfrak{T}$. Based on this argument, it follows that $\varpi \in \mathfrak{P}\left(\varpi \right)$, and so, $\mathfrak{P}$ is closed-valued. According to the hypothesis of the theorem, $\tilde{\mathfrak{O}}$ is compact, so one can easily verify that $\mathfrak{P}\left(\varpi \right)$ is bounded for each $\varpi \in \mathfrak{W}$. Here, we want to show that $\mathfrak{P}$ is $\alpha$-$\psi$-contractive. To prove this claim, we introduce $\alpha :{\mathfrak{W}}^2\to [0,\infty )$ by $\alpha (\varpi ,{\varpi }^{\prime })=1$ if $\tilde{\epsilon }\left(\varpi \left(z\right),{\varpi }^{\prime }\left(z\right)\right)\ge 0$, and $\alpha (\varpi ,{\varpi }^{\prime })=0$ otherwise. Assume that $\varpi ,{\varpi }^{\prime }\in \mathfrak{W}$ and ${\mathsf{\Phi }}_1\in \mathfrak{P}\left({\varpi }^{\prime }\right)$ are arbitrary. Furthermore, we select ${\widehat{\kappa }}_1\in {\left(\mathbb{SEL}\right)}_{\tilde{\mathfrak{O}},{\varpi }^{\prime }}$ so that

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_1\left(z\right)& =\frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\widehat{\kappa }}_1\left(r\right)\mathrm{d}r\mathrm{d}q\hfill \\ \hfill & +\frac{1-e^{-{\mu }_2^{*}z}-{\mu }_2^{*}z}{{\mu }_1^{*}{\widehat{\mathsf{\Omega }}}_{*}}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\widehat{\kappa }}_1\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\widehat{\kappa }}_1\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q]\hfill \end{array}$$

for any $z\in \mathfrak{T}$. Based on the hypothesis (16), we obtain

$${\mathbb{PH}}_{d_{\mathfrak{W}}}\left(\tilde{\mathfrak{O}}\left(z,\varpi \left(z\right)\right),\tilde{\mathfrak{O}}\left(z,{\varpi }^{\prime }\left(z\right)\right)\right)\le \beta \left(z\right)\psi \left(|\varpi \left(z\right)-{\varpi }^{\prime }\left(z\right)|\right)\frac{1}{{\mathbb{M}}_{*}\parallel \beta \parallel },$$

for each $\varpi ,{\varpi }^{\prime }\in \mathfrak{W}$ having the property $\tilde{\epsilon }\left(\varpi \left(z\right),{\varpi }^{\prime }\left(z\right)\right)\ge 0$ for any $z\in \mathfrak{T}$. Hence, $b\in \tilde{\mathfrak{O}}\left(z,\varpi \left(z\right)\right)$ exists with ${\displaystyle |{\widehat{\kappa }}_1\left(z\right)-b|\le \psi \left(|\varpi \left(z\right)-{\varpi }^{\prime }\left(z\right)|\right)\frac{\beta \left(z\right)}{{\mathbb{M}}_{*}\parallel \beta \parallel }}$. Next, we introduce ${\mathcal{Q}}^{*}:\mathfrak{T}\to \mathcal{P}\left(\mathfrak{W}\right)$ given by

$${\mathcal{Q}}^{*}\left(z\right)=\left\{b\in \mathfrak{W}:|{\widehat{\kappa }}_1\left(z\right)-b|\le \psi \left(|\varpi \left(z\right)-{\varpi }^{\prime }\left(z\right)|\right)\frac{\beta \left(z\right)}{{\mathbb{M}}_{*}\parallel \beta \parallel }\right\}$$

for each $z\in \mathfrak{T}$. Because of the measurability of both ${\widehat{\kappa }}_1$ and ${\displaystyle \varrho =\beta \psi \left(|\varpi -{\varpi }^{\prime }|\right)\frac{1}{{\mathbb{M}}_{*}\parallel \beta \parallel }}$, it follows that the intersection ${\mathcal{Q}}^{*}(·)\cap \tilde{\mathfrak{O}}\left(·,\varpi (·)\right)$ is measurable. In this direction, we select ${\widehat{\kappa }}_2$, which belongs to $\tilde{\mathfrak{O}}\left(z,\varpi \left(z\right)\right)$ so that

$$|{\widehat{\kappa }}_1\left(z\right)-{\widehat{\kappa }}_2\left(z\right)|\le \psi \left(|\varpi \left(z\right)-{\varpi }^{\prime }\left(z\right)|\right)\frac{\beta \left(z\right)}{{\mathbb{M}}_{*}\parallel \beta \parallel }$$

for all $z\in \mathfrak{T}$. Now, we regard the member ${\mathsf{\Phi }}_2\in \mathfrak{P}\left(\varpi \right)$ by

