On boundary value problems of Caputo fractional differential equation of variable order via Kuratowski MNC technique

In this manuscript, we examine both the existence and the stability of solutions to the boundary value problem of Caputo fractional differential equations of variable order by converting it into an equivalent standard Caputo boundary value problem of the fractional constant order with the help of the generalized intervals and the piece-wise constant functions. All results in this study are established using Darbo’s fixed point theorem combined with the Kuratowski measure of noncompactness. Further, the Ulam–Hyers stability of the given problem is examined; and finally, we construct an example to illustrate the validity of the observed results.


Introduction
The idea of fractional calculus is to replace the natural numbers in the derivative's order with the rational ones. Although it seems an elementary consideration, it has an exciting correspondence explaining some physical phenomena. While several research studies have been performed on investigating the solutions existence of the fractional constantorder problems (we refer to [1-5, 8, 11, 21, 22, 24]), the solutions' existence of the variableorder problems is rarely discussed in the literature (we refer to [7,16,27,28,30]). Therefore, all our results in this work are novel and worthwhile.
In relation to the study of the existence theory to boundary value problems of fractional variable order, we point out some of them. In [19], Jiahui and Pengyu studied the uniqueness of solutions to the initial value problem of Riemann-Liouville fractional differential equations of variable order. Zhang and Hu [35] established the existence of solutions and generalized Lyapunov-type inequalities of variable-order Riemann-Liouville boundary value problems.
Recently, Bouazza et al. [15] studied a Riemann-Liouville variable-order boundary value problem, and Benkerrouche et al. [14] presented the existence results and Ulam-Hyers stability for implicit nonlinear Caputo fractional differential equations of variable order.
In 2021, Hristova et al. [18] and Refice et al. [23] turned to the investigation of boundary value problems of Hadamard fractional differential equations of variable order via the Kuratowski measure of noncompactness technique; for more studies, we refer to [26,30,34].
In this paper, we shall look for a solution of (1). Further, we study the stability of the obtained solution of (1) in the sense of Ulam-Hyers (UH).

Preliminaries
This section introduces some important fundamental definitions that will be needed for obtaining our results in the next sections.
Recall the following pivotal observation.
So, we get 2s -1 + (s -1) 3 3 ds = 21 10 , Therefore, we obtain , the variable order fractional integral I u(t) 0 + f 2 (t) exists for any points on J.  A finite set P is called a partition of I if each x in I lies in exactly one of the generalized intervals E in P.
A function g : I → R is called piecewise constant with respect to partition P of I if, for any E ∈ P, g is constant on E.

Measure of noncompactness
This subsection discusses some necessary background information about the Kuratowski measure of noncompactness (KMNC).

Definition 2.2 ([9]
) Let X be a Banach space and X be the bounded subsets of X. The (KMNC) is a mapping ζ : X → [0, ∞] which is constructed as follows: The following properties are valid for (KMNC). i.e., is k-set contractions.

Lemma 2.4 ([17]) If U ⊂ C(J, X) is an equicontinuous and bounded set, then
Then has at least one fixed point in . (1) is (UH) stable if there exists c f 1 > 0 such that for any > 0 and for every solution z ∈ C(J, R) of the following inequality

Definition 2.3 ([13]) The equation of
there exists a solution x ∈ C(J, R) of equation (1) with

Existence of solutions
Let us introduce the following assumption: 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 Further, for a given set U of functions u : J → X, let us denote For each ∈ {1, 2, . . . , n}, the symbol E = C(J , R) indicates the Banach space of continuous functions x : J → R equipped with the norm Then, for any t ∈ J , = 1, 2, . . . , n, the (CFD) of variable order u(t) for function x(t) ∈ C(J, R), defined by (3), could be presented as a sum of left Caputo fractional derivatives of constant orders u , = 1, 2, . . . , n: Thus, according to (5), (BVP)(1) can be written, for any t ∈ J , = 1, 2, . . . , n, in the form In what follows we shall introduce the solution to BVP (1). Let the function x ∈ C(J, R) be such that x(t) ≡ 0 on t ∈ [0, T -1 ] and it solves integral equation (6). Then (6) is reduced to We shall deal with the following BVP: For our purpose, the upcoming lemma will be a corner stone of the solution of BVP (7). Then the function x ∈ E is a solution of BVP (7) if and only if x solves the integral equation Proof We presume that x ∈ E is a solution of BVP (7). Employing the operator I u T + -1 to both sides of (7) and regarding Lemma 2.1, we find By x(T -1 ) = 0, we get ω 1 = 0. Let x(t) satisfy x(T ) = 0. So, we observe that Then we find Conversely, let x ∈ E be a solution of integral equation (8). Regarding the continuity of function t δ f 1 and Lemma 2.1, we deduce that x is the solution of BVP (7).
We are now in a position to prove the existence of solution for (BVP) (7) based on the concept of (MNCK) and (DFPT).
for any y 1 , y 2 ∈ R, t ∈ J , and the inequality holds. Then BVP (7) possesses at least one solution in E .
Proof We construct the operator W : E → E as follows: It follows from the properties of fractional integrals and from the continuity of function t δ f 1 that the operator W : We consider the set Clearly, B R is nonempty, closed, convex, and bounded. Now, we demonstrate that W satisfies the assumption of Theorem 2.1. We shall prove it in four phases.
Step 1: . For x ∈ B R and by (H2), we get Step 2: Claim: W is continuous. We presume that the sequence (x n ) converges to x in E and t ∈ J . Then i.e., we obtain (Wx n ) -(Wx) E → 0 as n → ∞.
Ergo, the operator W is continuous on E .
Step 3: Claim: W is bounded and equicontinuous. By Remark 3.1 According to the remark of [12] page 20, we can easily show that inequality (9) and the following inequality are equivalent for any bounded sets B ⊂ X and for each t ∈ J .
Step 4: Claim: W is k-set contractions.
Then Remark 3.1 implies that, for each s ∈ J i , Thus, Therefore, all conditions of Theorem 2.1 are fulfilled, and thus BVP (7) has at least solution x ∈ B R . Since B R ⊂ E , the claim of Theorem 3.1 is proved. Now, we will prove the existence result for BVP (1). Introduce the following assumption: (H2) Let f 1 ∈ C(J × R, R), and there exists a number δ ∈ (0, 1) such that t δ f 1 ∈ C(J × R, R) and there exists a constant K > 0 such that t δ |f 1 (t, y 1 )f 1 (t, y 2 )| ≤ K|y 1y 2 | for any y 1 , y 2 ∈ R and t ∈ J.
is a solution of BVP (1) in C(J, R). Proof Let > 0 be an arbitrary number and the function z(t) from z ∈ C(J , R) satisfy inequality (4).

Example 2
Let us consider the following fractional boundary value problem: Let Then we have Hence condition (H2) holds with δ = 1 2 and K = 1 5 .

Conclusion
In this paper, we presented results about the existence of solutions to the BVP of Caputo fractional differential equations of variable order u(t), where u(t) : [0, T] → (1, 2] is a piecewise constant function. All our results are based on Darbo's fixed point theorem combined with the Kuratowski measure of noncompactness (Theorem 3.1), and we studied Ulam-Hyers stability of solutions to our problem (Theorem 3.3). Finally, we illustrated the theoretical findings by a numerical example. All results in this work show a great potential to be applied in various of sciences. Moreover, with the help of our results in this research paper, investigations on this open research problem could be also possible, and one could extend the proposed BVP to other complicated fractional models.
In the near future we want to study these BVPs with different boundary problem (implicit, resonance, thermostat model, etc.) value conditions involving integral conditions or integro-derivative conditions.