Problem Involving the Riemann–Liouville Derivative and Implicit Variable-Order Nonlinear Fractional Differential Equations

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Introduction
Te subject of fractional calculus has gained much attention and importance among the society of researchers.Te existing diferential equations in this theory are determined by generalizing integer-order derivatives to arbitrary order ones.For the sake of the efective memory of the fractional derivation operator, such classes of equations have been widely used in mathematical modeling including parameter identifcation in the 2D fractional system and modeling of heat distribution in porous aluminum (see [1][2][3][4][5]).
Ulam-type stability is often studied in the context of various types of diferential equations, including ordinary, partial, fractional, and integrodiferential equations, among others.Tis concept has broad applications in various felds of science and engineering, such as the control theory, signal processing, and physics, where the stability of solutions under small perturbations is a crucial consideration [40].
In [41,42], the presence of implicit nonlinear diferential equations involving fractional of constant order is studied by Benchohra et al.
Te organization of this paper is outlined as follows.Basic and crucial topics, including defnitions and theorems, are covered in Section 2. Te main results, which ofer two main theorems of uniqueness and existence, are found in Section 3. Section 4 discusses stability in the sense of Ulam and Hyres.One example is presented in Section 5 to show the efciency and validity of the proposed results.Finally, some conclusion notes are given in Section 6.

Preliminaries
Tis section introduces a few crucial, essential defnitions that are necessary to understand in order to get our outcomes in the next sections.
Te representation of the space of continuous Banach functions In the case of Te function Ψ 1 (s) is, therefore, represented by the variableorder left Riemann-Liouville fractional integral (FIRL) u(s) (see [32,44,46]).
As expected, FDRL and FIRL correspond to the usual Riemann-Liouville fractional derivative and integral, together, in the case where u(s) and v(s) are constant; for example, see [43,44,46].
Remember the succeeding crucial fnding.
Defnition 5 (see [49][50][51]).If I of R is either an interval, a point a 1  , or the empty set, it is referred to as a generalized interval.A partition of I is a fnite set P if every x in I falls in precisely one of the generalized intervals E in P. If a function g: I ⟶ X is constant on E for every E ∈ P, it is referred to as a piecewise constant with regards to the partition of I.
Theorem 6 (see [43]).Let G be a Banach space and P be a convex subset of Gand L: P ⟶ P is compact and is the continuous map.Ten, L has at least one fxed point in P.
Defnition 7 (see [52]).If there is a real number c Z > 0 such that (for each) ϵ > 0 and (for each) solution χ ∈ C(c m , R), then problem (1) is Ulam-Hyers stable.For the price of the inequality

Main Results
Let us state the underlying presumptions: where 1 < u m ≤ 2 are constants, and We frst provide an essential study concerning (2) in order to arrive at our main fndings.
For any s ∈ (Υ m− 1 , Υ m ], m � 1, . . ., n, the FDRL of the variable order u(s) for κ(s) ∈ C(c, R), given by ( 5), is the sum of the FDRLs of the constant orders u 1 , . . ., u m , i.e., Tus, according to (19), the (2) of the variable order can be written for any s Here is a defnition of the solution to PVB (2).Defnition 8.If there are functions κ m (m � 1, 2, . . ., n), then we can say that the boundary value problem (1) has a solution such that κ m ∈ C([η, Υ m ], R) satisfying equation (5) and According to the abovementioned analysis, the equation of PVB (2) can be expressed as (??), which is translated as (20) (20) is written as follows: Now, we consider the following boundary value problem: We utilize auxiliary lemma to prove that there are solutions to problem (22).

