Noninstantaneous Impulsive Fractional Quantum Hahn Integro-Difference Boundary Value Problems

In this paper, we study the existence and uniqueness results for noninstantaneous impulsive fractional quantum Hahn integro-difference boundary value problems with integral boundary conditions, by using Banach contraction mapping principle and Leray–Schauder nonlinear alternative. Examples are included illustrating the obtained results. To the best of our knowledge, no work has reported on the existence of solutions to the Hahn-difference equation with noninstantaneous impulses.

where ω > 0 is a fixed constant. The Hahn difference operator is used for constructing families of orthogonal polynomials and investigating some approximation problems, (cf. [25][26][27]). In 2013, Tariboon and Ntouyas [28], presented the new concepts of quantum calculus on [a, b] by defining With the help of definition (4), a series of quantum initial and boundary value problems contain impulses were studied. We refer the interested reader to the recent monograph [29] for details.
Let 0 < q i < 1, ω i ≥ 0 be quantum constants and θ i be the point defined by such that θ i ∈ [s i , t i+1 ), i = 0, 1, 2, . . . , m. The quantum shifting operator θ i Φ q i (t) is defined by Please note that if θ i = s i and θ i = 0 then (1) is reduced to and respectively. The power function (t − r) where (5) and (6) in (7), then we obtain respectively. The Gamma function in quantum calculus is defined by is the quantum number or q-number. However, in our work, the number q will be replaced by q i on an interval [s i , t i+1 ), i = 0, 1, 2, . . . , m.
In the following definitions, we give the Riemann-Liouville fractional derivative and integral in Hahn calculus as well as the Caputo fractional derivative which can be found in [31][32][33].

Definition 1. The fractional quantum Hahn difference operator of a Riemann-Liouville type of order
where n is the smallest integer greater than or equal to α i .

Definition 2.
Let α i ≥ 0 and f be a function defined on [s i , t i+1 ). The fractional quantum Hahn integral operator of Riemann-Liouville type is given by ( s i I 0 q i ,ω i f )(t) = f (t) and where n is the smallest integer greater than or equal to α i .
In our work, the problem (1) and (2) is based on fractional quantum Hahn calculus in Definitions 2 and 3. If ω i = 0, that is θ i = s i , then Definitions 1-3 are reduced to quantum calculus on the finite interval in the framework of Tariboon and Ntouyas [28] (see [29,30] for more details). Now we present some properties of fractional calculus on any interval [s i , t i+1 ), i = 0, 1, 2, . . . , m.
The rest of the paper is organized as follows. In Section 2, we first prove a basic lemma helping us to convert the boundary value problem (1) and (2) into an equivalent integral equation. Then we prove the main results, one existence and uniqueness result, via Banach contraction mapping principle and one existence result by using Leray-Schauder nonlinear alternative. Some special cases are discussed. Examples are also constructed to illustrate the main results in Section 3. The paper closes with a conclusion Section 4.

