Mixed nonlocal boundary value problem for implicit fractional integro-differential equations via ψ-Hilfer fractional derivative

In this paper, we investigate the existence and uniqueness of a solution for a class of ψ-Hilfer implicit fractional integro-differential equations with mixed nonlocal conditions. The arguments are based on Banach’s, Schaefer’s, and Krasnosellskii’s fixed point theorems. Further, applying the techniques of nonlinear functional analysis, we establish various kinds of the Ulam stability results for the analyzed problem, that is, the Ulam–Hyers stability, generalized Ulam–Hyers stability, Ulam–Hyers–Rassias stability, and generalized Ulam–Hyers–Rassias stability. Finally, we provide some examples to illustrate the applicability of our results.


Introduction
Fractional calculus is a generalization of ordinary differentiation and integration of arbitrary order, which can be noninteger. Differential equations of fractional order have attracted the attention of several researchers; see the monographs [1][2][3][4][5][6][7][8] and references therein. In the literature, there exist several definitions of fractional integrals and derivatives, from the most popular Riemann-Liouville and Caputo-type fractional derivatives to the other ones such as Hadamard fractional derivative, the Erdélyi-Kober fractional derivative, and so forth. A generalization of both Riemann-Liouville and Caputo derivatives was given by Hilfer [9], which is known as the Hilfer fractional derivative D α,β x(t) of order α and type β ∈ [0, 1]. The Hilfer fractional derivative interpolates between the Riemann-Liouville and Caputo derivatives as it reduces to the Riemann-Liouville and Caputo fractional derivatives for β = 0 and β = 1, respectively. The Hilfer fractional derivative is used in theoretical simulation of dielectric relaxation in glass-forming materials and in fractional diffusion equations; see [10,11]. Some properties and applications of the Hilfer derivative can be found in [12][13][14][15][16] and references therein.
The fractional derivative with another function, in the Hilfer sense, called the ψ-Hilfer fractional derivative and introduced in [17], generalizes the Hilfer fractional derivative [9].
The ψ-Hilfer fractional derivative is defined with respect to another function and unifies several definitions of fractional derivatives available in the literature. Thus the ψ-Hilfer fractional derivative covers a wide class of fractional derivatives and provides a platform to obtain a particular one by fixing the function ψ; see Remark 2.4. For some recent results on the existence and uniqueness of solutions of initial value problems and on the Ulam-Hyers-Rassias stability, see [10,11,[18][19][20][21][22][23][24][25][26][27] and references therein.
Nonlocal boundary value problems have become a rapidly growing area of research. The study of this type of problems is driven not only by theoretical interest, but also by the fact that several phenomena in engineering, physics, and life sciences can be modeled in this way. The idea of nonlocal conditions dates back to the work of Hilb [28]. However, the systematic investigation of a certain class of spatial nonlocal problems was carried out by Bitsadze and Samarskii [29]. We refer the reader to [30,31] and references therein for a motivation regarding nonlocal conditions.
The paper is organized as follows: In Sect. 2, we recall some basic and essential definitions and lemmas. In Sect. 3, we obtain the existence and uniqueness results for problem (1.5) via Banach's, Schaefer's, and Krasnosel'skiȋ's fixed point theorems. In Sect. 4, we discuss the Ulam-Hyers, generalized Ulam-Hyers, Ulam-Hyers-Rassias, and generalized Ulam-Hyers-Rassias stability results. Finally, in Sect. 5, we give some examples to illustrate the benefit of our main results.

Background material and auxiliary results
In this section, we introduce some notation, spaces, definitions, and some useful fundamental lemmas.
We denote C[J, R] the Banach space of all continuous functions from an interval J into R with the norm defined by The weighted space C γ ,ψ [J, R] of continuous functions f on J is defined by with the norm where is the gamma function.

Definition 2.2 ([2]
) Let ψ (x) = 0, α > 0, and n ∈ N. The Riemann-Liouville fractional derivatives of a function f with respect to another function ψ of order α is defined by where n = [α] + 1, and [α] represents the integer part of the real number α.
The following lemma presents the semigroup properties of the ψ-Hilfer fractional integral and derivative.
The composition of the ψ-Hilfer fractional integral and derivative operators is given by the following lemma.
Next, we take into account some important properties of the ψ-fractional derivative and integral operators.
On the other hand, for n ≥ k, By Definition 2.1 we obtain Then for all 0 < γ < 1 and t ∈ J.
To transform problem (1.5) into a fixed point problem, problem (1.5) must be converted to an equivalent Volterra integral equation. We provide the following lemma, which is important in our main results and concerns a linear variant of problem (1.5).