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_2\left(z\right)& =\frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\widehat{\kappa }}_2\left(r\right)\mathrm{d}r\mathrm{d}q\hfill \\ \hfill & +\frac{1-e^{-{\mu }_2^{*}z}-{\mu }_2^{*}z}{{\mu }_1^{*}{\widehat{\mathsf{\Omega }}}_{*}}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\widehat{\kappa }}_2\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\widehat{\kappa }}_2\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q]\hfill \end{array}$$

for any $z\in \mathfrak{T}$. Then, by computing the following estimates, we have

$$\begin{array}{cc}\hfill |{\mathsf{\Phi }}_1\left(z\right)& -{\mathsf{\Phi }}_2\left(z\right)|\le \frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}|{\widehat{\kappa }}_1\left(r\right)-{\widehat{\kappa }}_2\left(r\right)|\mathrm{d}r\mathrm{d}q\hfill \\ \hfill & +\frac{|1-e^{-{\mu }_2^{*}z}|+{\mu }_2^{*}z}{{\mu }_1^{*}\left|{\widehat{\mathsf{\Omega }}}_{*}\right|}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}|{\widehat{\kappa }}_1\left(r\right)-{\widehat{\kappa }}_2\left(r\right)|\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}|{\widehat{\kappa }}_1\left(r\right)-{\widehat{\kappa }}_2\left(r\right)|\mathrm{d}r\mathrm{d}m\mathrm{d}q]\hfill \\ \hfill & {\displaystyle \le \left[\frac{1-e^{-{\mu }_2^{*}}}{{\mu }_1^{*}{\mu }_2^{*}\mathsf{\Gamma }\left({\eta }^{*}\right)}+\frac{|1-e^{-{\mu }_2^{*}}|+{\mu }_2^{*}}{{\mu }_1^{*}\left|{\widehat{\mathsf{\Omega }}}_{*}\right|}\left(\frac{{\mu }_2^{*}+e^{-{\mu }_2^{*}}-1}{{\mu }_2^{{*}^2}\mathsf{\Gamma }\left({\theta }_1^{*}\right)\mathsf{\Gamma }\left({\eta }^{*}\right)}+\frac{{\sigma }^{{\theta }_2^{*}+{\eta }^{*}-1}({\mu }_2^{*}\sigma +e^{-{\mu }_2^{*}\sigma }-1)}{{\mu }_2^{{*}^2}\mathsf{\Gamma }\left({\theta }_2^{*}\right)\mathsf{\Gamma }\left({\eta }^{*}\right)}\right)\right]}\hfill \\ \hfill & \times \parallel \beta \parallel \psi \left(\parallel \varpi -{\varpi }^{\prime }\parallel \right)\frac{1}{{\mathbb{M}}_{*}\parallel \beta \parallel }={\mathbb{M}}_{*}\parallel \beta \parallel \psi \left(\parallel \varpi -{\varpi }^{\prime }\parallel \right)\frac{1}{{\mathbb{M}}_{*}\parallel \beta \parallel }=\psi (\parallel \varpi -{\varpi }^{\prime }\parallel )\hfill \end{array}$$

for all $z\in \mathfrak{T}$. Therefore, we obtain

$$\parallel {\mathsf{\Phi }}_1-{\mathsf{\Phi }}_2\parallel =\underset{z\in \mathfrak{T}}{sup}|{\mathsf{\Phi }}_1\left(z\right)-{\mathsf{\Phi }}_2\left(z\right)|\le \psi (\parallel \varpi -{\varpi }^{\prime }\parallel )$$

and so, we obtain

$$\alpha (\varpi ,{\varpi }^{\prime }){\mathbb{PH}}_{d_{\mathfrak{W}}}(\mathfrak{P}\left(\varpi \right),\mathfrak{P}\left({\varpi }^{\prime }\right))\le \psi (\parallel \varpi -{\varpi }^{\prime }\parallel )$$

for every $\varpi ,{\varpi }^{\prime }\in \mathfrak{W}$. This indicates that $\mathfrak{P}$ is $\alpha$-$\psi$-contractive. Now, by following the proof, we consider $\varpi \in \mathfrak{W}$ and ${\varpi }^{\prime }\in \mathfrak{P}\left(\varpi \right)$ coupled with $\alpha (\varpi ,{\varpi }^{\prime })\ge 1$. Then, by taking into account the definition of $\tilde{\epsilon }$, we obtain $\tilde{\epsilon }\left(\varpi \left(z\right),{\varpi }^{\prime }\left(z\right)\right)\ge 0$, and so, there exists $b\in \mathfrak{P}\left({\varpi }^{\prime }\right)$ so that $\tilde{\epsilon }\left({\varpi }^{\prime }\left(z\right),b\left(z\right)\right)\ge 0$. Hence, $\alpha ({\varpi }^{\prime },b)\ge 1$, and the latter inequality demonstrates that $\mathfrak{P}$ is $\alpha$-admissible. To complete the proof process, we consider ${\varpi }_0\in \mathfrak{W}$ and ${\varpi }^{\prime }\in \mathfrak{P}\left({\varpi }_0\right)$ so that $\tilde{\epsilon }\left({\varpi }_0\left(z\right),{\varpi }^{\prime }\left(z\right)\right)\ge 0$ for all z. Thus, we obtain $\alpha ({\varpi }_0,{\varpi }^{\prime })\ge 1$.

