Boundary value problem of weighted fractional derivative of a function with a respect to another function of variable order

This study aims to resolve weighted fractional operators of variable order in specific spaces. We establish an investigation on a boundary value problem of weighted fractional derivative of one function with respect to another variable order function. It is essential to keep in mind that the symmetry of a transformation for differential equations is connected to local solvability, which is synonymous with the existence of solutions. As a consequence, existence requirements for weighted fractional derivative of a function with respect to another function of constant order are necessary. Moreover, the stability with in Ulam–Hyers–Rassias sense is reviewed. The outcomes are derived using the Kuratowski measure of non-compactness. A model illustrates the trustworthiness of the observed results.


Introduction
The fractional calculus has gained prominence in recent decades due to the variety of applications in diverse areas of science and engineering [1,10,12,19].The Riemann-Liouville and Caputo fractional derivatives exist in the majority of commonly used fractional operators (with singular kernels).Nevertheless, there are additionally different kinds of fractional operators that help researchers in their endeavors to grasp many phenomena in the world around us, we refer to the ones in citations [6, 14-16, 18, 25].Lately, fractional integration and derivation of variable orders has also been explored.See, for instance, [20,29].
The solvability of differential equations represents one of the most important issues in differential equations.There are multiple techniques for analyzing the existence, such as Lie group symmetry [9,24,26].Throughout this document, we use integral equivalence to confirm the existence result for the bvp ψ-wfd with variable order.Many authors have set up and studied bvps for numerous forms of fractional differential equations [2,21].
While many other research works on the existence of solutions to fractional constant order problems have been carried, the existence of solutions to variable-order problems is infrequently mentioned in the literature, and there have been only a few research papers on the stability of solutions; we refer to [13,22,23,27,29].As a result of investigating this intriguing special research topic, our findings are novel and notable.
The weighted fractional differential of a function with constant order operators have recently gained popularity. Refs [3,4,17].In this paper, we will study the boundary value problem for ψ-wfd of variable order (Bvpwfdvo) The ψ-wfi of variable order σ (ζ ) : L → (n -1, n] for a function f has the form The corresponding derivative in Riemann-Liouville settings is where the weight w(ζ

Preliminaries
Before we begin, let us notate and make some abbreviation to avoid repetition.
K-mnc-Kuratowski measure of non-compactness; ψ-wfd-weighted fractional differential equation of function with respect to function ψ, ψ-wf-weighted fractional integral equation of function with respect to function ψ; bvp-boundary value problem; and UHRs stads for Ulam-Hyers-Rassias stable.
In this section, we begin by introducing several terms and conceptual results, which will be employed across the document.
Let L = [1, ] be a compact interval and denote by C(L, R) the Banach space of continuous functions y : L → R with the usual norm We define the weighted Banach space Remark 2.1 It is worth noting that the semigroup property is satisfied for a standard ψwfd for constant orders, but not for the general case with variable orders σ (ζ ), (ζ ), i.e., In what follow, for all δ ∈ [0, 1] and ζ , s ∈ (0, ] with ζ ≥ s, we pose (1,2]) and there exists a number δ ∈ [0, 1] such that h ∈ C w (L, R), then the fractional integral variable order From the definition (1.1), applying that the function ψ δ (•, 0) is an increasing function on L for δ ∈ (0, 1], we obtain which confirms that the ψ-wfi of variable order for the function h (I w h) exists for any ζ ∈ L.
• The set I is called a generalized interval if it is either an interval or a point or the empty set.
• The finite set P of generalized intervals is called a partition of I if each x in I lies in exactly one of the generalized intervals E in P.
• The function g : I → R is called a piecewise constant with respect to partition P of I if for any E ∈ P, g is constant on E.
In the following, we recall some important and necessary information about the K-mnc.

Definition 2.7 ([7])
Let M X be the bounded subsets of a Banach space X.The K-mnc ϑ is a mapping ϑ : M X → [0, ∞] initially derived from a construction as laid out in the following format where The K-mnc satisfied the following properties: Proposition 2.8 ([7, 8]).Let D, D 1 , D 2 be a bounded subsets of a Banach space X, then: Lemma 2.9 ([11]) Let X be a Banach space.If U is a bounded and equicontinuous subset of the the space C(L, X) of continuous functions, then: where ϑ(U) is the K-mnc on the space C(L, X).