Lemma 2.10
is a solution of the linear ψ-Hilfer fractional differential equation equipped with mixed nonlocal conditions

8)
if and only if x satisfies the integral equation Proof Let x be a solution of problem (2.8). By Lemma 2.9 we have where c 1 ∈ R is an arbitrary constant.
Fixed point theorems play a major role in establishing the existence theory for problem (1.5). We collect here some well-known fixed point theorems used in this paper.

Existence results
In this section, we present results on the existence of a solution of problem (1.5).
For simplicity, we set In this paper, the expression I α,ρ 0 + F x (s)(c) means that In view of Lemma 2.10, the operator Q : Note that problem (1.5) has solutions if and only if the operator Q has fixed points. In the following subsection, we establish the existence of solutions for the problem (1.5) by applying Banach's, Schaefer's, and Krasnosel'skiȋ's fixed point theorems.
We list here the necessary assumptions to prove our main results.

Existence and uniqueness via Banach contraction mapping principle
We will first prove the existence and uniqueness of a solution for problem (1.5) by using the Banach contraction mapping principle (Banach's fixed point theorem).
For any x ∈ B ϒ 1 , we have Consider It follows from condition (H 1 ) that Then Thus we get which implies that Qx C 1-γ ,ψ ≤ ϒ 1 . Therefore QB ϒ 1 ⊂ B ϒ 1 .

Existence result via Schaefer's fixed point theorem
The next existence result is based on Schaefer's fixed point theorem.

Theorem 3.2 Let f : J × R 3 → R be a continuous function satisfying (H 2 ). Then problem (1.5) has at least one solution on J.
Proof We show that the operator Q defined in (3.1) has at least one fixed point in C 1-γ ,ψ . The proof is divided into four steps.
Step I. The operator Q is continuous. Let x n be a sequence such that x n → x in C 1-γ ,ψ . Then for each t ∈ J, we obtain Since f is a continuous, this implies that F x is also continuous. Hence we obtain Qx n -Qx C 1-γ ,ψ → 0 as n → ∞.
Step II. The operator Q maps bounded sets into bounded sets in C 1-γ ,ψ .
For ϒ 2 > 0, there exists a constant μ > 0 such that, for each x ∈B ϒ 2 = {x ∈ C 1-γ ,ψ : Indeed, for any t ∈ J and x ∈B ϒ 2 , we have It follows from condition (H 2 ) that and thus (3.17) Then by substituting (3.17) into (3.16) we get from which we get Step III. The operator Q maps bounded sets into equicontinuous sets of C 1-γ ,ψ . For 0 ≤ t 1 < t 2 ≤ T and x ∈B ϒ 2 whereB ϒ 2 is as defined in Step II, since f is bounded on the compact set J ×B 3 ϒ 2 , we have Set sup (t,u,v,w)∈J×B 3 creasing function on t ∈ (0, T), it follows that This inequality is independent of x and tends to zero as t 2 → t 1 , which implies that (Qx)(t 2 ) -(Qx)(t 1 ) C 1-γ ,ψ → 0 as t 2 → t 1 . Thus, Steps I to III, together with the Arzelá-Ascoli theorem, we conclude that the operator Q is completely continuous.

It follows from
Step II that for each t ∈ J, Qx C 1-γ ,ψ ≤ μ < ∞. This implies that the set C 1-γ ,ψ is bounded. By all hypotheses of Theorem 3.2 we conclude that there exists a positive constant N such that x C 1-γ ,ψ ≤ N < ∞. By Schaefer's fixed point theorem (Lemma 2.12) the operator Q has at least one fixed point, which is a solution of problem (1.5). This completes the proof.

