Nonlocal q-Symmetric Integral Boundary Value Problem for Sequential q-Symmetric Integrodifference Equations

In this paper, we prove the sufficient conditions for the existence results of a solution of a nonlocal q-symmetric integral boundary value problem for a sequential q-symmetric integrodifference equation by using the Banach’s contraction mapping principle and Krasnoselskii’s fixed point theorem. Some examples are also presented to illustrate our results.


Introduction
Quantum difference operators dealing with sets of nondifferentiable functions have been extensively studied as they can be used as a tool to understand complex physical systems.There are several kinds of difference operators.The q-difference operator was first introduced by Jackson [1] and was studied in intensive work especially by Carmichael [2], Mason [3], Adams [4] and Trjitzinsky [5].The studies of quantum problems involving q-calculus have been presented.The recent works related to q-calculus theories can be found in [6][7][8] and the references cited therein.
The q-symmetric difference operators are a useful tool in several fields, especially in quantum mechanics [9].However, there are few research works [10][11][12][13] involving the development of q-symmetric difference operators.
In 2012, A.M.C. Brito da Cruz and N. Martins [10] studied the q-deformed theory, in which the standard q-symmetric integral must be generalized to the basic integral defined.
Recently, Sun, Jin and Hou [12] introduced basic concepts of fractional q-symmetric integral and derivative operator.Moreover, Sun and Hou [13] introduced basic concepts of fractional q-symmetric calculus on a time scale.
In particular, the boundary value problem for q-symmetric difference equations has not been studied.The results mentioned are the motivation for this research.In this paper, we devote our attention to the estabished existence results for a nonlocal q-symmetric integral boundary value for sequential q-symmetric integrodifference equation of the form Dq Dp u(t) = F t, u(t), (S θ u)(t), (Z ω u)(t) , t ∈ I T where This paper is organized as follows.In Section 2, we provide basic definitions, properties of q-symmetric difference operator and lemmas used in this paper.In Section 3, the existence results of problem (1) will be proved by employing Banach's contraction mapping principle and Krasnoselskii's fixed point theorem.In Section 4, we give some examples to illustrate our results.

Preliminaries
We introduce some basic definitions and properties of q-symmetric difference calculus as follows.
Definition 1.For q ∈ (0, 1), the q-symmetric difference of function f : R → R is defined by The higher order q-symmetric derivatives of f is defined by We note that D0 q f (t) = f (t).Next, if f is a function defined on the interval I, q-symmetric integral is defined by where the above infinite series is convergent.
We next discuss the following lemmas used to simplify our calculations.
x (qs) dq r dq s.
Proof.Using the definition of symmetric q-integral, we have To study the solution of the boundary value problem (1), we first consider the solution of a linear variant of the boundary value problem (1) as follows.
Lemma 3. Let p, q ∈ (0, 1), p = p 1 p 2 , q = q 1 q 2 , r = r 1 r 2 and κ = 1 LCM p 2 ,q 2 ,r 2 are the simplest form of proper fractions; λ ∈ R; g ∈ C I T κ , R + and h ∈ C I T κ , R are given functions.The solution for the problem is in the form where Proof.We first take the q-symmetric integral for (2) to obtain Next, taking the p-symmetric integral for (6), we have To find C 1 and C 2 , we first take the r-symmetric integral for g(t)u(t).We find that We apply condition (3) to (7).Then, we have We next apply condition (4) to (8).Then we have Constants C 1 and C 2 are obtained by solving the system of Equations ( 9) and (10) as follows Employing these results in (7), we get the solution (5).

Main Results
To study the existence and uniqueness of solution of (1), we transform the boundary value problem (1) into a fixed point problem.Let C = C(I T χ , R) denote the Banach space of all functions u.The norm is defined by u = sup where are the simplest form of proper fractions and χ = 1 LCM p 2 ,q 2 ,r 2 ,ω 2 ,θ 2 We note that problem (1) has solutions if and only if the operator T has fixed points.To present our results, we establish the following theorem based on Banach's fixed point theorem. where where Λ is given by (15), then the boundary value problem (1) has a unique solution.
Proof.For any u, v ∈ C and for each t ∈ I T χ , we have and From ( 14) and ( 15), we have As Ξ < 1, T is a contraction.Therefore, the proof is done based on Banach's contraction mapping principle.Furthermore, we prove the existence of a solution to the boundary value problem (1) by using the Krasnoselskii's fixed point theorem.Theorem 2. [14] Let K be a bounded closed convex and nonempty subset of a Banach space X.Let A, B be operators such that: (i) Ax + By ∈ K whenever x, y ∈ K, (ii) A is compact and continuous, (iii) B is a contraction mapping.
Then there exists z ∈ K such that z = Az + Bz.Theorem 3. Assume that (H 1 ) and (H 2 ) hold.In addition we suppose that: where Θ is given by (13), then the boundary value problem (1) has at least one solution on I T χ .
Proof.We let sup where Based on the results of Lemma 3, we define the operators T 1 and T 2 on the ball B R as We proceed similarly to Theorem 1 for u, v ∈ B R .Then we have which implies that T 1 x + T 2 y ∈ B R .Using (17), we find that T 2 is a contraction mapping.
From the continuity of F and the assumption (H 3 ), we see that an operator T 1 is continuous and uniformly bounded on B R .For t 1 , t 2 ∈ I T χ with t 1 ≤ t 2 and u ∈ B R , we have We observe that the above inequality tends to zero when t 2 − t 1 → 0. Therefore T 1 is relatively compact on B R .Hence, we can conclude by the Arzelá-Ascoli Theorem that T 1 is compact on B R .We find that the assumptions of Theorem 2 are satisfied implying that the boundary value problem (1) has at least one solution on I T χ .The proof is complete.

Examples
In this section, we provide some examples to illustrate our main results.Consider the following boundary value problem of sequential q-symetric difference equations We can show that ϕ 0 = 1 100 , φ 0 = 1 400 and Clearly, Λ = 0.0419 and Θ = 1231.332.  .

Conclusions
In this article, we consider a nonlocal q-symmetric integral boundary value problem for sequential q-symmetric difference-sum equation.We study the condition under which the problem has existence and a unique solution by using Banach's contraction mapping principle.Furthermore, we provide the condition for the case of at least one solution by using Krasnoselskii's fixed point theorem.A further extension of this article is the study of stability, behaviour under perturbation and possible applications in economics and engineering.
)|.The operator T : C → C is defined by )| = µ and choose a constant