Moreover, consider ${\left\{{\varpi }_n\right\}}_{n\ge 1}$ in $\mathfrak{W}$ via ${\varpi }_n\to \varpi$ and $\alpha ({\varpi }_n,{\varpi }_{n+1})\ge 1$ for all n. In this phase, we reach $\tilde{\epsilon }\left({\varpi }_n\left(z\right),{\varpi }_{n+1}\left(z\right)\right)\ge 0$. Now, by using the assumption $\left(\mathrm{HPS}8\right)$, we figure out that there exists a subsequence ${\left\{{\varpi }_{n_l}\right\}}_{l\ge 1}$ of $\left\{{\varpi }_n\right\}$ so that $\tilde{\epsilon }\left({\varpi }_{n_l}\left(z\right),\varpi \left(z\right)\right)\ge 0$ for each $z\in \mathfrak{T}$. As a consequence, $\alpha ({\varpi }_{n_l},\varpi )\ge 1$ for $l\ge 1$, and it is deduced that the space $\mathfrak{W}$ has the $\left({\mathcal{C}}_{\alpha }\right)$-property. In the final step, by considering Lemma 3, we can easily realize that $\mathfrak{P}$ includes a fixed point and the sequential nonhybrid multi-order inclusion BVP (3) has at least one solution. □

As a final criterion for the existence of the solution, we review the following theorem based on a new interesting condition attributed to Amini Harandi [32]. On this basis, we shall employ the APPX-endpoint property for $\mathfrak{P}$ illustrated in (15).

Theorem 3.

Let $\tilde{\mathfrak{O}}:\mathfrak{T}\times \mathfrak{W}\to {\mathcal{P}}_{cmp}\left(\mathfrak{W}\right)$ be compact. Further:

(HPS11)

$\psi :[0,\infty )\to [0,\infty )$ is a nondecreasing map having the upper semi-continuity property via $lim{inf}_{z\to \infty }(z-\psi \left(z\right))>0$ and $\psi \left(z\right)<z$ for all $z>0$;

(HPS12)

$\tilde{\mathfrak{O}}:\mathfrak{T}\times \mathfrak{W}\to {\mathcal{P}}_{cmp}\left(\mathfrak{W}\right)$ is bounded integrable so that $\tilde{\mathfrak{O}}(·,\varpi ):\mathfrak{T}\to {\mathcal{P}}_{cp}\left(\mathfrak{W}\right)$ is measurable for every $\varpi \in \mathfrak{W}$;

(HPS13)

$\mathrm{Y}\in {\mathcal{C}}_{{\mathbb{R}}^{\ge 0}}\left(\mathfrak{T}\right)$ exists such that for all $z\in \mathfrak{T}$ and $\varpi ,{\varpi }^{\prime }\in \mathfrak{W}$, we have

$${\mathbb{PH}}_{d_{\mathfrak{W}}}\left(\tilde{\mathfrak{O}}(z,\varpi )-\tilde{\mathfrak{O}}(z,{\varpi }^{\prime })\right)\le \mathrm{Y}\left(z\right)\psi \left(|\varpi -{\varpi }^{\prime }|\right)\frac{1}{{\mathbb{M}}_{*}\parallel \mathrm{Y}\parallel },$$

(17)

where ${sup}_{z\in \mathfrak{T}}\left|\mathrm{Y}\left(z\right)\right|=\parallel \mathrm{Y}\parallel$ and ${\mathbb{M}}_{*}$ is defined by (14);

(HPS14)

$\mathfrak{P}$ is an operator having the APPX-endpoint property, where $\mathfrak{P}$ is formulated by (15).

Then, the Caputo sequential nonhybrid multi-order inclusion BVP (3) has at least one solution.

Proof. 

To begin the proof, we want to check the existence of at least one endpoint for given multifunction $\mathfrak{P}:\mathfrak{W}\to \mathcal{P}\left(\mathfrak{W}\right)$. To reach this objective, we have to verify that $\mathfrak{P}\left(\varpi \right)$ is a closed set for each $\varpi \in \mathfrak{W}$. By $\left(\mathrm{HPS}12\right)$, due to the measurability of $z\mapsto \tilde{\mathfrak{O}}\left(z,\varpi \left(z\right)\right)$ and also since it is closed-valued for all $\varpi \in \mathfrak{W}$, it follows that $\tilde{\mathfrak{O}}$ has a measurable selection and, thus, ${\left(\mathbb{SEL}\right)}_{\tilde{\mathfrak{O}},\varpi }$ is nonempty for every $\varpi \in \mathfrak{W}$. In this case, similar to the argument of Theorem 2, one can simply see that the subset $\mathfrak{P}\left(\varpi \right)$ of $\mathfrak{W}$ is closed, and so, we omit it. From another angle, we know that $\mathfrak{P}\left(\varpi \right)$ is a set having the boundedness property for each $\varpi \in \mathfrak{W}$ due to the compactness of $\tilde{\mathfrak{O}}$. Eventually, to end our proof, we must control whether the inequality ${\mathbb{PH}}_{d_{\mathfrak{W}}}(\mathfrak{P}\left(\varpi \right),\mathfrak{P}\left({\varpi }^{\prime }\right))\le \psi (\parallel \varpi -{\varpi }^{\prime }\parallel )$ holds or not. To check this, let $\varpi ,{\varpi }^{\prime }\in \mathfrak{W}$ and ${\mathsf{\Phi }}_1\in \mathfrak{P}\left({\varpi }^{\prime }\right)$. Let us select ${\widehat{\kappa }}_1\in {\left(\mathbb{SEL}\right)}_{\tilde{\mathfrak{O}},{\varpi }^{\prime }}$ such that

