Implicit nonlinear fractional differential equations of variable order

In this manuscript, we examine both the existence and the stability of solutions to the implicit boundary value problem of Caputo fractional differential equations of variable order. 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 rational ones. Although it seems an elementary consideration, it has an exciting relevance explaining some physical phenomena. Especially in the last two decades, significant numbers of papers appeared on this topic, some papers deal with the existence of solutions to problems of variable order; see e.g. [3,4,9,10,12].
In particular, [2] Benchohra et al. studied the existence and uniqueness results for the following nonlinear implicit fractional differential equations: x(t), c D u 0 + x(t)), t ∈ [0, T], 0 < T < +∞, 1 < u ≤ 2, where f is a given function, x 0 , x 1 ∈ , and c D u 0 + is the Caputo fractional derivative of order u.
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.
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 → is called piecewise constant with respect to partition P of I if for any E ∈ P, g is constant on E. Theorem 2.1 (Krasnoselskii fixed point theorem [6]) Let S be a closed, bounded and convex subset of a real Banach space E and let W 1 and W 2 be operators on S satisfying the following conditions: (1) is (UH) stable if there exists c f 1 > 0, such that, for any > 0 and for every solution z ∈ C(J, ) of the following inequality:

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 For each ∈ {1, 2, . . . , n}, the symbol E = C(J , ), indicates the Banach space of continuous functions x : J → equipped with the norm Then, for any t ∈ J , = 1, 2, . . . , n, the left Caputo fractional derivative of variable order u(t) for the function x(t) ∈ C(J, ), 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), the BVP (1) can be written for any t ∈ J , = 1, 2, . . . , n in the form In what follows we shall introduce the solution to the BVP (1). Let the function x ∈ C(J, ) be such that x(t) ≡ 0 on t ∈ [0, T -1 ] and such that it solves the 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 the BVP (7).
Then the function x ∈ E is a solution of the BVP (7) if and only if x solves the integral equation where Proof We presume that x ∈ E is solution of the BVP (7) and we take c D 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 where Conversely, let x ∈ E be a solution of the integral equation (8). Regarding the continuity of the function t δ f 1 and Lemma 2.1, we deduce that x is the solution of the BVP (7).
We will prove the existence result for the BVP (7). This result is based on Theorem 2.1.
Proof We construct the operators as follows: where It follows from the properties of fractional integrals and from the continuity of the function t δ f 1 that the operators W 1 , W 2 : E → E defined in (10) are well defined.
We consider the set Clearly B R is nonempty, closed, convex and bounded. Now, we demonstrate that W 1 , W 2 satisfy the assumption of Theorem 2.1. We shall prove it in four phases.

y(s), y(s) ds
STEP 2: Claim: W 1 is continuous. We presume that the sequence (y n ) converges to y in E and t ∈ J . Then i.e., we obtain Ergo, the operator W 1 is a continuous on E . STEP 3: W 1 is compact Now, we will show that W 1 (B R ) is relatively compact, meaning that W 1 is compact. Clearly W 1 (B R ) is uniformly bounded because by Step 1, we have W 1 (B R ) = {W 1 (y) : y ∈ B R } ⊂ W 1 (B R ) + W 2 (B R ) ⊆ (B R ) thus for each y ∈ B R we have W 1 (y) E ≤ R , which means that W 1 (B R ) is bounded. It remains to show that W 1 (B R ) is equicontinuous.
Consequently by (9), the operator W 2 is a strict contraction.
Therefore, all conditions of Theorem 2.1 are fulfilled and thus there exists x ∈ B R , such which is a solution of the BVP (7). Since B R ⊂ E , the claim of Theorem 3.1 is proved.
Now, we will prove the existence result for the BVP (1).