Integro-Differential Boundary Conditions to the Sequential ψ 1 -Hilfer and ψ 2 -Caputo Fractional Differential Equations

: In this paper, we introduce and study a new class of boundary value problems, consisting of a mixed-type ψ 1 -Hilfer and ψ 2 -Caputo fractional order differential equation supplemented with integro-differential nonlocal boundary conditions. The uniqueness of solutions is achieved via the Banach contraction principle, while the existence of results is established by using the Leray–Schauder nonlinear alternative. Numerical examples are constructed illustrating the obtained results

In addition, the given constants λ i , δ j ∈ R and some points η i , ξ j ∈ (0, x 1 ), I µ j ;ψ 2 is the Riemann-Liouville fractional integral of order µ j > 0, with respect to a function ψ 2 , for i Existence and uniqueness are established via Banach's fixed point theorem and the Leray-Schauder nonlinear alternative.
The novelty of this study lies in the fact that we introduce a new class of nonlocal boundary value problems in which we combine ψ 1 -Hilfer and ψ 2 -Caputo fractional derivative operators and as far as we know, this is the only paper dealing with this combination.By fixing the parameters in the nonlocal integro-differential fractional boundary value problem (1), we obtain some new results as special cases.For example, we get to: (i) Hilfer and Caputo fractional nonlocal integro-differential boundary value problem if The remaining part of this article is organized as follows: in Section 2, some preliminary definitions and results that will be applied in the next sections are recalled.In addition, an auxiliary result is proved to convert the problem (1) into a fixed point problem.In Section 3, the main results for the nonlocal integro-differential boundary value problem (1) are established, while in Section 4, these results are discussed for some special cases.Section 5 includes some numerical examples illustrating the main results.
Remark 1 ([22]).The following relations hold: Lemma 1 ([14]).Let α, µ > 0 and δ > 1 be constants.Then, we have: (i) I α;ψ I µ;ψ ĥ(t) = I α+µ;ψ ĥ(t); we denote the space of k times absolutely continuous functions on [0, x 1 ].)Then, we have where ĥ A linear variant of the sequential Hilfer-Caputo fractional integro-differential boundary value problem (1) is investigated in the next lemma.Lemma 3. Let h ∈ C([0, x 1 ], R) be a given function and all constants are as in boundary value problem (1).Then, the sequential Hilfer-Caputo fractional integro-differential linear boundary value problem is equivalent to the integral equation where it is assumed that Proof.Operating the fractional integral I α;ψ 1 to both sides of the first equation in (2) and applying Lemma 2, we obtain for t ∈ [0, where we have c 0 = 0. Therefore, we get which leads to Acting I γ;ψ 2 in (5) yields In addition, we have From the second boundary condition (2) with ( 6) and ( 8), we get where A is defined in (4).Substituting the value of c 1 in ( 7), we get the solution (3).On the other hand, by taking the fractional differential operator of ψ 2 -Caputo and ψ 1 -Hilfer of orders γ and α, respectively, we get the first equation in problem (2).By direct computation, it is easy to see that (3) satisfies the two boundary conditions in (2).Therefore, the proof is completed.