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_1\left(z\right)& =\frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\widehat{\kappa }}_1\left(r\right)\mathrm{d}r\mathrm{d}q\hfill \\ \hfill & +\frac{1-e^{-{\mu }_2^{*}z}-{\mu }_2^{*}z}{{\mu }_1^{*}{\widehat{\mathsf{\Omega }}}_{*}}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\widehat{\kappa }}_1\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\widehat{\kappa }}_1\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q]\hfill \end{array}$$

for all $z\in \mathfrak{T}$ (a.e.). In light of the inequality (17) demonstrated in the assumption $\left(\mathrm{HPS}13\right)$, we know that

$${\mathbb{PH}}_{d_{\mathfrak{W}}}\left(\tilde{\mathfrak{O}}(z,\varpi )-\tilde{\mathfrak{O}}(z,{\varpi }^{\prime })\right)\le \mathrm{Y}\left(z\right)\psi \left(|\varpi -{\varpi }^{\prime }|\right)\frac{1}{{\mathbb{M}}_{*}\parallel \mathrm{Y}\parallel }$$

for any $z\in \mathfrak{T}$; thus; there exists $b^{*}\in \tilde{\mathfrak{O}}\left(z,\varpi \left(z\right)\right)$ for which one can write

$$|{\widehat{\kappa }}_1\left(z\right)-b^{*}|\le \mathrm{Y}\left(z\right)\psi \left(|\varpi \left(z\right)-{\varpi }^{\prime }\left(z\right)|\right)\frac{1}{{\mathbb{M}}_{*}\parallel \mathrm{Y}\parallel }$$

for each $z\in \mathfrak{T}$. Now, we introduce ${\mathfrak{R}}^{*}:\mathfrak{T}\to \mathcal{P}\left(\mathfrak{W}\right)$, which is illustrated by

$${\mathfrak{R}}^{*}\left(z\right)=\left\{b^{*}\in \mathfrak{W}:|{\widehat{\kappa }}_1\left(z\right)-b^{*}|\le \mathrm{Y}\left(z\right)\psi \left(|\varpi \left(z\right)-{\varpi }^{\prime }\left(z\right)|\right)\frac{1}{{\mathbb{M}}_{*}\parallel \mathrm{Y}\parallel }\right\}.$$

For the sake of the measurability of ${\widehat{\kappa }}_1$ and $\varrho =\mathrm{Y}\psi \left(|\varpi -{\varpi }^{\prime }|\right)\frac{1}{{\mathbb{M}}_{*}\parallel \mathrm{Y}\parallel }$, we can easily realize that ${\mathfrak{R}}^{*}(·)\cap \tilde{\mathfrak{O}}(·,\varpi (·))$ is measurable. At this moment, choose the member ${\widehat{\kappa }}_2\left(z\right)\in \tilde{\mathfrak{O}}\left(z,\varpi \left(z\right)\right)$ such that

$$|{\widehat{\kappa }}_1\left(z\right)-{\widehat{\kappa }}_2\left(z\right)|\le \mathrm{Y}\left(z\right)\psi \left(|\varpi \left(z\right)-{\varpi }^{\prime }\left(z\right)|\right)\frac{1}{{\mathbb{M}}_{*}\parallel \mathrm{Y}\parallel }$$

for each $z\in \mathfrak{T}$. Now, we can choose ${\mathsf{\Phi }}_2\in \mathfrak{P}\left(\varpi \right)$, provided that

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_2\left(z\right)& =\frac{1}{{\mu }_1^{*}}{\int }_0^z{e}^{-{\mu }_2^{*}(z-q)}{\int }_0^q\frac{{(q-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\widehat{\kappa }}_2\left(r\right)\mathrm{d}r\mathrm{d}q\hfill \\ \hfill & +\frac{1-e^{-{\mu }_2^{*}z}-{\mu }_2^{*}z}{{\mu }_1^{*}{\widehat{\mathsf{\Omega }}}_{*}}[{\int }_0^1\frac{{(1-q)}^{{\theta }_1^{*}-1}}{\mathsf{\Gamma }\left({\theta }_1^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\widehat{\kappa }}_2\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{\sigma }\frac{{(\sigma -q)}^{{\theta }_2^{*}-1}}{\mathsf{\Gamma }\left({\theta }_2^{*}\right)}{\int }_0^q{e}^{-{\mu }_2^{*}(q-m)}{\int }_0^m\frac{{(m-r)}^{{\eta }^{*}-2}}{\mathsf{\Gamma }({\eta }^{*}-1)}{\widehat{\kappa }}_2\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q]\hfill \end{array}$$

for all $z\in \mathfrak{T}$. Hence, if we repeat the same process implemented in Theorem 2, then we arrive at inequality

$$\parallel {\mathsf{\Phi }}_1-{\mathsf{\Phi }}_2\parallel =\underset{z\in \mathfrak{T}}{sup}|{\mathsf{\Phi }}_1\left(z\right)-{\mathsf{\Phi }}_2\left(z\right)|\le {\mathbb{M}}_{*}\parallel \mathrm{Y}\parallel \psi (\parallel \varpi -{\varpi }^{\prime }\parallel )\frac{1}{{\mathbb{M}}_{*}\parallel \mathrm{Y}\parallel }=\psi (\parallel \varpi -{\varpi }^{\prime }\parallel ).$$

