Uniqueness of solutions for a ψ - Hilfer fractional integral boundary value problem with the p - Laplacian operator

: In this article, we discuss the existence of a unique solution to a ψ - Hilfer fractional di ﬀ erential equation involving the p - Laplacian operator subject to nonlocal ψ - Riemann - Liouville fractional integral boundary conditions. Banach ’ s ﬁ xed point theorem is the main tool of our study. Examples are given for illustrating the obtained results.


Introduction
Fractional differential operators are found to be of great utility in the mathematical modeling of natural and engineering phenomena, for example, see [1][2][3][4][5][6] and the references cited therein.In contrast to the integerorder differential operators, these operators are nonlocal in nature and account for the history of the physical phenomena under consideration.In the literature, there do exist several kinds of fractional integrals and derivatives, for instance, see [7][8][9][10].Hilfer in [11] proposed a generalized fractional derivative of order α and type β, which is known as the Hilfer fractional derivative, and it interpolates between the Riemann-Liouville and Caputo derivatives.This two-parameter fractional derivative operator appeared in the modeling of diffusion models, dielectric relaxation in glass-forming materials, etc. [12,13].For some recent works on Hilfer fractional differential equations, we refer the reader to the articles [14][15][16][17][18].In [19], Sousa and Capelas de Oliveira generalized the Hilfer fractional derivative by introducing the concept of ψ-Hilfer fractional derivative.One of the advantages of this derivative is that it covers a wide class of fractional derivatives, which can be fixed by choosing the function ψ appropriately.Later, this derivative gained much attention, and many researchers turned to investigate it, for example, see [20][21][22][23][24][25][26].
On the other hand, differential equations with the p-Laplacian operator appeared for the first time when Leibenson [27] was attempting to derive an accurate formula to model turbulent flow in the porous medium.Keeping in mind the application of differential equations involving the p-Laplacian operator in the areas of mechanics, nonlinear dynamics, glaciology, nonlinear elasticity, flow through porous media, and so on, many authors focused on this topic.For details and examples, one can see the article [28][29][30][31][32].As far as we know, there is no work dealing with ψ-Hilfer fractional differential equations with the p-Laplacian operator.
The objective this study is to investigate a ψ-Hilfer fractional boundary value problem involving the p-Laplacian operator ( ) ⋅ ϕ p and ψ-Riemann-Liouville fractional integral boundary conditions given as follows: where , , denotes the ψ-Hilfer fractional derivative operator of order is a continuous function.The rest of the article is organized as follows: Section 2 presents the background material related to our work, Section 3 contains the main results for problem (1), and Section 4 presents examples to illustrate the obtained results., into .In the forthcoming analysis, it is assumed that ψ is an increasing and positive monotone function on ( ] 0, 1 possessing a continuous derivative ( ) ′ ≠ ψ t 0 on ( ) 0, 1 .

Preliminaries
, is defined as follows: , , is given as follows: n is defined as follows: , the left-sided ψ-Hilfer frac- tional derivative for a function n is defined as follows: which can alternatively be written as where In the following lemma, we solve the linear variant of problem (1).
) ∈ h C 0, 1 , , the integral representation of the solution for the following linear ψ-Hilfer p-Laplacian fractional integral boundary value problem: is as follows: where it is assumed that , problem (9) can be split into two problems: and Uniqueness of solutions for a ψ-Hilfer fractional integral boundary value problem  3 Solving (11), we obtain ( ) ( ) . Applying + I α ψ 0 , to the ψ-Hilfer p-Laplacian fractional differential equation in (12), we obtain where c 0 and c 1 are arbitrary constants.Using the condition ( ) = y 0 0 in (13) yields = c 0 0 .Then, from (13), we have and Inserting ( 14) and (15) in the condition: Substituting the values of c 0 , c 1 , and ( ) ( ) 13), we obtain the solution (10).This completes the proof. □ The following lemma provides bounds for the p-Laplace operator, which can easily be proved by using the mean value theorem when the function Moreover, , and 2 , and

Main results
This section is devoted to the uniqueness results for problem (1) for different values of p. Consider the following composition operator: where is defined as follows: and is given as follows: is a continuous operator.For computational convenience, we set Now, we present our main results, which will be proved with the aid of the Banach contraction mapping principle and Lemma 2.
Theorem 1. Assume that < ≤ p 1 2 and the following conditions hold: (A 1 ) There exist a nonnegative integrable function g on [ ] 0, 1 and a positive constant M such that (A 2 ) There exists a positive constant k such that Then, the boundary value problem (1) has a unique solution provided that where Ω is given by (20).
Proof.By the assumption (A 1 ), we have , where the operator is defined by (17).For ∈ y B r , it follows by the definition of ( )

Consequently, we obtain
Uniqueness of solutions for a ψ-Hilfer fractional integral boundary value problem  5 Then, using Lemma 3 and the fact that q t q t q t t q In consequence, we obtain Taking the norm of the above inequality for which together with the condition (21) implies that the operator is a contraction.Hence, by the Banach's contraction mapping principle, the operator has a unique fixed point.Hence, problem (1) has a unique solution on [ ] 0, 1 .This completes the proof.□ Theorem 2. Assume that > p 2 and the following conditions hold: (A 3 ) There exist constants > m 0 and δ with < ≤ − δ 0 (A A ) There exists a constant > k 0 such that where Ω is given by (20), then there exists a unique solution to the boundary value problem (1) on [ ] 0, 1 .

Proof. By ( )
A 3 , we have ¯, where the operator is defined by (17).Using the definition of ( ) ⋅ ϕ q and the values of r ānd q, we have for ∈ y B r ¯that As in the previous theorem, one can obtain Next, it will be shown that is a contraction.Let q t q t δ q t t q δ q q Thus, it follows from condition (22) that the operator is a contraction.Hence, by Banach's contraction principle, the operator has a unique fixed point, which is indeed a unique solution to problem (1) on [ ] 0, 1 .This finishes the proof.□ Theorem 3. Let > p 2. Assume that ( ) A 4 and the following condition are satisfied: (A 5 ) There exist a number > m 0 and < ≤ − δ 0 Proof.Since the proof is similar to that of the last theorem, we omit it.□

Examples
Consider the following problem: all n-times continuously differential functions from [ ] a b

δ 1 Then, the boundary value problem ( 1 )
has a unique solution on [ ] 0, 1 if k satisfies