Existence result via Krasnosel'skiȋ's fixed point theorem
By using Krasnosel'skiȋ's fixed point theorem, we obtain the final existence theorem. Proof Let sup t∈J |q(t)| = q * . By choosing a suitable B ϒ 3 := {x ∈ C 1-γ ,ψ : where ϒ 3 ≥ |A| | | + q * , we define the operators Q 1 and Q 2 on B ϒ 3 by Note that Q = Q 1 + Q 2 . For any x, y ∈ B ϒ 3 , we have This implies that Q 1 x + Q 2 y ∈ B ϒ 3 , which satisfies assumption (i) of Lemma 2.13. We now show that assumption (ii) of Lemma 2.13 is satisfied. Let x n be a sequence such that x n → x in C 1-γ ,ψ . Then for each t ∈ J, we have Since f is continuous, this implies that the operator F x is also continuous. Hence we obtain This shows that the operator Q 1 x is continuous, since Q 1 x n -Q 1 x 1-γ ,ψ → 0 as n → ∞. Also, the set Q 1 B ϒ 3 is uniformly bounded as Next, we prove the compactness of Q 1 . For each t 1 , t 2 ∈ J with 0 ≤ t 1 < t 2 ≤ T, we have (see Step III of Theorem 3.2) Obviously, the right-hand side in this inequality is independent of x and tends to zero as t 2 → t 1 . Therefore the operator Q 1 is equicontinuous, and so by the Arzelà-Ascoli theorem, Q 1 is relatively compact. Moreover, it is easy to prove using condition (3.18) that the operator Q 2 is a contraction mapping, and thus assumption (iii) of Lemma 2.13 holds. Thus all the assumptions of Lemma 2.13 are satisfied. So the conclusion of Lemma 2.13 implies that the boundary value problem (1.5) has at least one solution on J. The proof is completed.
Before stating the main theorem, we need the following definitions. Let > 0, and let B : [0, T] → [0, ∞) be a continuous function. We consider the following inequalities:  Remark 4.6 A function z ∈ C 1 1-γ ,ψ (J, R) is a solution of inequality (4.1) if and only if there exists a function w ∈ C 1-γ ,ψ (J, R) (dependent on z) such that: Firstly, we present an important lemma that will be used in the proofs of UH and GUH stability.
is a solution of inequality (4.1), then z is a solution of the inequality where and is given by (3.7).
Proof Let z be a solution of inequality (4.1). So, in view of Remark 4.6(ii) and Lemma 2.10, Thus the solution of (4.9) is of the form Then by using (i) of Remark 4.6 it follows that from which we obtain inequality (4.8). The proof is completed.
Now we present the UH and GUH stability results. Proof Suppose that > 0 and z ∈ C 1 1-γ ,ψ (J, R) is any solution of inequality (4.1), that is, Let x ∈ C 1 1-γ ,ψ (J, R) be the unique solution of problem (1.5). Then we have By applying the triangle inequality we get (4.10) By using Lemma 4.7 with (4.10) we obtain This implies that Hence problem (1.5) is UH stable. Now setting B = τ such that B(0) = 0 yields that problem (1.5) is GUH stable. The proof is completed. Remark 4.9 Let B ∈ C 1-γ ,ψ (J, R + ) be an increasing function. Then there exists λ B > 0 such that for each t ∈ J, we have the integral inequality (4.11) Lemma 4.10 Let α ∈ (0, 1] and ρ ∈ [0, 1). If z ∈ C 1 1-γ ,ψ (J, R) is a solution of inequality (4.2), then z is a solution of the inequality where Proof From Lemma 4.7, using Remarks 4.6(i) and 4.9, we obtain from which we obtain inequality (4.12). This completes the proof.
Next, we are ready to prove UlHR and GUHR stability results. Proof Let z ∈ C 1 1-γ ,ψ (J, R) be a solution of inequality (4.2), and let x be the unique solution of problem (1.5). By applying the triangle inequality and Lemma 4.10 we get where 1 is defined by (3.6), which implies that By setting we get the inequality (4.14) Hence problem (1.5) is UHR stable. Moreover, if we set = 1 in (4.14) with B(0) = 0, then problem (1.5) is GUHR stable. The proof is completed.

Examples
In this section, we present two examples, which illustrate the validity and applicability of main results.
Hence by Theorem 4.8 problem (5.1) is both UH and GUH stable.

Conclusion
We have proved the existence and uniqueness of a solution for a class of ψ-Hilfer fractional integro-differential equations with mixed nonlocal conditions. We emphasize that the nonlocal boundary condition is very general, including multipoint, fractional derivative multiorder, and fractional integral multiorder conditions. We used the fixed point theorems of Banach, Schaefer, and Krasnosel'ski to investigate the existence and uniqueness of solutions. Our results are not only new in the given setting but also provide some new special cases by fixing the parameters involved in the problem at hand. For instance, by fixing ω j = 0, λ k = 0 for all j = 1, 2, . . . , n, k = 1, 2, . . . , r our results correspond to boundary value problems for ψ-Hilfer nonlinear fractional integro-differential equations supplemented with multipoint boundary conditions. In case we take δ i = 0, λ k = 0 for all i = 1, 2, . . . , m, k = 1, 2, . . . , r, we obtain results for boundary value problems for ψ-Hilfer nonlinear fractional integro-differential equations equipped with multiterm integral boundary conditions.
We also investigated different kinds of Ulam stability, such as the Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability, and generalized Ulam-Hyers-Rassias stability. The obtained results are well illustrated by examples.
The work accomplished in this paper is new and enriches the literature on boundary value problems for nonlinear ψ-Hilfer fractional differential equations.