On a boundary value problem for fractional Hahn integro-di ﬀ erence equations with four-point fractional integral boundary conditions

: In this paper, we study a boundary value problem consisting of Hahn integro-di ﬀ erence equation supplemented with four-point fractional Hahn integral boundary conditions. The novelty of this problem lies in the fact that it contains two fractional Hahn di ﬀ erence operators and three fractional Hahn integrals with di ﬀ erent quantum numbers and orders. Firstly, we convert the given nonlinear problem into a ﬁxed point problem, by considering a linear variant of the problem at hand. Once the ﬁxed point operator is available, we make use the classical Banach’s and Schauder’s ﬁxed point theorems to establish existence and uniqueness results. An example is also constructed to illustrate the main results. Several properties of fractional Hahn integral that will be used in our study are also discussed.


Introduction
The topic of fractional differential equations has gained considerable attention and has evolved as an interesting field of research, mainly due to the fact that the tools of fractional calculus are found to be more practical and effective than the corresponding ones of classical calculus in the mathematical modeling real wold problems. In fact, fractional calculus has numerous applications in various disciplines of science and engineering such as mechanics, chemistry, biology, economics, electricity, control theory, signal and image processing, regular variation in thermodynamics, biophysics, considered a nonlocal boundary value problem for second-order nonlinear Hahn integro-difference equation with integral boundary condition.
In quantum calculus, there are apparently few research works related to boundary value problems of fractional Hahn difference equations (see [46]- [48]). Motivated by the above discussion, to fill the gap on contributions concerning boundary value problems of fractional Hahn difference equations, the goal of this paper is to enrich this new research area. So, in this paper, we introduce and study a four-point fractional Hahn integral boundary value problems for fractional Hahn integrodifference equation of the form D α q,ω u(t) = F t, u(t), Ψ γ r,ρ u(t), Υ ν m,χ u(t) , t ∈ I T q,ω , u(ξ) = φ 1 (u) + λ 1 I β 1 p 1 ,θ 1 g 1 (η 1 )u(η 1 ), ξ, η 1 ∈ I T q,ω − {ω 0 , T }, ξ > η 1 , ,ω , R + and given functions, φ 1 , φ 2 : C I T q,ω , R → R are given functionals, and for ϕ ∈ C I T r,ρ × I T r,ρ , [0, ∞) and ψ ∈ C I T m,χ × I T m,χ , [0, ∞) , we define We emphasize that our problem contains two fractional Hahn difference operators and three fractional Hahn integrals with different quantum numbers and orders. To the authors' best knowledge, this is the new development on the topic as the quantum number and order of the problems studied in the literature are the same.
We aim to show the existence and uniqueness of a solution to the problem (1.1) by using the Banach fixed point theorem, and the existence of at least one solution by using the Schauder's fixed point theorem. In addition, an example is provided to illustrate our results in the last section.
The rest of this paper is organized as follows: We present our existence and uniqueness result in Section 3, and our existence result in Section 4, while Section 2 contains some preliminary concepts related to our problem. An example is constructed to illustrate the main results in Section 5. Finally, Section 6 is a conclusion section.

