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In this paper, we study a coupled system of nonlinear proportional fractional differential equations of the Hilfer-type with a new kind of multi-point and integro-multi-strip boundary conditions. Results on the existence and uniqueness of the solutions are achieved by using Banach’s contraction principle, the Leray–Schauder alternative and the well-known fixed-point theorem of Krasnosel’skiĭ. Finally, the main results are illustrated by constructing numerical examples.
Fractional-order differential equations arise in the mathematical modeling of several engineering and scientific phenomena. Examples include physics, chemistry, robotics, signal and image processing, control theory and viscoelasticity (see the monographs in [1,2,3,4,5]). In particular, nonlinear coupled systems of fractional-order differential equations appear often in investigations connected with anomalous diffusion , disease models  and ecological models . Unlike the classical derivative operator, one can find a variety of its fractional counterparts, such as the Riemann–Liouville, Caputo, Hadamard, Erdeyl–Kober, Hilfer and Caputo–Hadamard counterparts. Recently, a new class of fractional proportional derivative operators was introduced and discussed in [9,10,11]. The concept of Hilfer-type generalized proportional fractional derivative operators was proposed in . For the detailed advantages of the Hilfer derivative, see  and a recent application in calcium diffusion in .
Many researchers studied initial and boundary value problems for differential equations and inclusions, including different kinds of fractional derivative operators (for examples, see [15,16,17,18,19,20]). In , the authors studied a nonlocal initial value problem of an order within involving a Hilfer generalized proportional fractional derivative of a function with respect to another function. Recently, in , the authors investigated the existence and uniqueness of solutions for a nonlocal mixed boundary value problem for Hilfer fractional -proportional-type differential equations and inclusions of an order within In , the authors discussed the existence of solutions for a nonlinear coupled system of Hilfer fractional differential equations of different orders within complemented with coupled Riemann–Liouville fractional integral boundary conditions given by
Here, is the Hilfer fractional proportional derivative operator of the order and type , is a continuous function (or is a multi-valued map), is the fractional integral operator of the order and , , Very recently, in , the authors considered a new boundary value problem consisting of a Hilfer fractional -proportional differential equation and nonlocal integro-multi-strip and multi-point boundary conditions of the form
where denotes the Hilfer fractional proportional derivative operator of the order and type , is an increasing function with for all and is a continuous function.
Motivated by the foregoing work on boundary value problems involving Hilfer-type fractional -proportional derivative operators, in this paper, we aim to establish existence and uniqueness results for a class of coupled systems of nonlinear Hilfer-type fractional -proportional differential equations equipped with nonlocal multi-point and integro-multi-strip coupled boundary conditions. To be precise, we investigate the following problem:
where , denote the Hilfer fractional -proportional derivative operator of the order and type , is an increasing function with for all and are continuous functions.
Here we emphasize that system (1) is novel, and its investigation will enhance the scope of the literature on nonlocal Hilfer-type fractional -proportional boundary value problems. It is worthwhile to mention that the Hilfer fractional -proportional derivative operators are of a more general nature and reduce to the Hilfer generalized proportional fractional derivative operators  when and which unify the classical Riemann–Liouville and Caputo fractional derivative operators. Our strategy to deal with system (1) is as follows. First of all, we solve a linear variant of system (1) in Lemma 3, which plays a pivotal role in converting the nonlinear problem in system (1) into a fixed-point problem. Afterward, under certain assumptions, we apply different fixed-point theorems to show that the fixed-point operator related to the problem at hand possesses fixed points. The first result (Theorem 1) shows the existence of a unique solution to system (1) by means of Banach’s contraction mapping principle. In the second result (Theorem 2), the existence of at least one solution to system (1) is established via the Leray–Schauder alternative. The last result (Theorem 3), relying on Krasnosel’skiĭ’s fixed-point theorem, deals with the existence of at least one solution to system (1) under a different hypothesis. We illustrate all the obtained results with the aid of examples in Section 4. In the last section, we describe the scope and utility of the present work by indicating that several new results follow as special cases by fixing the parameters involved in system (1).
The rest of this paper is organized as follows. In the following section, some necessary definitions and preliminary results related to our study are outlined. Section 3 contains the main results for system (1), while numerical examples illustrating these results are presented in Section 4. The paper concludes with some interesting observations.
Let us begin this section with some basic definitions.
([10,11]).For and the fractional proportional integral of with respect to of an order ρ is given by
([10,11]).Let with and The fractional proportional derivative for with respect to of an order is given by
where and denotes the integer part of the real number ρ.
().Let be positive and strictly increasing with for all and The Hilfer fractional proportional derivative for with respect to another function of an order ρ and type is defined by
where and In addition, , and is the fractional proportional integral operator defined in Equation (2).
