Nonlocal integro-multistrip-multipoint boundary value problems for ψ ∗ -Hilfer proportional fractional di ff erential equations and inclusions

: In the present paper, we establish the existence criteria for solutions of single valued and multivalued boundary value problems involving a ψ ∗ -Hilfer fractional proportional derivative operator, subject to nonlocal integro-multistrip-multipoint boundary conditions. We apply the fixed-point approach to obtain the desired results for the given problems. The obtained results are well-illustrated by numerical examples. It is important to mention that several new results appear as special cases of the results derived in this paper (for details, see the last section)


Introduction
Fractional differential equations have been of great interest during the last few years as such equations provide appropriate mathematical models for real world problems arising in physics, engineering, economics, robotics, control theory, etc.A systematic development of fractional calculus and fractional differential equations can be found in the monographs [1][2][3][4][5].Unlike the classical derivative operator, one can find a variety of its fractional counterparts, such as the Riemann-Liouville, Caputo, Hadamard, Erdelyi-Kober, Hilfer, Caputo-Hadamard, etc.In [6][7][8], the authors discussed the concept of a proportional fractional derivative and its generalizations.The authors in [9] proposed a Hilfer type generalized proportional fractional derivative.In [10], the authors discussed a generalization of the ψ * -Hilfer fractional derivative.
The objective of the present work is to enrich the literature on boundary value problems involving ψ * -Hilfer fractional proportional derivative operators.In precise terms, we consider and investigate a new problem consisting of a ψ * -Hilfer fractional proportional differential equation and nonlocal integro-multistrip-multipoint boundary conditions given by denotes the ψ * -Hilfer fractional proportional derivative operator of order ρ ∈ (1, 2] and type φ ∈ [0, 1], ϑ * ∈ (0, 1], a 1 < ζ j < ξ i < η i < b 1 , φ i , θ j ∈ R, i = 1, 2, . . ., n, j = 1, 2, . . ., m, ψ * : [a 1 , b 1 ] → R is an increasing function with ψ accomplished in the present study are novel and give rise to some new results as special cases (for details, see Section 5).We also demonstrated the application of the main results by constructing numerical examples.
The rest of the paper is constructed as follows.In Section 2, some basic definitions and preliminary results related to our work are recalled.Section 3 contains the existence and uniqueness results for the single valued problem (1.1), while the existence results for the multi-valued analogue of the problem (1.1) are proved in Section 4. The paper concludes with some interesting observations.

Single-valued case
In this section we will establish existence and uniqueness results for the boundary value problem (1.1).
Then, σ is a solution of the linear nonlocal integro-multistrip-multipoint ψ * -Hilfer generalized proportional fractional boundary value problem if and only if Proof.From Lemma 2.2 with n = 2, we have where and Using (3.4) in the condition σ(a 1 ) = 0, we get c 1 = 0 since γ 1 ∈ [ρ, 2].Hence, (3.4) takes the form: Inserting (3.5) in the condition: ds which, together with notation (3.1), yields Substituting the above value of c 0 in (3.5) leads to the solution (3.3).The converse of the lemma can be established by direct computation.□ Denote by the Banach space of all continuous functions from [a 1 , b 1 ] to R endowed with the norm In view of Lemma 3.1, we define an operator S: X → X as For convenience, in the sequel, the following notation is used: .

Uniqueness result
Here, we establish the existence of a unique solution for the nonlinear nonlocal integro-multistrip-multipoint ψ * -Hilfer generalized proportional fractional boundary value problem (1.1) by using Banach's fixed point theorem [19].
Theorem 3.1.Assume that: where Ω is given by (3.Proof.Let Then, by (H 1 ), we have We give the proof in two steps.
Step I: Consider B r = {σ ∈ X : ∥σ∥ < r} with r ≥ Ψ 0 Ω/(1 − Λ 0 Ω).Then, we show that S(B r) ⊂ B r.For σ ∈ B r and using Step II: We show that the operator S is a contraction.For σ 1 , σ 2 ∈ X and for any z ∈ [a 1 , b 1 ], we have which, by the assumption Λ 0 Ω < 1, shows that the operator S is a contraction.Hence, by Banach's contraction mapping principle, the operator S has a unique fixed point.Therefore, the nonlocal integro-multistrip-multipoint ψ * -Hilfer fractional proportional boundary value problem (1.1) has a unique solution on [a 1 , b 1 ].□

