On Periodic Fractional ( p , q )-Integral Boundary Value Problems for Sequential Fractional ( p , q )-Integrodifference Equations

: We study the existence results of a fractional ( p , q )-integrodifference equation with periodic fractional ( p , q )-integral boundary condition by using Banach and Schauder’s ﬁxed point theorems. Some properties of ( p , q )-integral are also presented in this paper as a tool for our calculations.


Introduction
The studies of quantum calculus with integer order were presented in the last three decades, and many researchers extensively studied calculus without a limit that deals with a set of nondifferentiable functions, the so-called quantum calculus. Many types of quantum difference operators are employed in several applications of mathematical areas, such as the calculus of variations, particle physics, quantum mechanics, and theory of relativity. The q-calculus, one type of quantum initiated by Jackson [1][2][3][4][5], was employed in several fields of applied sciences and engineering such as physical problems, dynamical system, control theory, electrical networks, economics, and so on [6][7][8][9][10][11][12][13][14].
Recently, Soontharanon and Sitthiwirattham [37] introduced the fractional (p, q)difference operators and its properties. Now, this calculus was used in the inequalities [38,39] and the boundary value problems [40][41][42]. However, the study of the boundary value problems for fractional (p, q)-difference equation in the beginning, there are a few literature on this knowledge. In [40], the existence results of a fractional (p, q)-integrodifference p,q , R + are given functions; ϕ : C I T p,q , R → R is given functional; and for φ ∈ C I T p,q × I T p,q , [0, ∞) , we define an operator of the (p, q)-integral of the product of functions φ and u as We aim to show the existence results to the problem (1). Firstly, we convert the given nonlinear problem (1) into a fixed point problem related to (1), 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 results.
The paper is organized as follows: Section 2 contains some preliminary concepts related to our problem. We present the existence and uniqueness result in Section 2, and the existence of at least one solution in Section 4. To illustrate our results, we provide some examples in Section 5. Finally, Section 6 discusses our conclusions.

Preliminaries
In this section, we provide some basic definitions, notations, and lemmas as follows. For 0 < q < p ≤ 1, we define The (p, q)-forward jump and the (p, q)-backward jump operators are defined as σ k p,q (t) := q p k t and ρ k p,q (t) := p q k t, for k ∈ N, respectively.
The q-analogue of the power function (a − b) n q with n ∈ N 0 := {0, 1, 2, . . .} is given by The (p, q)-analogue of the power function (a − b) n p,q with n ∈ N 0 is given by Generally, for α ∈ R, we define The (p, q)-gamma and (p, q)-beta functions are defined by respectively.
Observe that the function

Definition 2.
Let I be any closed interval of R containing a, b and 0. Assuming that f : provided that the series converges at x = a and x = b and f is called An operator I N p,q is defined as The relations between (p, q)-difference and (p, q)-integral operators are given by Fractional (p, q)-integral and fractional (p, q)-difference of Riemann-Liouville type are defined as follows.
Definition 3. For α > 0, 0 < q < p ≤ 1 and f defined on I T p,q , the fractional (p, q)-integral is defined by Definition 4. For α > 0, 0 < q < p ≤ 1 and f defined on I T p,q , the fractional (p, q)-difference operator of Riemann-Liouville type of order α is defined by
The following lemma, dealing with a linear variant of problem (1), plays an important role in the forthcoming analysis.

Lemma 6.
Let Ω = 0, α, β, θ ∈ (0, 1], 0 < q < p ≤ 1, h ∈ C I T p,q , R and g ∈ C I T p,q , R + be given functions, ϕ : C I T p,q , R → R be given functional. Then, the problem has the unique solution: and the constants A T , A η , B η and Ω are defined by Proof. Taking fractional (p, q)-integral of order α for (2) and using Lemma 1, we then have Next, taking fractional (p, q)-difference of order β for (12), we have Substituting t = 0, T p into (13) and employing the condition (3), we have By taking fractional (p, q)-integral of order θ for (13), we have From the condition (4) we have Solving the system of linear Equations (14) and (16),we obtain where P[h], Q[h], A T , A η , B η and Ω are defined by (6)-(11), respectively. After substituting C 0 , C 1 into (13), we obtain (5). We can prove the converse by direct computation. The proof is complete.

Existence and Uniqueness Result
In this section, we prove the existence and uniqueness result for problem (1) by using Banach fixed point theorem as follows.

Lemma 7 ([43]
Banach fixed point theorem). Let a nonempty closed subset C of a Banach space X, then there is a unique fixed point for any contraction mapping P of C into itself.
Let C = C I T p,q , R be a Banach space of all function u with the norm defined by where u = max t∈I T p,q |u(t)| and D ν p,q u C = max t∈I T p,q D ν p,q u(t) .
By Lemma 6, replacing h(t) by F t, u(t), Ψ γ p,q u(t), D ν p,q u(t) , we define an operator where the functionals P * [F u ] and Q * [F u ] are defined by p,q u s p α−1 d p,q s d p,q x d p,q y (19) and the constants A T , A η , B η and Ω are defined by (8)- (11), respectively. We see that the problem (1) has solution if and only if the operator A has fixed point.
is continuous with φ 0 = max φ(t, s) : (t, s) ∈ I T p,q × I T p,q , and ϕ : C I T p,q , R → R is given functional. Suppose that the following conditions hold: (H 1 ) There exist positive constants L 1 , L 2 , L 3 such that for each t ∈ I T p,q and u i , v i ∈ R, i = 1, 2, 3, (H 2 ) There exists a positive constant ω such that for each u, v ∈ C, Then, problem (1) has a unique solution in I T p,q .
By using Lemma 5(a), we obtain and by using Lemma 5(b), we have

Existence of at Least One Solution
In this section, we prove the existence of at least one solution to (1). The following lemmas reviewing the Schauder's fixed point theorem are also provided. Lemma 9 ([43]). If a set is closed and relatively compact, then it is compact.

Lemma 10 ([44]
Schauder's fixed point theorem). Let (D, d) be a complete metric space, U be a closed convex subset of D, and T : D → D be the map such that the set Tu : u ∈ U is relatively compact in D. Then, the operator T has at least one fixed point u * ∈ U: Tu * = u * . Theorem 2. Assume that F : I T p,q × R × R × R → R is continuous, and ϕ : C I T p,q , R → R is given functional. Suppose that the following conditions hold: (H 5 ) There exists a positive constant M such that for each t ∈ I T p,q and u i ∈ R, i = 1, 2, 3, (H 6 ) There exists a positive constant N such that for each u ∈ C, |ϕ(u)| ≤ N.
Then, problem (1) has at least one solution on I T p,q .
Proof. To prove this theorem, we proceed as follows.
Step I. Verify A maps bounded sets into bounded sets in B R = {u ∈ C : u C ≤ R}. Let us prove that for any R > 0, there exists a positive constant L such that for each x ∈ B R , we have Au C ≤ L. By using lemma 5, for each t ∈ I T p,q and u ∈ B R , we have 1 100e 2+2π D ν p,q u − D ν p,q v .
For all u, v ∈ C, Hence, by Theorem 1 this problem has a unique solution.