Main Results
In this section, we present our main results. The next lemma deals with a linear variant of the boundary value problem (1) and (2).
. . , m, β, γ and λ be given constants which satisfy the problem (1) and (2). Assume that Then the following linear noninstantaneous impulsive fractional quantum Hahn difference equations with integral boundary conditions has a unique solution x(t), t ∈ J, of the form Proof. Firstly, we take the quantum fractional Hahn integral of order α 0 to the first equation of (9) on an interval [s 0 , t 1 ) and set x(s + 0 ) = C, then For On the first jump interval, [t 1 , s 1 ), calling the second equation of (9) with (11), we have Next consecutive subinterval [s 1 , t 2 ), again by applying the quantum fractional Hahn integral of order α 1 , we get and then by (12).
Similar to the above method, we have for t ∈ [t 2 , s 2 ) as and for t ∈ [s 2 , t 3 ) as Generally, we compute that Now, taking the quantum Hahn fractional integral of order δ z to (13) over [s z , t z+1 ), multiplying constant µ z and summing for z = 0, 1, 2, . . . , m, we have From (14) and (15) and boundary integral condition in (9), we get Substituting (16) into (13) we get (10). The converse follows by direct computation. The proof is completed. Now, we establish the existence and uniqueness results for the boundary value problem (1) and (2) on J = [0, T]. We first define the Banach space C = C(J, R) equipped with the norm x = sup{|x(t)| : t ∈ J}. Based on the Lemma 1, we define the operator A : C → C by where the notation f xy = f (t, x, y), y = s i I κ i q i ,ω i x (t) is used, that is, which is convenient in our computations. The first result concerns the existence of a unique solution of the problem (1) and (2), and will be proved by using the Banach contraction mapping principle, involving the following constants: Theorem 3. Let f : J × R 2 → R and g i : J → R, i = 1, 2, . . . , m, be given continuous functions such that where L 1 , L 2 > 0, ∀t ∈ J and x 1 , x 2 , y 1 , y 2 ∈ R. If then the boundary value problem of the noninstantaneous impulsive fractional quantum Hahn integro-difference in Equations (1) and (2) has a unique solution in J.
Proof. First of all, we will show that AB r ⊆ B r ,where B r = {x ∈ C : x ≤ r}, and the radius r is defined by Setting M 1 = sup{| f (t, 0, 0)| : t ∈ J} and M 2 = sup{|g i (t)| : t ∈ J, i = 1, 2, . . . , m}, and using (18) we have Then, for any x ∈ B r , we obtain which holds from (19) and (20). This shows that AB r ⊆ B r . To show that A is a contraction operator, we let x 1 , x 2 ∈ B r and y k = s i I Therefore, the operator A satisfies Ax 1 − Ax 2 ≤ (L 1 + L 2 Λ 2 ) x 1 − x 2 . As, (L 1 + L 2 Λ 2 ) < 1, we can conclude that the operator A is a contraction mapping which has a unique fixed point in B r . Hence the problem (1) and (2) has a unique solution on J. This completes the proof.
Corollary 1. Suppose that f and g satisfy the conditions of Theorem 3. If then the initial value problem of the noninstantaneous impulsive fractional quantum Hahn integro-difference Equations (1)-(21) has a unique solution on J.
Then the boundary value problem of the noninstantaneous impulsive fractional quantum Hahn integro-difference Equations (1) and (2) has at least one solution on J.
Proof. Let A be the operator defined in (17) and now we are going to prove that the operator A is compact on a bounded ball B ρ , where B ρ = {x ∈ C : x ≤ ρ}. For any x ∈ B ρ , we have Finally, we will show that there exists an open set U ⊆ B ρ and 0 ∈ U such that x = λAx, where λ ∈ (0, 1) and x ∈ ∂U. Let x ∈ C and x = λAx for some λ ∈ (0, 1). Then for any t ∈ J, using the computation in the first step, we obtain which can be written as From (A 2 ), there exists a positive constant K such that x = K. Then we define U = {x ∈ C : x < K}. From above, the operator A : U → C is continuous and completely continuous. Therefore, there is no x ∈ ∂U such that x = λAx with 0 < λ < 1. Applying Lemma 2 we get that the operator A has a fixed point x ∈ U, which obviously is a solution of problem (1)-(2) on J. The proof is completed.
Please note that the nonlinear condition (22) of functions f and g is a very general condition. However, we can reduce it to be a linear one by Corollary 2. If f and g i , i = 1, 2, 3, . . . , m, satisfy (24) and if K 1 Λ 7 < 1, then the problem (1) and (2) has at least one solution on J.
Finally, we state the corresponding existence result for the initial value problem discussed in Corollary 1.

Corollary 4.
Assume that f and g satisfy condition (22). If there exists a positive constant K satisfying with p 2 Λ 2 Λ 1 < 1, then the initial value problem (1)-(21) has at least one solution on J.

Examples
In this section, we give some examples to illustrate the usefulness of our main results.

Conclusions
In this paper, we establish existence and uniqueness of solutions for a boundary value problem for fractional quantum Hahn integro-difference equations with noninstantaneous impulses, supplemented with integral boundary conditions. The classical Banach fixed point theorem is used to prove the existence and uniqueness result, while the existence result is proved via Leray-Schauder nonlinear alternative. Examples are included illustrating the obtained results.