Existence results
Here we present three existence results which are proved with the aid of Krasnosel'ski ȋ's fixed point theorem [20], Schaefer's fixed point theorem [21] and Leray-Schauder nonlinear alternative [22].Theorem 3.2.Suppose that the continuous function . In addition, we suppose that Then, the nonlinear nonlocal integro-multistrip-multipoint ψ * -Hilfer fractional proportional boundary value problem (1.1) has at least one solution on [a 1 , b 1 ], provided that we consider a ball The operator S given by (3.6) can be decomposed as S = S 1 + S 2 , where For any σ, y ∈ B r 0 , we have where Ω is given by (3.7).Therefore, ∥S 1 σ + S 2 y∥ ≤ r 0 , which shows that S 1 σ + S 2 y ∈ B r 0 .As in the proof of Theorem 3.1, it can be shown by using the condition (3.9) that the operator S 2 is a contraction mapping.Since ψ * is continuous, the operator S 1 is continuous.Moreover, S 1 is uniformly bounded on B r 0 , since ∥ϕ * ∥.
In the final step we will prove that the operator S 1 is completely continuous.For which tends to zero independently of σ ∈ B r 0 when z 1 → z 2 .Consequently, we deduce that S 1 is equicontinuous.Hence, by the Arzelá-Ascoli theorem, it is compact on B r 0 .Thus, the hypotheses of Krasnosel'ski ȋ's fixed point theorem [20] are verified, and hence, its conclusion implies that there exists at least one solution for the nonlinear nonlocal integro-multistrip-multipoint ψ * -Hilfer fractional proportional boundary value problem Then, there exists at least one solution for the nonlinear nonlocal integro-multistrip-multipoint ψ * -Hilfer fractional proportional boundary value problem Proof.We will give the proof in two steps.In the first step, we establish the complete continuity of the operator S: X → X given by (3.6).For the continuity of S, let {σ n } be a sequence such that σ n → σ in X.Then, for each z ∈ [a 1 , b 1 ], we get which proves that S is continuous.Now, we show that S transforms bounded sets into bounded sets in X.For r 0 > 0, let which leads to ∥S(σ)∥ ≤ ΩM.
Finally, we show that S transforms bounded sets into equicontinuous sets.For which tends to zero, independently of σ ∈ B r 0 , as z 1 → z 2 .Thus, by the Arzelá-Ascoli theorem, the operator S: X → X is completely continuous.
In the second step, it will be established that the set Therefore, ∥σ∥ ≤ ΩM, ) and Y: R + → R + is a continuous nondecreasing function; where Ω is defined by (3.7). Then ∥p∥Y(∥σ∥).

Therefore, we have ∥σ∥
where Ω is given in (3.7).By (H 4 ), we have that ∥σ∥ K. Consider the set The operator S: ) is continuous and completely continuous.By the definition of U, we cannot find any σ ∈ ∂U satisfying σ = µS(σ) for some µ ∈ (0, 1).In consequence, by the application of the Leray-Schauder nonlinear alternative [22], we deduce that there exists a fixed point σ ∈ U for the operator S, which is a solution of the problem (1.1).This finishes the proof.□

Multi-valued case
In this section, we study the multi-valued variant of the nonlinear nonlocal integro-multistrip-multipoint ψ * -Hilfer fractional proportional boundary value problem (1.1) given by where H: [a 1 , b 1 ] × R → P(R) is a multi-valued map, P(R) denotes the family of all nonempty subsets of R, and the other symbols are the same as defined in the problem (1.1).
Let (X, ∥•∥) be a normed space.We denote, respectively, the classes of all closed, bounded, compact, and compact and convex sets in X by P cl , P b , P cp and P cp,c .

