Multiterm boundary value problem of Caputo fractional differential equations of variable order

In this manuscript, the existence, uniqueness, and stability of solutions to the multiterm boundary value problem of Caputo fractional differential equations of variable order are established. All results in this study are established with the help of the generalized intervals and piece-wise constant functions, we convert the Caputo fractional variable order to an equivalent standard Caputo of the fractional constant order. Further, two fixed point theorems due to Schauder and Banach are used, the Ulam–Hyers stability of the given Caputo variable order is examined, and finally, we construct an example to illustrate the validity of the observed results. In literature, the existence of solutions to the variable-order problems is rarely discussed. Therefore, investigating this interesting special research topic makes all our results novel and worthy.


Introduction
The main idea of fractional calculus is to constitute the natural numbers in the order of derivation operators with rational ones. Although this idea is preliminary and simple, it involves remarkable effects and outcomes which describe some physical, dynamics, modeling, control theory, bioengineering, and biomedical applications phenomena. For this reason, recently, a significant number of papers have appeared on this topic (see for example [8,9] and the references therein); on the contrary, few papers deal with the existence of solutions to problems via variable order, see, e.g., [4,15,16,18,19].
In general, it is usually difficult to solve boundary value problems of fractional variable order (FBVPs) and obtain their analytical solution. Therefore, some methods are introduced for the approximation of solutions to different FBVPs of variable order. In relation to the study of the existence theory to FBVPs of variable order, we point out some of them. In [20], Zhang studied solutions of a two-point boundary value problem of fractional variable order involving singular fractional differential equations (FDEs). After some years, Zhang and Hu [22] established the existence results for approximate solutions of variable order fractional initial value problems on the half line. Recently, Bouazza et al. [3] considered a multiterm FBVP variable order and derived their results by terms of fixed point methods. In 2021, Hristova et al. [5] turned to investigation of the Hadamard FBVP of variable order by means of Kuratowski MNC method. For more details on other instances, refer to [10,14] and the references therein.
In [1] Bai et al. investigate the existence for nonlinear fractional differential equations of constant order 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 < ∞.
Some existence and Ulam stability properties for FDEs have been studied by many authors (see [2,13] and the references therein).
In this paper, we 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.

Lemma 2.3 ([25])
Let u : J → (1, 2] be a continuous function, then 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. Theorem 2.1 (Schauder fixed point theorem, [7]) Let E be a Banach space, Q be a convex subset of E, and F : Q − → Q be a compact and continuous map. Then F has at least one fixed point in Q. (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 2.2 ([2]) The equation of
there exists a solution x ∈ C(J, R) of Eq. (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 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 left Caputo fractional derivative 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 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 cornerstone 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 where G (t, s) is the Green's function defined by where = 1, 2, . . . , n.
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 by the continuity of Green's function which implies that 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).
Since ϕ(t, s) is nonincreasing with respect to t, then ϕ(t, s) ≤ ϕ(s, s) for T -1 ≤ s ≤ t ≤ T . On the other hand, for T -1 ≤ t ≤ s ≤ T , we get These assure that Hence, for =1, . . . , n, We will prove the existence results for BVP (7). The first result is based on Theorem 2.1.

holds. Then BVP (7) possesses at least one solution in E .
Proof We construct the operator 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 three phases.
We presume that the sequence (x n ) converges to x in E and t ∈ J . Then i.e., we obtain Ergo, the operator W is continuous on E .
Step 3: W is compact. Now, we will show that W (B R ) is relatively compact, meaning that W is compact.
For t 1 , t 2 ∈ J , t 1 < t 2 , and x ∈ B R , we have x(s)f 1 (s, 0, 0) ds by the continuity of Green's function G . Hence (Wx)(t 2 ) -(Wx)(t 1 ) E → 0 as |t 2t 1 | → 0. It implies that W (B R ) is equicontinuous. Therefore, all conditions of Theorem 2.1 are fulfilled, and thus there exists x ∈ B R such that W x = x , which is a solution of BVP (7). Since B R ⊂ E , the claim of Theorem 3.1 is proved.

Example
Let us consider the following fractional boundary value problem: Let .

Conclusion
In this work we presented two results on the existence, uniqueness of solutions to the multiterm BVP boundary value problem of Caputo fractional differential equations of variable order, which is a piecewise constant function based on the essential difference about the variable order. The first one is based on Schauder's fixed point theorem (Theorem 3.1) and the second one on the Banach contraction principle (Theorem 3.2). By a numerical example, we illustrated the theoretical findings. Finally, we study Ulam-Hyers stability (Theorem 4.1) of solutions to our problem. Therefore, all results in this work show a great potential to be applied in various applications of multidisciplinary sciences. The variable order BVPs are important and interesting to all researchers. In other words, 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.