Preliminaries
In this section, we briefly recall some definitions and lemmas used in this research work. In this work, we use the Banach space C = C I T q,ω , R of all function u with the norm defined as The forward jump operator and the backward jump operator are defined as respectively. In generally, if α ∈ R, we get The q-gamma and q-beta functions are defined as respectively.
Definition 2.1. For q ∈ (0, 1), ω > 0 and f defined on an interval I ⊆ R which containing ω 0 := ω 1−q , the Hahn difference of f is defined as and D q,ω f (ω 0 ) = f (ω 0 ), provided that f is differentiable at ω 0 . We call D q,ω f the q, ω-derivative of f , and say that f is q, ω-differentiable on I.
The Hahn difference operator has the following properties: [30] Let f, g : I → R are q, ω-differentiable on I. Then we have: Definition 2.2. Let I be any closed interval of R that contains a, b and ω 0 . If f : I → R is a given function, we define the q, ω-integral of f from a to b by where x ω 0 and the series converges at x = a and x = b. We say f is q, ω-integrable on [a, b] and the sum to the right hand side of this equation is called the Jackson-Nörlund sum.
Notice that the actual domain of function f is defined on [a, b] q,ω ⊂ I.
Next, we introduce the fundamental theorem of Hahn calculus.
[29] Let f : I → R be continuous at ω 0 and define Then, F is continuous at ω 0 . In addition, D q,ω 0 F(x) exists for every x ∈ I and [38] Let q ∈ (0, 1) and ω > 0. Then Now, we give the definitions of fractioanal Hahn integral and fractional Hahn difference of Riemann-Liouville type, as follows: For α, ω > 0, q ∈ (0, 1) and f : I T q,ω → R, the fractional Hahn integral is defined by and Letting α > 0, q ∈ (0, 1), ω > 0 and f : I T q,ω → R, Next, we give some auxiliary lemmas to use in simplifying calculations.
The following lemma, dealing with a linear variant of problem (1.1), plays an important role in the forthcoming analysis.
has the unique solution

Existence and uniqueness result
In this section, we show the existence and uniqueness result for problem (1.1). In view of Lemma 2.8 we define an operator A : C → C as and the constants A ξ,η 1 , A T,η 2 , B ξ,η 1 , B T,η 2 and Ω are defined by (2.5)-(2.9), respectively.
Obviously the problem (1.1) has solutions if and only if the operator A has fixed points.
Proof. For each t ∈ I T r,ρ , we have Similarly, for each t ∈ I T m,χ , we obtain To show that F is contraction, we denote that for each t ∈ I T q,ω and u, v ∈ C. We find that Similarly, we get Taking fractional Hahn m, χ-difference of order ν to (3.1), we obtain By the same expression as above, we obtain

From (3.16) and (3.18), we get
By (H 4 ) and Banach fixed point theorem, we get that A is a contraction and hence A has a fixed point. Consequently problem (1.1) has a unique solution of on I T q,ω .

Existence of at least one solution
In this section, we prove an existence result for the problem (1.1) via Schauder's fixed point theorem.
for each t ∈ I T q,ω and u ∈ B R , we obtain Similarly, Then, we have From (4.4) and (4.5), we obtain Au C ≤ R. Hence A is uniformly bounded.
Step II. By the continuity of F, the operator A is continuous on B R .
Step III. Next, we show that A is equicontinuous on B R . For any t 1 , t 2 ∈ I T q,ω with t 1 < t 2 , we have and Clearly the right-hand side of (4.6) and (4.7) tend to be zero when |t 2 − t 1 | → 0. So A is relatively compact on B R .
Hence the set F (B R ) is an equicontinuous set. As a result of Steps I to III and the Arzelá-Ascoli theorem, we can conclude that A : C → C is completely continuous. By Schauder's fixed point theorem, we obtain that problem (1.1) has at least one solution.
Therefore, by Theorem 3.1, problem (5.1) has a unique solution. Moreover, by Theorem 4.1, this problem has at least one solution.

Conclusions
In the present research we considered a boundary value problem for Hahn integro-difference equation subject to four-point fractional Hahn integral boundary conditions. Notice that the problem at hand contains two fractional Hahn difference operators and three fractional Hahn integrals with different quantum numbers and orders. We note that if we let q = r = m = p 1 = p 2 and ω = ρ = θ 1 = θ 2 , our results reduce to the results obtained in [46]- [48]. After proving an auxiliary result concerning a linear variant of the considered problem, the problem at hand is transformed into a fixed point problem. Existence and uniqueness results are established via Banach's and Schauder's fixed point theorems. The main results are illustrated by a numerical example. Some properties of fractional Hahn integral needed in our study are also discussed. The results of the paper are new and enrich the subject of boundary value problems for Hahn integro-difference equations. In the future work, we may extend this work by considering new boundary value problems.