Now, we recall some known results.
().The Hilfer fractional proportional derivative can be expressed as
Before proceeding for the existence and uniqueness results for the system (1), we consider the following lemma associated with the linear variant of the coupled system of Hilfer-type fractional -proportional differential equations considered in system (1).
Let and Then, is a solution to the following coupled, linear, nonlocal integro-multi-strip and multi-point, Hilfer generalized proportional fractional system:
if and only if
From Lemma 2 with we have
where and Using Equations (9) and (10) in the conditions and we obtain and since and Hence, Equations (9) and (10) take the forms
By inserting Equations (11) and (12) into the conditions and we obtain
In light of the notation (8), we can express Equations (13) and (14) in the form of the following system:
Since under the condition in Equation (20), the operator is a contraction. Therefore, the conclusion of Banach’s contraction mapping principle applies, and hence the operator has a unique fixed point. As a consequence, there exists a unique solution to the nonlocal integro-multi-strip and multi-point Hilfer generalized proportional fractional system (1). □
The following result is based on the Leray–Schauder alternative :
Let be continuous functions such that the following condition holds:
There exist for and such that for any , we have
If and , where are given in Equation (19), then the nonlocal integro-multi-strip and multi-point Hilfer generalized proportional fractional system (1) has at least one solution on .
Observe that the operator defined in Equation (16) is continuous, owing to the continuity of functions and on Next, we show that the operator is complete continuous. We define . Then, for all , there exist such that and . Therefore, for all , we have
which implies that
Similarly, we can obtain
Consequently, we have
Thus, we deduce that the operator is uniformly bounded.
Now, we establish that the operator is equicontinuous. Let with . Then, we have
which implies that as independent of Thus, the operator is completely continuous under the Arzelá–Ascoli theorem.
Similarly, it can be shown that
as independent of Hence, the operator is completely continuous.
Lastly, we verify that the set is bounded. Let . Then, . Hence, for all we have
Under assumption we have
which imply that
Consequently, we have
where Hence, the set is bounded. Under the Leray–Schauder alternative, the operator has at least one fixed point. Therefore, the nonlocal integro-multi-strip and multi-point Hilfer generalized proportional fractional system (1) has at least one solution on □
Our second existence result is based on Krasnosel’skiĭ’s fixed-point theorem :
Let be continuous functions satisfying condition In addition, the following assumption holds:
There exist non-negative functions such that, for all
Then, the nonlocal integro-multi-strip and multi-point Hilfer generalized proportional fractional system (1) has at least one solution on provided that
In order to verify the hypothesis of Krasnosel’skiĭ’s fixed-point theorem , we decompose the operator as follows:
Let us set and and introduce the set , with
As in the proof of Theorem 2, we can obtain that
As a consequence, it follows that
Now, it will be proven that the operator is a contraction mapping. For and for any we have
Similarly, we can obtain
Consequently, we obtain
which, according to Equation (25), implies that is a contraction.
It remains to be verified that the operator is completely continuous. Under the continuity of functions and , we deduce that the operator is continuous. For all , following the arguments employed in the proof of Theorem 2, we find
Similarly, we have that
Consequently, we have
Thus, set is uniformly bounded.
Lastly, we show that set is equicontinuous. Let such that . For all , due to the equicontinuous property of operators and , we can show that , as independent of Consequently, set is equicontinuous. Now, under the Arzelá–Ascoli theorem, the compactness property of operator on is established. Hence, under the conclusion of Krasnosel’skiĭ’s fixed-point theorem, the nonlocal integro-multi-strip and multi-point Hilfer generalized proportional fractional system (1) has at least one solution on □
4. Illustrative Examples
Let us consider a coupled system of nonlinear proportional fractional differential equations of the Hilfer type:
supplemented with multi-point and integro-multi-strip boundary conditions of the form
Here, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , and . Using these values, we find that , , , , , , , , , , , and .
For illustrating Theorem 1, let us take the Lipschitzian functions and on defined by
for all , , and . By setting the Lipschitz constants to , , and , we obtain
Clearly, all the assumptions of Theorem 1 are fulfilled, and hence its conclusion implies that the system (30) with multi-point and integro-multi-strip boundary conditions (31) and the functions and given in Equation (32) has a unique solution on
We demonstrate the application of Theorem 2 by considering the following nonlinear non-Lipschitzian functions:
Note that and are bounded as
for all and . By fixing , , , , and , we obtain and Therefore, it follows with the conclusion of Theorem 2 that there exists at least one solution on the interval of the system (30) with multi-point and integro-multi-strip boundary conditions (31) and two nonlinear functions and given in Equation (33).
Let us use the following functions for explaining the application of Theorem 3: