A fractional differential equation with multi-point strip boundary condition involving the Caputo fractional derivative and its Hyers–Ulam stability

In this work, we investigate the existence, uniqueness, and stability of fractional differential equation with multi-point integral boundary conditions involving the Caputo fractional derivative. By utilizing the Laplace transform technique, the existence of solution is accomplished. By applying the Bielecki-norm and the classical fixed point theorem, the Ulam stability results of the studied system are presented. An illustrative example is provided at the last part to validate all our obtained theoretical results.


Introduction
In the last few decades, a special consideration has been paid to fractional differential equations (FDEs) due to their wide range applications into real world phenomena (see [1][2][3][4]). Various attempts have been made in order to present these phenomena in a superior way and to explore new fractional derivatives with different approaches such as Riemann-Liouville, Caputo, Hadamard, Hilfer-Hadamard, and Grünwald-Letnikov [5][6][7][8][9][10][11]. In fact, FDEs are nonlocal in nature, and they describe many nonlinear phenomena very precisely, so they have a huge impact on different disciplines of science like hydrodynamics, control theory, signal processing, and image processing. More applications in these multidisciplinary sciences can be traced in [12][13][14][15]. In literature, there exist many complex differential systems that cannot be solved analytically, and obtaining the solution of such a type of systems is a big challenge to mathematicians; therefore, in such a situation, a solution can be traced through its properties, which are known as qualitative properties. One of the interesting examples is to investigate the unique solution's existence of an elliptic partial differential equation provided that there is a known average value [16].
In addition, another interesting example is the nonlinear problem of implicit FDEs with impulsive and integral boundary conditions which was investigated in [17] with the help of Schaefer's fixed point theorem and Banach's contraction principle. In fact, the qualitative properties like the existence and uniqueness theory for the solutions of fractional models with boundary value problem have attracted great attention among the researchers [18][19][20][21][22][23][24][25][26][27].
Another remarkable area which has recently received a considerable attention as one of the central topics in mathematical analysis is the stability of FDEs. In 1940, Ulam [28] initiated the stability of functional equations, which was improved by Hyers [29] in 1941 via Banach spaces. That is the reason why this stability is named Hyers-Ulam stability (HU S). After that, Rassias [30] introduced the Hyers-Ulam-Rassias stability (HU RS) by generalizing the concept of HUS. Moreover, a number of mathematicians have spread the idea of HUS to different classes of functional equations [31][32][33].
Abbas et al. [34] studied the Ulam stability and existence and uniqueness of solutions (·) are the Hilfer-Hadamard fractional derivative and the Hadamard fractional integral, respectively.

Fundamental results
Let C 2 [0, 1] denote a set of differentiable functions and its derivatives that are continuous on [0, 1] with the norm

The Laplace transform of Caputo derivatives is
where U(s) is the Laplace transform of the function u(σ ).

Definition 3
The Mittag-Leffler (ML) functions for arbitrary ∈ C are The Laplace transforms of ML functions are defined by

Theorem 5 ([40] Krasnosel'skii fixed point theorem) Let Λ be a closed convex and nonempty subset of a Banach space
Then there exists y ∈ Λ such that y = T 1 y + T 2 y.

Theorem 6 ([41]
Generalized Banach's fixed point theorem) Let (X, d) be a generalized complete metric space. Assume that T : X → X is a strictly contractive operator with Lipschitz constant L < 1. If there exists κ ≥ 0 such that d(T κ+1 x, T κ x) < ∞ for some x ∈ X, then the following propositions hold:

Theorem 7 ([39])
The Laplace transform of any function u (L{u(σ )}) exists and converges absolutely for (s) > if u is of exponential order.

Main results
Here, we use assumptions for the existence and uniqueness of solution to the considered problem (1) under Krasnosel'skii and generalized Banach fixed point theorems. We also discuss the HUS and HURS for the solution of considered system (1). The following hypotheses need to hold for the upcoming results:

Existence and uniqueness
Here, we examine the existence and uniqueness as follows.
Also, let g be any continuous function and is given by where G(σ , s) is given by and We have that u and c D α 0 + are of exponential order, where σ ∈ J. Taking the Laplace transform of (4), by (2), we acquire Applying the inverse Laplace transform, by (3), we have Further, we acquire Applying the boundary conditions, we get and Thus, substituting (7) and (8) into (6), we deduce that Hence, the proof is completed.
Remark 1 Using the definition of ML function, we obtain this implies that the series is convergent. Thus, there exists a constant E 2-α,4 > 0 such that Moreover, by the continuity of ML function and (5), there exists a constant ℵ > 0 such that σ 0 G(σ , s) ds ≤ ℵ, σ ∈ J.

the given system (1) has a unique solution.
Proof Using Lemma 8, the corresponding system (1) has the following solution: where G(σ , s) is given by (5).

Theorem 10 Under hypotheses [H 1 ]-[H 2 ] and the inequality
the given system (1) has at least one solution.
For any u, v ∈ S r , using [H 1 ]-[H 2 ], Remark 1, and the definitions of the operators T 1 and T 2 , we get that Thus, we obtain T 1 u + T 2 v ∈ S r . Using Theorem 9, the operator T 2 is a contraction mapping. By the continuity of φ(σ , u(σ )), u ∈ S r , and the two-parameter ML function, the operator T 1 is continuous.
Let u ∈ S r , from [H 2 ] and Remark 1, we have Hence, T 1 is uniformly bounded on S r .
Therefore, T 1 is relatively compact on S r . So, by Theorem 5 and Arzelà-Ascoli theorem, the operator T 1 is compact on S r . Thus, system (1) has at least one solution on J.

Ulam's stability results
In this subsection, using the Banach fixed point theorem and Bielecki metric, we investigate HUS and HURS results in C 2 [0, 1] for system (1).
Proof Using Lemma 8, the corresponding system (1) has the following solution: which follows from the proof of (6). We conclude that u(σ ) satisfies (1) iff u(σ ) satisfies (14). Consider the operator : Since φ and the two-parameter ML function are continuous, this implies that the operator is continuous. From (11), for any v, w ∈ C 2 [0, 1], we obtain Since Q φ μ < 1, the operator is strictly contractive. Also, letû ∈ C 2 [0, 1] satisfy (12). Then, we get thatû satisfies the following inequality: Using (11), (15), and the definition of the operator , we get Therefore, we conclude that Since (14) is the likewise integral equation of (1), so u(σ ) is a solution of (1). Also, using [A 3 ] of heorem 6 and (16), we have By the definition of d, we get that (13) holds.
Proof The first segment of the proof is obtained by performing the same steps as in Theorem 13. Let the operator : C 2 [0, 1] → C 2 [0, 1] be defined by For any v, w ∈ C 2 [0, 1], we have Since Q φ μ < 1, from Theorem 13, this implies that the operator is strictly contractive in (C 2 [0, 1], d).
Suppose thatû ∈ C 2 [0, 1] satisfies (18). Using Remark 1, we get Now, by the definition of the operator , we get Since z is a nondecreasing positive function, we have which implies u(σ ) is a solution of (1).

Conclusion
Banach's contraction principle and Krasnoselskii's fixed point theorem have been successfully used in this work to accomplish the essential conditions for investigating the existence and uniqueness of solution of our proposed system. In this manner, under specific assumptions and conditions, the HUS and HURS results have been demonstrated to study the solution of our proposed system. An illustrative example is given at the end to apply our theoretical results and show its validity. Some possible future directions of our work can be dedicated to applying our obtained results to study some interesting and important phenomena in physics and engineering such as elastic beam equation and fluid flow problems. Now possible extensions and generalizations of our obtained results can also be our future directions.