Case 1: the upper semicontinuous case
Applying the nonlinear alternative for Kakutani maps [22] together with a closed graph operator theorem [26], we establish an existence result for the nonlocal integro-multistrip-multipoint ψ * -Hilfer fractional proportional inclusion boundary value problem (4.1).
Theorem 4.1.Let the following assumptions be satisfied: Proof.Let us introduce a multi-valued operator W: It will be verified through several steps that the operator W satisfies the hypotheses of the Leray-Schauder nonlinear alternative for Kakutani maps [22].
Let σ ∈ B ε and ϖ ∈ W(σ).Then, there exists v ∈ S H,σ such that ) is completely continuous by virtue of the Arzelá-Ascoli theorem.
Consider a continuous linear operator Φ: Clearly, ∥ϖ n − ϖ * ∥ → 0 as n → ∞, and consequently, by the closed graph operator theorem [26], Φ • S H,σ is a closed graph operator.Also, we have ϖ n ∈ Φ(S H,σ n ) and for some v * ∈ S H,σ * .Thus, W has a closed graph.By [23,Proposition 1.2], that is, if a completely continuous operator has a closed graph, then it is upper semicontinuous, we deduce that the operator W is upper semicontinuous.
By (A 3 ), ∥σ∥ M for some M. We define a set Obviously W: ) is a compact, convex valued and upper semicontinuous multivalued map.By the definition of Θ, we cannot find any σ ∈ ∂Θ for some κ ∈ (0, 1) satisfying σ ∈ κW(σ).Therefore, by the Leray-Schauder nonlinear alternative for Kakutani maps [22], the operator W has a fixed point σ ∈ Θ.So, the nonlocal integro-multistrip-multipoint ψ * -Hilfer fractional proportional inclusion boundary value problem Here, we discuss the existence of solutions for the integro-multistrip-multipoint ψ * -Hilfer fractional proportional inclusion boundary value problem (4.1) with a possible non-convex valued multi-valued map.The main tool of our study in this case is a fixed point theorem for contractive multivalued maps due to Covitz and Nadler [27].Theorem 4.2.Assume that Then, the nonlocal integro-multistrip-multipoint ψ * -Hilfer fractional proportional inclusion boundary value problem (4.1) has at least one solution on where Ω is given by (3.7).
Proof.Consider the operator defined by (4.2).We complete the proof in two steps.
Step I. W is nonempty and closed for every v ∈ S H,σ .By the measurable selection theorem ([29, Theorem III.6]), the set-valued map H(•, σ(•)) is measurable and admits a measurable selection v: [a 1 , b 1 ] → R. By the assumption (B 2 ), we get Since H has compact values, we can obtain a subsequence (if necessary) Thus, u ∈ W(σ).
Step II.Here, we establish that there exists 0 < m 0 < 1 (m 0 = Ω∥ϱ∥) such that By (B 2 ), we have Since the multivalued operator V(z) ∩ H(z, σ(z)) is measurable by Proposition III.4 in [29], there exists a function v 2 (z) which is a measurable selection of V. Thus, v 2 (z) ∈ H(z, σ(z)), and for each In consequence, we obtain On switching the roles of σ and σ, we have which verifies that W is a contraction.Hence, it follows by Covitz-Nadler's fixed point theorem [27] that the operator W has a fixed point σ, which is indeed a solution of the nonlocal integro-multistripmultipoint ψ * -Hilfer fractional proportional inclusion boundary value problem (4.1).□

Illustrative examples: the multi-valued case
Let us consider the ψ * -Hilfer fractional proportional inclusion boundary value problem:

Conclusions
In this paper, we have presented the criteria ensuring the existence and uniqueness of solutions for a ψ * -Hilfer fractional proportional differential equation complemented with nonlocal integro-multistrip-multipoint boundary conditions.The desired results for the given problem are derived by applying the fixed point theorems due to Banach, Krasnosel'ski ȋ, Schaefer and Leray-Schauder alternative.Also, two existence results for the ψ * -Hilfer fractional proportional differential inclusion problem with nonlocal nonlocal integro-multistrip-multipoint boundary conditions are proved when the multivalued map takes convex as well as non-convex values.Examples are constructed for illustrating all the abstract results presented in this paper.We emphasize that our results are new and contribute to the literature on the nonlocal integro-multistrip-multipoint boundary value problems involving ψ * -Hilfer fractional proportional differential equations and inclusions.
In the future, we plan to investigate the systems of ψ * -Hilfer fractional proportional differential equations and inclusions equipped with nonlocal integro-multistrip-multipoint boundary conditions.
[21]equently the set E is bounded.Hence, by Schaefer's fixed point theorem[21], the operator S has at least one fixed point which is a solution for the nonlinear nonlocal integro-multistrip-multipoint ψ * -Hilfer fractional proportional boundary value problem (1.1) on [a 1 , b 1 ].This completes the proof.