These findings verify that ${\mathbb{PH}}_{d_{\mathfrak{W}}}(\mathfrak{P}\left(\varpi \right),\mathfrak{P}\left({\varpi }^{\prime }\right))\le \psi (\parallel \varpi -{\varpi }^{\prime }\parallel )$ holds for any $\varpi ,{\varpi }^{\prime }\in \mathfrak{W}$. In addition, from the hypothesis $\left(\mathrm{HPS}14\right)$, we are sure that $\mathfrak{P}$ is an operator having the APPX-endpoint property, so by referring to Lemma 4, we derive the conclusion, which shows the existence of ${\varpi }^{*}\in \mathfrak{W}$ uniquely such that $\mathfrak{P}\left({\varpi }^{*}\right)=\left\{{\varpi }^{*}\right\}$. Therefore, it is concluded that ${\varpi }^{*}$ is a solution of the sequential multi-order inclusion FBVP (3). □

4. Examples

This part of the current paper is devoted to supporting the obtained theoretical results by proposing two simulative examples to demonstrate the correctness and the applicability of these outcomes.

Example 1.

In view of the given boundary problem (1) and (2), we here formulate a Caputo sequential hybrid integro-differential inclusion as the following form

$$\begin{array}{cc}\hfill \frac{3}{1000}\left({}^C{\mathcal{D}}_{0^{+}}^{2.34}+\frac{5}{1000}{}^C{\mathcal{D}}_{0^{+}}^{1.34}\right)& {\displaystyle \left(\frac{\varpi \left(z\right)}{z\left(\frac{sin\varpi \left(z\right)+arcsin\left({}^{RL}{\mathcal{I}}_{0^{+}}^{0.71}\varpi \left(z\right)\right)+arctan\left({}^{RL}{\mathcal{I}}_{0^{+}}^{0.21}\varpi \left(z\right)\right)}{100000}\right)+\frac{37}{10000}}\right)}\hfill \\ \hfill & \in \left[0,\left(0.0002z^{4999}+\frac{1}{5000}\right)cos\varpi \left(z\right)\right]\hfill \end{array}$$

(18)

subject to hybrid multi-order integro-derivative boundary conditions

$$\left\{\begin{array}{c}{\displaystyle \left(\frac{\varpi \left(z\right)}{z\left(\frac{sin\varpi \left(z\right)+arcsin\left({}^{RL}{\mathcal{I}}_{0^{+}}^{0.71}\varpi \left(z\right)\right)+arctan\left({}^{RL}{\mathcal{I}}_{0^{+}}^{0.21}\varpi \left(z\right)\right)}{100000}\right)+\frac{37}{10000}}\right){|}_{z=0}=0,}\hfill \\ {\displaystyle {}^C{\mathcal{D}}_{0^{+}}^1\left(\frac{\varpi \left(z\right)}{z\left(\frac{sin\varpi \left(z\right)+arcsin\left({}^{RL}{\mathcal{I}}_{0^{+}}^{0.71}\varpi \left(z\right)\right)+arctan\left({}^{RL}{\mathcal{I}}_{0^{+}}^{0.21}\varpi \left(z\right)\right)}{100000}\right)+\frac{37}{10000}}\right){|}_{z=0}=0,}\hfill \\ {\displaystyle {}^{RL}{\mathcal{I}}_{0^{+}}^{1.02}\left(\frac{\varpi \left(z\right)}{z\left(\frac{sin\varpi \left(z\right)+arcsin\left({}^{RL}{\mathcal{I}}_{0^{+}}^{0.71}\varpi \left(z\right)\right)+arctan\left({}^{RL}{\mathcal{I}}_{0^{+}}^{0.21}\varpi \left(z\right)\right)}{100000}\right)+\frac{37}{10000}}\right){|}_{z=1}}\hfill \\ {\displaystyle +{}^{RL}{\mathcal{I}}_{0^{+}}^{1.08}\left(\frac{\varpi \left(z\right)}{z\left(\frac{sin\varpi \left(z\right)+arcsin\left({}^{RL}{\mathcal{I}}_{0^{+}}^{0.71}\varpi \left(z\right)\right)+arctan\left({}^{RL}{\mathcal{I}}_{0^{+}}^{0.21}\varpi \left(z\right)\right)}{100000}\right)+\frac{37}{10000}}\right){|}_{z=0.65}=0,}\hfill \end{array}\right.$$

