Existence Results of a Coupled System of Caputo Fractional Hahn Difference Equations with Nonlocal Fractional Hahn Integral Boundary Value Conditions

In this article, we propose a coupled system of Caputo fractional Hahn difference equations with nonlocal fractional Hahn integral boundary conditions. The existence and uniqueness result of solution for the problem is studied by using the Banach’s fixed point theorem. Furthermore, the existence of at least one solution is presented by using the Schauder fixed point theorem.

Jackson q-difference operator, the forward (delta) difference operator and the backward (nabla) difference operator are the well-known operators used in many research works.In 1949, the Hahn difference operator developed from the forward difference operator and the Jackson q-difference operator was proposed by Hahn [15] as We note that D q,ω f (t) = ∆ ω f (t) := f (t + ω) − f (t) ω whenever q = 1, D q,ω f (t) = D q f (t) := f (qt) − f (t) t(q − 1) whenever ω = 0, and D q,ω f (t) = f (t) whenever q = 1, ω → 0.
Hahn's operator has been used in the determination of new families of orthogonal polynomials and in approximation problems (see [16][17][18]).Recently, the Hahn difference calculus has become a favourite topic for analysis.Later, the right inverse of Hahn difference operator was introduced by Aldwoah [19,20].This operator is formulated from Nörlund sum and Jackson q-integral [21].In 2010, Malinowska and Torres proposed Hahn variational calculus [22,23].In 2013, Hamza et al. [24,25] proved the existence and uniqueness of solution for the initial value problems for Hahn difference equations.In addition, they obtained a mean value theorems for this calculus, and established Gronwall's and Bernoulli's inequalities with respect to the Hahn difference operator.In the same year, Malinowska and Martins [26] proposed Hahn variational problem to generalize the Hahn calculus of variations.They obtained transversality conditions.
In 2016, Sitthiwirattham [27] studied the nonlocal boundary value problem for a second-order Hahn difference equation, their problem contains two Hahn difference operators with different numbers of q and ω, the existence and uniqueness result was proved by using the Banach fixed point theorem, and the existence of a positive solution was established by using the Krasnoselskii fixed point theorem.In 2017, Sriphanomwan et al. [28] developed the above problem by including an Hahn integro-difference term and integral boundary condition, the existence and uniqueness of solutions was obtained by using the Banach fixed point theorem, and the existence of at least one solution was established by using the Leray-Schauder nonlinear alternative and Krasnoselskii's fixed point theorem.
Since the boundary value problem for systems of fractional Hahn difference equations have never been presented before, we devote our attention to study this kind of problem.In this paper, we consider the boundary value problem for the system of Caputo fractional Hahn difference equations of the form with the nonlocal three-point fractional Hahn integral boundary value conditions where , R are given functions, and φ i : C I T q,ω , R × C I T q,ω , R → R are given functionals.We organize the paper as follows.We provide some definitions and lemmas in Section 2. We present the existence and uniqueness of a solution for system (1) in Section 3. The Banach fixed point theorem is the tool to get the result.In addition, we prove the existence of at least one solution for system (1) by employing the Schauder's fixed point theorem in Section 4. Finally, we present some examples of the main results.
Definition 2. Let I be any closed interval of R which contain a, b and ω 0 .Assume that f : provided that the series converges at x = a and x = b.We call f q, ω-integrable on [a, b].The above summation is called the Jackson-Nörlund sum.
We note that f is defined on [a, b] q,ω ⊂ I.We next provide the following lemma introducing the fundamental theorem of Hahn calculus.
Then, F is continuous at ω 0 .Furthermore, D q,ω 0 F(x) exists for every x ∈ I and D q,ω F(x) = f (x).
Conversely, b a D q,ω F(t)d q,ω t = F(b) − F(a) for all a, b ∈ I.
Lemma 5. Letting α, β > 0, p, q ∈ (0, 1) and ω > 0, In order to study the existence and uniqueness results of solution of the nonlinear problem (1), we first consider the linear variant of problem (1) and its solution in the following lemma.
has the unique solution where and Proof.For i, j ∈ {1, 2} and i = j, by using Lemma 4 and the fractional Hahn integral of order α for (3), we have Using the boundary condition (3), we find that Therefore, Taking the fractional Hahn integral of order 0 < θ i ≤ 1 for (11), we get for t ∈ I T q,ω .From the boundary condition (3), we have Finally, the constants C 11 and C 12 are investigated by solving the system of Equations (13) and (14) as and where Λ, P (h 1 , h 2 ) and Q(h 1 , h 2 ) are defined as Equations ( 8)- (10), respectively.Substituting C 11 and C 12 into (11), we then obtain ( 4) and ( 5).

Existence and Uniqueness Result
In this section, we aim to prove the existence result for Problems (1) and ( 2).Here, we let E : C I T q,ω , R be the Banach space for all continuous functions on I T q,ω , and clearly that the product space C = E × E is the Banach space.We set the spaces Define the norm as follows: where C D Obviously, the space C 1 ∩ C 2 , (u 1 , u 2 ) C 1 ∩ C 2 is also the Banach space with the norm Next, we let U = C 1 ∩ C 2 and we define the operator T : U → U by and where Λ is defined as (7), and the functionals P ) q,ω u 2 (s) .
We note that Problems (1) and ( 2) have solutions if and only if the operator T has fixed points.

Theorem 1.
For each i, j ∈ {1, 2}; i = j, we assume that (H 1 ) There exist constants M 1 , M 2 , N 1 , N 2 > 0 such that, for each t ∈ I T q,ω , Then, Problems (1) and ( 2) have a unique solution provided that where Proof.The goal is to prove that T is a contraction mapping.Letting t ∈ I T q,ω and (u 1 , u 2 ), (v 1 , v 2 ) ∈ U , we obtain and From ( 29) and ( 30), we find that Therefore, it implies that Next, taking the Caputo fractional Hahn difference of order 0 < β 1 ≤ 1 for (16), we have Hence, It implies that Similarly, we obtain and and From ( 43) and (44), we find that Similarly to Theorem 1, we obtain Furthermore, we have From (46) and (47), we can show that Similarly, from (45) and (48), we have Therefore, from (49) and (50), we can conclude that From (51), we get T (u 1 , u 2 ) U ≤ R. Therefore, T is uniformly bounded.
Step II.Show that T is continuous on B R .Letting > 0, there exists δ = max{δ 1 , δ 2 , δ 3 , δ 4 } > 0 such that, for each t ∈ I T q,ω and and From ( 52) and ( 53), we find that Therefore, it implies that Similarly to the above proof and Theorem 1, we obtain and From (56) and (57), we can show that Similarly, from (55) and (58), we have Therefore, from (59) and (60), we can conclude that T (u 1 , u 2 )T (v 1 , v 2 ) U < .This means that T is continuous on B R .
Step III.For this step, we prove that T is equicontinuous with B R .For any t 1 , t 2 ∈ I T q,ω with t 1 < t 2 , we have Furthermore, by (61) and (62), we have and When |t 2 − t 1 | → 0, the right-hand side of inequalities ( 61) and (64) tends be zero.Thus, T is relatively compact on B R .Therefore, G(B R ) is an equicontinuous set.From the result of Steps I to III together with the Arzelá-Ascoli theorem, we find that T : U → U is completely continuous.Therefore, we can conclude from the Schauder fixed point theorem that Problems (1) and ( 2) have at least one solution.
Then, we find that Therefore, we can conclude from Theorem 1 that Problem (65) has a unique solution.