Theorem 2.10 ([7] (DFPT))
If is nonempty, bounded, convex and closed subset of a Banach space X, and : − → is a continuous operator satisfying i.e., is k-set contractions, then has at least one fixed point in .Definition 2.11 Let the function ρ ∈ C(L, R + ).The (Bvpwfdvo) is UHRs with respect to ρ if there exists a constant c f > 0 such that for any ε > 0 and for every z ∈ C(L, R) such that there exists a solution h ∈ C(L, R) for (Bvpwfdvo) satisfying

Existence solutions of (Bvpwfdvo)
Let us proceed with the following assumption: Hypothesis 1 (H1) Let n ∈ N be such an integer and a finite point sequence {ζ j } n j=0 be given in such a way 0 , where 1 < σ l ≤ 2 are constants and 1 l is the indicator of the interval L l , l = 1, 2, . . ., n: Then, for any ζ ∈ L l , l = 1, 2, . . ., n, the ψ-wfd of variable order σ (ζ ) for function h ∈ C w (L, R), defined by (1.2), could be presented as a sum of ψ-wfd constant orders σ j , j = 1, 2, . . ., l.
Thus, the equation of the bvp of ψ-wfd of variable order can be written for any ζ ∈ L l , l = 1, 2, . . ., n in the form and it solves integral Equation (3.1).Then (3.1) is reduced to Taking into account the above for any = 1, 2, . . ., n, we consider the following auxiliary bvp for ψ-wfd of constant order , and there exists a number δ ∈ (0, 1)

.2)
Proof Let h ∈ E be a solution of the problem (Bvpwfdco).Using the operator ζ -1 I σ w to both sides of the equation in the problem (Bvpwfdco), we find (see Theorem 2.5) where a 1 , a 2 are two constants.
Based on the operating environment h as well as the boundary condition h(ζ -1 ) = 0, we conclude that a 2 = 0.
Based on the boundary condition h(ζ ) = 0, we obtain Then, we find h solves integral Equation (3.2).
In contrast, suppose h ∈ E be a solution of integral Equation (3.2).In respect of the continuity w(ζ )ψ δ (ζ , 0)f (ζ ), we deduce that h is the solution of problem (Bvpwfdco).Theorem 3.2 Let the conditions of Lemma 3.1 be satisfied and there are constants V , W > 0 such that holds, where Then, the (Bvpwfdco) does have at least one solution in E .
We introduce the operator F defined on E by Out from qualities of fractional integrals and from the continuity of function ψ δ (•, 0)w(•)f (•), the above operator F : E − → E is clearly defined.
From the definition of the operator F and Lemma 3.1, we perceive that the fixed points of F are solutions of problem (Bvpwfdco).For this reason, it suffices to verify the axioms of Theorem 2.10, it is done in four steps.
Step 3. F is bounded and equicontinuous.By the first step for h ∈ B , we obtain Fh E ≤ r , which confirm that As an outcome, we acquire For H ∈ B .We denote by ϑ w the K-mnc on E , by utilizing Lemma 2.9 and the third step, we get where According to Inequality (3.3), F is a k-set contraction.
As a matter of fact, all Theorem 2.10 requirements have been met, so as side effect F admits a fixed point F( h ) = h , where h ∈ B , which is a solution of the bvp for ψ-wfd of constant order.Since B ⊂ E , the claim of Theorem 3.2 is established.

Ulam-Hyers-Rassias stability of (Bvpwfdvo)
We present the underlying assertion: For any ∈ {1, 2, . . ., n}, according to Equality (1.2), for ζ ∈ L , we obtain Taking the ζ -1 I σ w of both sides of the Inequality (2.3) and applying (H3), we obtain According to Theorem 3.3, the (Bvpwfdvo) has a solution h ∈ C(L, R) defined by and h ∈ E is a solution of (Bvpwfdco).According to Lemma 3.1, the integral equation which implies that for any ζ ∈ L, we have Then the (Bvpwfdvo) is UHRs.