(19)

where $z\in \mathfrak{T}:=[0,1]$, ${\eta }^{*}=2.34$, ${\eta }^{*}-1=1.34$, ${\mu }_1^{*}=0.003$, ${\mu }_2^{*}=0.005$, $\sigma =0.65$, ${\theta }_1^{*}=1.02$. ${\theta }_2^{*}=1.08$, and for $k=2$, we have ${\delta }_1^{*}=0.71$ and ${\delta }_2^{*}=0.21$. In this case, we obtain ${\widehat{\mathsf{\Omega }}}_{*}\simeq 0.000006$. Here, one can specify the map ${\tilde{S}}^{*}:\mathfrak{T}\times {\mathbb{R}}^3\to \mathbb{R}\setminus \left\{0\right\}$ defined continuously as follows:

$${\tilde{S}}^{*}(z,{\varpi }_1\left(z\right),{\varpi }_2\left(z\right),{\varpi }_3\left(z\right))=z\left(\frac{sin{\varpi }_1\left(z\right)+arcsin\left({}^{RL}{\mathcal{I}}_{0^{+}}^{0.71}{\varpi }_2\left(z\right)\right)+arctan\left({}^{RL}{\mathcal{I}}_{0^{+}}^{0.21}{\varpi }_3\left(z\right)\right)}{100000}\right)+\frac{37}{10000}$$

in which ${\tilde{\mathbb{S}}}^{*}={sup}_{z\in \mathfrak{T}}\left|{\tilde{S}}^{*}(z,0,0,0)\right|=0.0037$. We claim that ${\tilde{S}}^{*}$ is Lipschitzian. To confirm this claim, for each $\varpi ,{\varpi }^{\prime }\in \mathbb{R}$, we may write

$$\begin{array}{cc}\hfill |{\tilde{S}}^{*}\left(z,\varpi \left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}\varpi \left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_2^{*}}\varpi \left(z\right)\right)& -{\tilde{S}}^{*}\left(z,{\varpi }^{\prime }\left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_1^{*}}{\varpi }^{\prime }\left(z\right),{}^{RL}{\mathcal{I}}_{0^{+}}^{{\delta }_2^{*}}{\varpi }^{\prime }\left(z\right)\right)|\hfill \\ \hfill & \le \tilde{\upsilon }\left(z\right)\left[1+\frac{z^{{\delta }_1^{*}}}{\mathsf{\Gamma }({\delta }_1^{*}+1)}+\frac{z^{{\delta }_2^{*}}}{\mathsf{\Gamma }({\delta }_2^{*}+1)}\right]|\varpi \left(z\right)-{\varpi }^{\prime }\left(z\right)|\hfill \\ \hfill & =\frac{z}{100000}\left[1+\frac{z^{0.71}}{\mathsf{\Gamma }\left(1.71\right)}+\frac{z^{0.21}}{\mathsf{\Gamma }\left(1.21\right)}\right]|\varpi \left(z\right)-{\varpi }^{\prime }\left(z\right)|,\hfill \end{array}$$

where ${\displaystyle \tilde{\upsilon }\left(z\right)=\frac{z}{100000}}$ and ${\tilde{\upsilon }}^{*}={sup}_{z\in \mathfrak{T}}\left|\tilde{\upsilon }\left(z\right)\right|=0.00001$. Hence, one can observe that the Lipschitz constant of ${\tilde{S}}^{*}$ equals

$${\displaystyle {\tilde{\upsilon }}^{*}\left[1+\frac{1}{\mathsf{\Gamma }({\delta }_1^{*}+1)}+\frac{1}{\mathsf{\Gamma }({\delta }_2^{*}+1)}\right]=\frac{1}{100000}\left[1+\frac{1}{\mathsf{\Gamma }\left(1.71\right)}+\frac{1}{\mathsf{\Gamma }\left(1.21\right)}\right]\simeq 0.0000319>0.}$$

In the next stage, we formulate the multifunction $\tilde{\mathfrak{O}}:\mathfrak{T}\times \mathbb{R}\to \mathcal{P}\left(\mathbb{R}\right)$ in the following framework:

$$\tilde{\mathfrak{O}}\left(z,\varpi \left(z\right)\right)=\left[0,(0.0002z^{4999}+\frac{1}{5000})cos\varpi \left(z\right)\right].$$

As for every $v\in \tilde{\mathfrak{O}}\left(z,\varpi \left(z\right)\right)$, we have

$$\left|v\right|\le max\left[0,(0.0002z^{4999}+\frac{1}{5000})cos\varpi \left(z\right)\right]\le 0.0002z^{4999}+0.0002,$$

so we obtain

$$\parallel \tilde{\mathfrak{O}}\left(z,\varpi \left(z\right)\right)\parallel =sup\left\{\right|\widehat{\kappa }|:\widehat{\kappa }\in \tilde{\mathfrak{O}}\left(z,\varpi \left(z\right)\right)\}\le 0.0002z^{4999}+0.0002.$$

By setting $x^{*}\left(z\right)=0.0002z^{4999}+0.0002$ for any $z\in \mathfrak{T}$, we obtain $\parallel x^{*}\parallel =0.0004.$ Furthermore, by the above obtained values, ${\mathbb{M}}_{*}\simeq 721419.86984$. In view of the above results and by some direct calculations, we select the positive number $r_{*}$ so that $r_{*}>1.0776$. Then, the following inequality holds:

$${\displaystyle {\tilde{\upsilon }}^{*}\left[1+\frac{1}{\mathsf{\Gamma }({\delta }_1^{*}+1)}+\frac{1}{\mathsf{\Gamma }({\delta }_2^{*}+1)}\right]{\mathbb{M}}_{*}\parallel x^{*}\parallel \simeq \left(0.0000319\right)\left(721419.86984\right)\left(0.0004\right)\simeq 0.0092053<\frac{1}{2}.}$$

The latter inequality and all the above numerical findings show that Theorem 1 is settled on the present BVP, and so, it is verified that the Caputo sequential hybrid multi-order integro-differential inclusion (18) and (19) has a solution.

