Nonlocal Boundary Value Problems of Nonlinear Fractional ( p , q ) -Difference Equations

: In this paper, we study nonlinear fractional ( p , q ) -difference equations equipped with separated nonlocal boundary conditions. The existence of solutions for the given problem is proven by applying Krasnoselskii’s ﬁxed-point theorem and the Leray–Schauder alternative. In contrast, the uniqueness of the solutions is established by employing Banach’s contraction mapping principle. Examples illustrating the main results are also presented.


Introduction
In recent years, fractional differential equations have been considered a popular field of research and have attracted many researchers' attention. This is mainly because fractional differential equations are found to be effective and more practical than classical differential equations, particularly in the mathematical modeling of dynamical systems, such as fractals and chaos. In the last two decades, the fractional differential equations have developed from theoretical aspects of the existence and uniqueness of solutions to analytic and numerical methods for finding solutions, which can be found in [1][2][3][4][5][6][7][8][9]. Moreover, in modern mathematical analysis, fractional differential equations have a range of applications, such as engineering and clinical disciplines, including biology, physics, chemistry, economics, signal and image processing, and control theory; see [10][11][12][13][14][15][16][17] for more details.
In 1910, F. H. Jackson initiated a study of the q-difference calculus or quantum calculus (briefly called q-calculus) in a symmetrical manner and introduced the q-derivative and q-integral, which can be found in [18,19]. With these results, q-calculus has arisen in a range of applications, such as combinatorics, orthogonal polynomials, basic hypergeometric functions, number theory, quantum theory, mechanics, and theory of relativity; see  and the references cited therein. The book by V. Kac and P. Cheung [41] covers the basic theoretical concepts of q-calculus.
Later on, the subject of (p, q)-calculus was generalized and developed from the qcalculus theory into the two-parameter (p, q)-integer, which is used effectively in many fields, and some results on the study of (p, q)-calculus can be found in .
Inspired and motivated by some of the above applications, in 2018, N. Kamsrisuk et al. [75] considered the existence and uniqueness of solutions for the first-order quantum (p, q)-difference equation subject to a nonlocal condition: Furthermore, C. Promsakon et al. [76] introduced the second-order (p, q)-difference equations with separated boundary conditions: where f ∈ C [0, T/p 2 ] × R, R and α i , β i (i = 1, 2, 3) are constants. D 2 p,q and D q are the second and first order of the (p, q)-difference operator, respectively.
However, the study of nonlocal boundary value problems of fractional (p, q)-difference equations is at its infancy, and much of the work on the topic is yet to be done.
In this paper, we study the existence and uniqueness of solutions of the nonlocal boundary value problem of nonlinear fractional (p, q)-difference equations given by where h ∈ C([0, T/p α ] × R, R) and β i , γ i , η i (i = 1, 2) are constants; c D α p,q and D p,q are the fractional (p, q)-derivative of the Caputo type and the first-order (p, q)-difference operator, respectively.

Preliminaries
In this section, we give some definitions and fundamental results of the q-calculus and (p, q)-calculus along with the fractional (p, q)-calculus, which can be found in [27,28,56,58]. We also give a lemma that will be used in obtaining the main results of the paper.

Definition 1 ([56]
). If f : [0, T] → R is a continuous function, then the (p, q)-derivative of f is defined by and D p,q f (0) = lim t→0 D p,q f (t). Observe that the function D p,q f (t) is defined on [0, T/p] provided that D p,q f (t) exists for all t ∈ [0, T/p].

Definition 2 ([56]
). If f : [0, T] → R is a continuous function, then the (p, q)-integral of f is defined as q n p n+1 f q n p n+1 t (12) provided that the right-hand side converges. Note that the function t 0 f (s)d p,q s is defined on [0, pT], which is extended from [0, T] of a function f (t).

Definition 4 ([77]
). The fractional (p, q)-derivative of Riemann-Liouville type of order α ≥ 0 of a continuous function f is defined by where [α] is the smallest integer greater than or equal to α.
p,q f (t), (15) where [α] is the smallest integer greater than or equal to α.

Lemma 2 ([77]
). Let f be a continuous function, α ≥ 0, and n ∈ N. Then, the following equality holds: ). Let f be a continuous function and α ≥ 0. Then, the following equality holds: In order to define the solution of the boundary value problem (4), we need the following lemma.
be constants and c D α p,q and D p,q be the fractional (p, q)derivative of the Caputo type and the first-order (p, q)-difference operator, respectively. For a given h ∈ C([0, T/p α ], R), a unique solution of the boundary value problem is given by Proof. By taking the fractional (p, q)-integral on (16) and applying Lemma 3, we have where c 0 , c 1 are constants and t ∈ [0, T].
Applying the above boundary condition of (16), we obtain Solving the above system of (19) to find the constants c 1 , c 2 , we have Substituting the values of c 0 , c 1 in (18), we obtain (17). We can prove the converse by direct computation. Therefore, the proof is completed.

Main Results
In view of Lemma 4, we define an operator Ψ : h(s, y(p α−1 s)) d p,q s h(s, y(p α−1 s)) d p,q s where Let Ω := C([0, T], R) denote the Banach space of all continuous functions from [0, T] to R endowed with norm defined by y = max{|y(t)| : t ∈ [0, T]}.
Observe that the boundary value problem (4) has a unique solution if the operator equation Ψy = y has a fixed point, where Ψ is given in (20). For convenience, we let where

Theorem 3.
Let h : [0, T/p α ] × R → R be a continuous function and note that there exists a (p, q)-integrable function λ where k is given by (21), then the boundary value problem (4) has a unique solution.
In case λ(t) = λ, where λ is a constant, the condition (22) becomes λA < 1 and Theorem 3 takes the form of the following results.

Remark 1. If h : [0, T/p α ] × R → R is a contraction function and there exists a constant
for each t ∈ [0, T/p α ] and y, z ∈ R, then the boundary value problem (4) has a unique solution, provided that k < 1.
Our next existence results are based on Krasnoselskii's fixed-point theorem.

Lemma 5 ([80]
). (Krasnoselskii's fixed-point theorem) Let M be a closed, bounded, convex, and non-empty subset of a Banach space X. Let A, B be two operators such that: (i) Ay + Bz ∈ M, whenever y, z ∈ M; (ii) A is compact and continuous; (iii) B is a contraction mapping.
Then, there exists x ∈ M such that x = Ax + Bx.

Theorem 4.
Let h : [0, T/p α ] × R → R be a continuous function satisfying (A 1 ). In addition, we assume that (A 2 ) there exists a function µ ∈ C([0, T/p α ], R + ) and a non-decreasing function φ ∈ C([0, T/p α ], |µ(t)| and define the operators P and Q on B r as h(s, y(p α−1 s)) d p,q s h(s, y(p α−1 s)) d p,q s h(s, y(p α−2 s)) d p,q s, where For y, z ∈ B r , we have Thus, P y + Qz ∈ B r . From (A 1 ) and (26), it follows that Q is a contraction mapping. By continuity of h, we obtain that the operator P is continuous. It is not difficult to verify that Therefore, the set P (B r ) is uniformly bounded. Next, we shall prove the compactness of the operator P. Now, for any y ∈ B r t 1 , t 2 ∈ [0, T] with t 1 < t 2 . Then, we obtain which is independent of x and tends to zero as t 1 → t 2 . Thus, the set P (B r ) is equicontinuous. By the Arzelá-Ascoli theorem, P is compact on B r . Therefore, the boundary value problem (4) has at least one solution on [0, T]. This completes the proof.
Then, the boundary value problem (4) has at least one solution on [0, T].
Proof. Consider the operator Ψ : C → C defined by (20). We first show that Ψ is continuous. Let {y n } be a sequence of function such that y n → y on [0, T/p α ]. Given that h is a continuous function on [0, T/p α ], we have h(t, y n (p α t)) → h(t, y(p α t)).
Thus, the operator F is continuous. Next, we show that Ψ maps a bounded set into a bounded set in C([0, T], R). For a positive number r > 0, let B r = {y ∈ C([0, T]) : y ≤ r}. Then, for any y ∈ B r , we have h(s, y(p α−1 s)) d p,q s h(s, y(p α−1 s)) d p,q s h(s, y(p α−1 s)) d p,q s h(s, y(p α−1 s)) d p,q s h(s, y(p α−1 s)) d p,q s h(s, y(p α−1 s)) d p,q s h(s, y(p α−1 s)) d p,q s It is not difficult to show that for each t ∈ [0, 8], and k = λ Γ 1/4,1/5 (3/2) Then, all the assumptions of Theorem 3 are satisfied. Accordingly, by Theorem 3, the boundary value problem (28) has a unique solution.

Conclusions
In this paper, we study nonlocal fractional (p, q)-difference equations with separated nonlocal boundary conditions. The existence of solutions for the problem is given by applying some well-known tools in fixed-point theory, such as Banach's contraction mapping principle, Krasnoselskii's fixed-point theorem, and the Leray-Schauder nonlinear alternative. Some illustrating examples are also presented. We hope that the paper will inspire interested readers working in this field to draw upon these ideas and techniques.