Boundary value problem for nonlinear fractional differential equations of variable order via Kuratowski MNC technique

In the present research study, for a given multiterm boundary value problem (BVP) involving the Riemann-Liouville fractional differential equation of variable order, the existence properties are analyzed. To achieve this aim, we firstly investigate some specifications of this kind of variable-order operators, and then we derive the required criteria to confirm the existence of solution and study the stability of the obtained solution in the sense of Ulam-Hyers-Rassias (UHR). All results in this study are established with the help of the Darbo’s fixed point theorem (DFPT) combined with Kuratowski measure of noncompactness (KMNC). We construct an example to illustrate the validity of our observed results.


Introduction
The idea of fractional calculus is replacing the natural numbers in the derivative order with rational ones. Although it seems an elementary consideration, it has an interesting correspondence in explaining some physical phenomena. In the last two decades, significant research studies appeared on this topic, and some papers dealt with the existence of solutions to the problems of variable order; see, for example, [1][2][3][4][5][6][7].
Whereas many researchers investigated the existence of solutions for fractional constantorder problems, the existence of solutions of variable-order problems is rarely mentioned in the literature (we refer to [8][9][10][11][12][13]).
As a result of our investigation in this interesting research field, our findings are unique and noteworthy.
Furthermore, all of the findings in this paper have a great potential to be applied in a variety of transdisciplinary science applications. With the support of our original findings in this research study, we are able to do further research on this open research topic. In other words, the proposed BVP can be extended to more sophisticated real mathematical fractional models in the future.
In particular, Bai et al. [14] studied the following problem: where c D u 0 + and I u 0 + stand for the Caputo-Hadamard derivative and Hadamard integral operators of order u, respectively, f is a given function, x a ∈ R, and 0 < a < b < ∞.
In this paper, we investigate the solution of (1). Further, we study the stability of the obtained solution of (1) in the Ulam-Hyers-Rassias (UHR) sense.

Preliminaries
In this section, we introduce some important fundamental definitions that will be needed for obtaining our results in the next sections.
Let us recall the following pivotal observation.
A function g : I → X is called piecewise constant with respect to partition P of I if for any E ∈ P, g is constant on E.

Measure of noncompactness
In this subsection, we discuss some necessary background information about KMNCs.

Definition 2.2 ([26]
) Let X be a Banach space, and let X be the bounded subsets of X. A KMNC is a mapping ζ : X → [0, ∞] constructed as follows: The following properties are valid for KMNCs. 26,27]) Let X be a Banach space, and let D, D 1 , and D 2 be bounded subsets of X. Then: is an equicontinuous and bounded set, then: Theorem 2.1 (DFPT [26]) Let be nonempty, closed, bounded, and convex subset of a Banach space X, and let : − → be a continuous operator satisfying Then has at least one fixed point in .
there exists c f > 0 such that for any > 0 and every solution z ∈ C(J, X) of the inequality there exists a solution x ∈ C(J, X) of equation (1) with

Existence of solutions
Let us introduce the following assumptions: where 1 < u ≤ 2 are constants, and I is the indicator of the interval J := (T -1 , T ], = 1, 2, . . . , n (with T 0 = 0, T n = T), such that Remark 3.1 According to the remark of [30] on page 20, we can easily show that condition (H2) and the inequality are equivalent for any bounded sets B 1 , B 2 ⊂ X and t ∈ J.
Further, for a given set U of functions u : J → X, let us denote Let us now prove the existence of solution for the BVP (1) via the concepts of MNCK and DFPT.
For 0 ≤ t ≤ T -1 , taking x(t) ≡ 0, we can write (5) as We will deal with the following BVP: For our purpose, the following lemma will be the basis of the solution of (6).

Lemma 3.1 A function x ∈ E forms a solution of (6) if and only if x fulfills the integral equation
Proof Let x ∈ E be solution of problem (6). Applying the operator I u T + -1 to both sides of (6), from Lemma 2.1 we find Due to the assumption on the function f 1 along with x(T -1 ) = 0, we conclude that ω 2 = 0. Let x satisfy x(T ) = 0. Observe that Then we find Conversely, let x ∈ E be a solution of integral equation (7), Regarding the continuity of the unction t δ f 1 and Lemma 2.1, we deduce that x is a solution of problem (6).
Our first existence result is based on Theorem 2.1.
Theorem 3.1 Assume that conditions (H1) and (H2) hold and Then problem (6) possesses at least one solution on J.
Proof We construct the operator as follows: It follows from the properties of fractional integrals and the continuity of function t δ f 1 that the operator W is well defined. Let We consider the set Clearly, B R is nonempty, closed, convex, and bounded. Now we demonstrate that W satisfies the assumptions of Theorem 2.1. We shall prove it in four phases.
Step 1: . For x ∈ B R , by (H2) we get: x(s) ds Step 2: W is continuous. Let a sequence (x n ) converge to x in E , and let t ∈ J . Then x(s) ds x(s) ds that is, Thus the operator W is continuous on E .
Step 3: W is bounded and equicontinuous.
For U ∈ B R and t ∈ J , we have: Then Remark 3.1 implies that, for each s ∈ J i , Therefore we have: Consequently, from (8) we deduce that W forms a set contraction. Hence by Theorem 2.1 problem (6) has at least a solution x in B R . Let We know that x ∈ C([0, T ], X) defined by (10) satisfies the equation for t ∈ J , which means that x is a solution of (5) with x (0) = 0 and x (T ) = x (T ) = 0. Then x , t ∈ J , forms a solution of BVP (1).
For t ∈ J 1 , problem (12) is equivalent to the problem , t ∈ J 1 , Next, we prove that condition (8) is fulfilled.
It is known that As a result, by Definition 3.1 the boundary value problem (12) has a solution x 2 (t), t ∈ J 2 .
In addition, by Theorem 4.1 the equation in (12) is UHR stable.

Conclusion
Our proposed multiterm BVP has been successfully investigated in this work via three theorems: The Darbo's fixed point theorem (DFPT), the Kuratowski measure of noncompactness (KMNC), and the Ulam-Hyers-Rassias stability (UHR) to prove the existence and stability of solutions for our proposed BVP. A numerical example is given at the end to support and validate the potentiality of all our obtained results. As a result of our investigation into this particular research subject, our results are new and novel. Furthermore, with the support of our new results in this work, further research works can be investigated on this open research subject. Our proposed BVP can be possibly extended to other fractional models.