Main Results
In this section, we establish existence and uniqueness of solutions to the sequential Hilfer-Caputo fractional integro-differential boundary value problem (1) on an interval J = [0, x 1 ].At first, we denote the Banach space of all continuous functions from J to R equipped with the norm π = sup{|π(t)| : t ∈ J} by C = C(J, R).Having in mind Lemma 3, we define an operator where For convenience, we put In the following theorem, we prove the existence and uniqueness of solutions of the fractional integro-differential boundary value problem of sequential Hilfer and Caputo fractional derivatives (1) by applying the Banach contraction mapping principle.
where A 1 is given by (11).Then, the fractional integro-differential boundary value problem of sequential Hilfer and Caputo fractional derivatives (1) has a unique solution on J.
Now, we will show that WB r * ⊆ B r * .For any π ∈ B r * , we obtain which holds from (14).This shows that WB r * ⊆ B r * .Next, we let π 1 , π 2 ∈ B r * , then we have Therefore, the operator W satisfies the inequality Wπ W is a contraction.Therefore, the operator W has a unique fixed point in the ball B r , by Banach's contraction mapping.Consequently, the sequential Hilfer-Caputo fractional integro-differential boundary value problem (1) has a unique solution on J.
Next, the nonlinear alternative of the Leray-Schauder-type [23] is used to prove the existence of at least one solution to the sequential Hilfer-Caputo fractional integrodifferential boundary value problem (1).
Theorem 2. Assume that Π : J × R → R is a continuous function satisfying the conditions: (H 2 ) There exists a continuous function Ω : [0, ∞) → (0, ∞) which is nondecreasing and u 1 , u 2 : J → R + two continuous functions such that for all t ∈ J and π ∈ R; (H 3 ) There exists a positive constant K such that Then, the sequential Hilfer-Caputo fractional integro-differential boundary value problem (1) has at least one solution on J.
Proof.We show that the operator W defined by ( 10) is compact on a bounded ball B ρ , when B ρ = {π ∈ C : π ≤ ρ}.For any π ∈ B ρ , we have which yields Wπ ≤ Φ.Therefore, the set W(B ρ ) is uniformly bounded.To show that W(B ρ ) is an equicontinuous set, we let t 1 and t 2 be the two points in J such that t 1 < t 2 .Thus, for any π ∈ B ρ , we have Observe that if t 1 → t 2 , then we have |Wπ(t 2 ) − Wπ(t 1 )| → 0 independently of π.Therefore, the set W(B ρ ) is an equicontinuous set.Hence, the set W(B ρ ) is relatively compact.By applying the Arzelá-Ascoli theorem, the operator W is completely continuous.Finally, we show that the set of all solutions to equations π = λWπ is bounded for λ ∈ (0, 1).Let π ∈ C and π = λWπ for some λ ∈ (0, 1).Then, for any t ∈ J, as in the first step, we obtain and, consequently, From (H 3 ), π = K.After that, we define U = {π ∈ B ρ : π < K}.Now, we can see that W : U → C is continuous and completely continuous.Thus, there is no π ∈ ∂U such that π = λWπ with 0 < λ < 1.By the nonlinear alternative of the Leray-Schauder-type, we get that the operator W has a fixed point π ∈ U, which is a solution of the nonlocal fractional integro-differential sequential Hilfer and Caputo boundary value problem (17) on J.The proof is completed.

Some Special Cases
In this section, we present some special cases and some interesting behavior of solutions to the investigated problem (1).If we put ψ 1 (t) = ψ 2 (t) = ψ(t), then the nonlocal fractional integro-differential sequential Hilfer and Caputo boundary value problem ( 17) is reduced to The following constants are used in the next corollaries.
then the nonlocal fractional integro-differential sequential Hilfer and Caputo boundary value problem (17) has at least one solutions on J.
If n = p + q, and µ w = 0 for w = 1, . . ., q, then the problem ( 17) can be reduced to the following problem with integro-differential multi-point boundary conditions as In addition, we put The existence and uniqueness results for the integro-differential multi-point boundary value problem (18) are similar to the Corollaries 2 and 3 by replacing Â1 with A * 1 .

Conclusions
In this paper, we have studied a new kind of boundary value problem consisting of a combination of two fractional derivative operators, one ψ 1 -Hilfer and one ψ 2 -Caputo, supplemented with nonlocal integro-differential boundary conditions.This combination, as far as we know, is new in the literature.Our uniqueness result is derived via Banach's contraction principle, while the Leray-Schauder nonlinear alternative is used to derive the existence result.The main results are well illustrated by constructing numerical examples.
where M is a positive constant, then the nonlocal fractional integrodifferential sequential Hilfer and Caputo boundary value problem (17) has at least one solution J. (b) If u 1 (t) = 1, Ω(u) = Bu + C, u 2 (t) = D, where B ≥ 0, C, D > 0, then the nonlocal fractional integro-differential sequential Hilfer and Caputo boundary value problem (17) has at least one solution where B ≥ 0, C, D > 0, then the nonlocal fractional integro-differential sequential Hilfer and Caputo boundary value problem (17) has at least one solution J, if 4A 2 1 B(C + D) < 1.
Hilfer and Caputo-type fractional nonlocal integro-differential boundary value problem if ψ 1 (t) = t; (iii) ψ 1 -Hilfer and Caputo-type nonlocal integro-differential boundary value problem if If f satisfies the Lipschitz condition in (H 1 ) and if A * 1 L < 1, then the nonlocal fractional integro-differential sequential Hilfer and Caputo boundary value problem (17) has a unique solution on J.If the continuous function f satisfies (H 2 ) in Theorem 2 and if there exists a positive constant M such that M