Lemma 9. If and only if a function κ ∈ E m holds the following integral equation:
it forms the solution to problem (22), where ) is the Green's function defned as follows: Proof.Let κ ∈ E m be a solution of the PVB (22).Now, we take operator to each of these sides of (22).According to Lemma 1, we obtain By κ(T m− 1 ) � 0 and the assumption of function ℏ, we could get Te problem's solution (22) is then provided by For the implied continuity of Green's function, On the other hand, consider κ ∈ E m as the integral equation's solution (23).Ten, it is evident that κ is the solution to problem (22) due to the continuity of the function t ρ ℏ and Lemma 1.
According to Teorem 6, our frst existence result is as follows: Theorem 11.Suppose the hypotheses (P1) and (P2) are valid and the following inequality is satisfed Ten, at least one solution for the PVB (6) exists on E m .
Proof.Consider the operator defned by where Te operator S 1 : E m ⟶ E m defned in ( 32) is well defned, as evidenced by the characteristics of fractional integrals and the continuity of the function s ρ ℏ.
Now, let us consider where We pay regard to the set Tere is no doubt that B R m is closed, convex, bounded, and nonempty.
We will now demonstrate that S 1 meets the Teorem 11's fundamental assumption.Tree steps will be taken to provide the proof.
Let κ ∈ B R m and s ∈ c m .From Proposition 10, we have where By P2, we have Ten, Tus, Step 13. S 1 is continuous.Te sequence (κ n ) is assumed to converge to κ in E m and s ∈ c m .Due to Proposition 10, we have where Ten, P2 gives us Tus, Hence, i.e., we obtain Te operator S 1 is hence a continuous on E m .
Step 14. S 1 (B R m ) is relatively compact.
We must now demonstrate that S 1 (B R m ) is relatively compact.Due to step 13, it is obvious that S 1 (B R m ) is uniformly bounded.As a result, we obtain For s 1 , s 2 ∈ c m , s 1 < s 2 and κ ∈ B R m , we have Complexity where Ten, considering Green's functions' continuity G m .Tus, | (S 1 κ) We arrived at the conclusion that S 1 is completely continuous due to steps 12 through 14 and the Arzela-Ascoli theorem.
Problem (22) has at least one solution (  κ m in B R m ) according to Teorem 11.
We let Tus, we know that for which s ∈ c m , indicating that κ m is a solution of (20) with which is a form of the solution of PVB (2).Now, the following result can be defned using the Banach contraction principle.

Theorem 15. Let us consider the assumptions (P1), (P2) hold and
Problem (22) then has a solution on E m .
Proof.Te Banach contraction principle demonstrates that S 1 has a singular fxed point as stated in (32).
We have for κ, y ∈ E m , and s ∈ c m . where By P2, we have Ten, 8 Complexity Tus, Consequently, by (55), the operator S 1 is condensed.As a result, according to the Banach contraction principle, S 1 has a singular fxed point  κ m ∈ E m , which is a singular solution to problem (22).
We let We are clear that the following equation, which is defned by (61), is satisfed by for s ∈ c m , which denotes that κ m is a singular solution of (20) with is the form of unique solution of PVB (1).
Proof.Let χ ∈ E m be a solution of the inequality.So, we have Let us use κ ∈ E m to represent the problem's singular solution (22).By using Lemma 9, we have where By integration of (64), we also obtain where Complexity 9 However, we also have According to P2, for any s ∈ c m , we have Ten, Tus, Hence, we get Ten, it is obtained that for each t ∈ c m .So, problem ( 22) is SUH.Consequently, the equation of ( 2) is SUH.

Example
Let us look at the following fractional boundary problem: Ten, we get As a result, condition P2 is satisfed when c � 1/2 and ϖ 1 � ϖ 2 � 1/2 are used.
Te equation for problem (75) is split into two parts as follows by (77):

Complexity
For s ∈ c 1 , problem (75) is equivalent to the following problem: We will ascertain whether or not condition (55) is met.
It can be seen that

Conclusion
Te semigroup properties of the Riemann-Liouville fractional integral have played a key role in dealing with the existence of solutions to diferential equations of fractional order.Based on some results of some experts, we know that the Riemann-Liouville variable order fractional integral does not have semigroup property, thus bringing us extreme difculties in considering the existence of solutions of variable order fractional diferential equations.In this work, we presented results about the existence and the uniqueness of solutions for implicit nonlinear fractional diferential equations of variable order u(t), where u(t): [η, Υ] ⟶ (1, 2] is a piecewise constant function.
All our results are based on the Schauder's fxed-point theorem and the Banach contraction principle.Lastly, we conducted a research on SUH, our problem's stability; fnally, we illustrated the theoretical fndings by an example.
All the results in this work show a great potential to be applied in various applications of sciences.Moreover, we will extend our studies in reducing chaos and stabilising the system of the utilisation of a Chua oscillator in the future.