Remark 1.

Note that in the above example, we can review other numerical results to check the correctness of the existence criteria of solutions for different values of order ${\eta }^{*}$. Indeed, for every ${\eta }^{*}\in [2,3)$, we compute the corresponding values of the constant ${\mathbb{M}}_{*}$ and the minimum value of $r_{*}$. Then, by assuming ${\displaystyle \Delta :={\tilde{\upsilon }}^{*}\left[1+\frac{1}{\mathsf{\Gamma }({\delta }_1^{*}+1)}+\frac{1}{\mathsf{\Gamma }({\delta }_2^{*}+1)}\right]{\mathbb{M}}_{*}\parallel x^{*}\parallel }$, we shall see that an inequality $\Delta <0.5$ holds in all cases. These numerical values can be observed in

Table 1. Numerical values for ${\mathbb{M}}_{*},min{r}_{*}$ and $\Delta$ corresponding to different values of order ${\eta }^{*}$.

${\mathit{\eta }}^{*}$	${\mathbb{M}}_{*}$	${\mathit{r}}_{*}>$	$\mathbf{\Delta }<0.5$
2	889,292.91961	1.3313	0.011347
2.05	865,898.41669	1.2958	0.011049
2.10	841,868.56165	1.2595	0.010742
2.15	817,328.27460	1.2224	0.010429
2.20	792,395.79444	1.1847	0.010111
2.25	767,182.51010	1.1467	0.0097892
2.30	741,792.85850	1.1083	0.0094653
2.34	721,419.86984	1.0776	0.0092053
2.35	716,324.28332	1.0699	0.0091403
2.40	690,867.24816	1.0316	0.0088155
2.45	665,505.29858	0.99338	0.0084918
2.50	640,315.16705	0.95547	0.0081704
2.55	615,366.91563	0.91795	0.0078521
2.60	590,724.11123	0.88091	0.0075376
2.65	566,444.02851	0.84444	0.0072278
2.70	542,577.87603	0.80861	0.0069233
2.75	519,171.04139	0.77350	0.0066246
2.80	496,263.35146	0.73915	0.0063323
2.85	473,889.34410	0.70562	0.0060468
2.90	452,078.54815	0.67296	0.0057685
2.95	430,855.76866	0.64119	0.0054977
2.99	414,314.74663	0.61644	0.0052867

and Figure 1.

In the last part, the analytical results obtained in Theorem 3 are investigated by the following numerical example.

Example 2.

In this example, we utilize the same given values ${\eta }^{*}=2.34$, ${\eta }^{*}-1=1.34$, ${\mu }_1^{*}=0.003$, ${\mu }_2^{*}=0.005$, $\sigma =0.65$, ${\theta }_1^{*}=1.02$. ${\theta }_2^{*}=1.08$. According to the given problem (3), we formulate the following Caputo sequential nonhybrid multi-order differential inclusion:

$$\frac{3}{1000}\left({}^C{\mathcal{D}}_{0^{+}}^{2.34}+\frac{5}{1000}{}^C{\mathcal{D}}_{0^{+}}^{1.34}\right)\varpi \left(z\right)\in \left[0,(z^{4999}+\frac{1}{5000})cos\varpi \left(z\right)\right]$$

(20)

furnished with multi-order integro-derivative boundary conditions

$$\varpi \left(0\right)=0,{}^C{\mathcal{D}}_{0^{+}}^1\varpi \left(0\right)=0,{}^{RL}{\mathcal{I}}_{0^{+}}^{1.02}\varpi \left(1\right)+{}^{RL}{\mathcal{I}}_{0^{+}}^{1.08}\varpi \left(0.65\right)=0$$

(21)

for all $z\in \mathfrak{T}$, where ${}^C{\mathcal{D}}_{0^{+}}^{\gamma }$ stands for the derivative of order $\gamma \in \{2.34,1.34\}$ of the Caputo type and ${}^{RL}{\mathcal{I}}_{0^{+}}^{\lambda }$ indicates the RL integral of order $\lambda \in \{1.02,1.08\}$. In a similar manner, we have ${\widehat{\mathsf{\Omega }}}_{*}\simeq 0.000006$ and ${\mathbb{M}}_{*}\simeq 2354320.0929$. To investigate the proposed problem precisely, we construct the Banach space as follows $\mathfrak{W}=\{\varpi \left(z\right):\varpi \left(z\right)\in {\mathcal{C}}_{\mathbb{R}}\left([0,1]\right)\}$ with ${\parallel \varpi \parallel }_{\mathfrak{W}}={sup}_{z\in \mathfrak{T}}\left|\varpi \left(z\right)\right|$. Now, we can introduce $\tilde{\mathfrak{O}}:\mathfrak{T}\times \mathfrak{W}\to \mathcal{P}\left(\mathfrak{W}\right)$ by

$$\tilde{\mathfrak{O}}\left(z,\varpi \left(z\right)\right)=\left[0,\frac{0.008(z+6)}{1628}\frac{|sin\varpi (z\left)\right|}{1+|sin\varpi (z\left)\right|}\right]$$

for each $z\in \mathfrak{T}$. In view of the above set-valued map $\tilde{\mathfrak{O}}$, we reach the function $\mathrm{Y}\in {\mathcal{C}}_{{\mathbb{R}}^{\ge 0}}\left(\mathfrak{T}\right)$, which is demonstrated by ${\displaystyle \mathrm{Y}\left(z\right)=\frac{0.008(z+6)}{814}}$ for any z along with ${\displaystyle \parallel \mathrm{Y}\parallel =\frac{0.056}{814}=0.0000687}$. In addition, the nondecreasing mapping $\psi :[0,\infty )\to [0,\infty )$ with the upper semi-continuity specification is regarded as ${\displaystyle \psi \left(z\right)=\frac{z}{2}}$ for $z>0$. Furthermore, it is evident that $lim{inf}_{z\to \infty }(z-\psi \left(z\right))>0$ and $\psi \left(z\right)<z$ for each $z>0$. For both arbitrary elements $\varpi ,{\varpi }^{\prime }\in \mathfrak{W}$, we estimate the following inequality:

$$\begin{array}{cc}\hfill {\mathbb{PH}}_{d_{\mathfrak{W}}}(\tilde{\mathfrak{O}}\left(z,\varpi \left(z\right)\right)& ,\tilde{\mathfrak{O}}\left(z,{\varpi }^{\prime }\left(z\right)\right))\le \frac{0.008(z+6)}{814}\frac{1}{2}\left(|\varpi -{\varpi }^{\prime }|\right)\hfill \\ \hfill & =\frac{0.008(z+6)}{814}\psi \left(|\varpi -{\varpi }^{\prime }|\right)\le \mathrm{Y}\left(z\right)\psi \left(|\varpi -{\varpi }^{\prime }|\right)\frac{1}{{\mathbb{M}}_{*}\parallel \mathrm{Y}\parallel },\hfill \end{array}$$

where ${\displaystyle \frac{1}{{\mathbb{M}}_{*}\parallel \mathrm{Y}\parallel }\simeq 0.006182}$. Finally, with due attention to the method implemented in Theorem 3, we regard $\mathfrak{P}:\mathfrak{W}\to \mathcal{P}\left(\mathfrak{W}\right)$ illustrated by

$$\mathfrak{P}\left(\varpi \right)=\left\{\mathsf{\Phi }\in \mathfrak{W}:thereis\widehat{\kappa }\in {\left(\mathbb{SEL}\right)}_{\tilde{\mathfrak{O}},\varpi }suchthat\mathsf{\Phi }\left(z\right)=b\left(z\right)foranyz\in \mathfrak{T}\right\},$$

where

$$\begin{array}{cc}\hfill b\left(z\right)& =\frac{1}{0.003}{\int }_0^z{e}^{-0.005(z-q)}{\int }_0^q\frac{{(q-r)}^{2.34-2}}{\mathsf{\Gamma }(2.34-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}q\hfill \\ \hfill & +\frac{1-e^{-0.005z}-0.005z}{\left(0.003\right)\left(0.000006\right)}[{\int }_0^1\frac{{(1-q)}^{1.02-1}}{\mathsf{\Gamma }\left(1.02\right)}{\int }_0^q{e}^{-0.005(q-m)}{\int }_0^m\frac{{(m-r)}^{2.34-2}}{\mathsf{\Gamma }(2.34-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q\hfill \\ \hfill & +{\int }_0^{0.65}\frac{{(0.65-q)}^{1.08-1}}{\mathsf{\Gamma }\left(1.08\right)}{\int }_0^q{e}^{-0.005(q-m)}{\int }_0^m\frac{{(m-r)}^{2.34-2}}{\mathsf{\Gamma }(2.34-1)}\widehat{\kappa }\left(r\right)\mathrm{d}r\mathrm{d}m\mathrm{d}q].\hfill \end{array}$$

Therefore, by referring to Theorem 3 based on a numerical method, it is realized that the Caputo sequential nonhybrid inclusion (20) along with multi-order integro-differential conditions (21) involves a solution. This means that the numerical outcomes of the current example are compatible with all theoretical arguments given in Theorem 3.

5. Conclusions

In this manuscript, we introduced and combined a new configuration of a Caputo sequential inclusion BVP with the hybrid integro-differential inclusion problem in which the boundary conditions are also formulated as the hybrid multi-order integro-derivative conditions. To reach the desired goal of this abstract general fractional model, we derived some theoretical existence results with the help of analytical techniques due to Dhage on the product operators for the given mixed sequential hybrid BVP (1) and (2). Next, based on some conditions on a defined space including the APPX-endpoint property, the $\left(C_{\alpha }\right)$-property, and the compactness, we investigated the nonhybrid structure of the suggested BVP (3). Lastly, two examples were designed in this regard. For the next works, one can study the qualitative behaviors and the stability of coupled systems of such sequential hybrid inclusion BVPs by using newly defined fractional operators. Specifically, we can conduct research on the fractal–fractional hybrid models of thermostat or pantograph systems by considering different cases for fractional orders and fractal dimensions.