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

Abstract

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.

On Solvability of Fractional (p,q)-Difference Equations with (p,q)-Difference Anti-Periodic Boundary Conditions

“…Let us first describe some fundamental concepts of q-calculus and (p, q)-calculus [16]. We also establish an auxiliary lemma that will be used in obtaining the main results of the paper.…”

“…For details of the above concepts, see [16] and the references cited therein. In order to define the solution for the problem in ( 1) and ( 2), we need the following lemma.…”

“…The existence of multiple positive solutions for a fractional (p, q)-difference equation under (p, q)-integral boundary conditions was discussed in [15]. More recently, the authors investigated the existence of solutions for a nonlinear fractional (p, q)-difference equation subject to separated nonlocal boundary conditions in [16], whereas some existence results for a sequential fractional Caputo (p, q)-integrodifference equation with three-point fractional Riemann-Liouville (p, q)-difference boundary conditions were obtained in [17]. However, one can notice that the study on the boundary value problems of fractional (p, q)-difference equations is at its initial phase and needs further attention for the enrichment of the topic.…”

Existence and Uniqueness Results for Fractional (p, q)-Difference Equations with Separated Boundary Conditions

“…In 2021, Neang et al [31] considered the nonlocal boundary value problem of nonlinear fractional (p, q)-difference equations with taking care of solutions of existence and uniqueness results obtained by c D α p,q u(t) = f (t, u(p α t)), t ∈ [0, T/p α ], 1 < α ≤ 2, (1)…”

“…However, even though Neang et al [31] investigated and proved the nonlocal boundary value problems by considering on existence results of a class of fractional (p, q)difference equations, it still was a bit complicated with the domain of a function when the authors applied the fractional (p, q)-integral operators. In this paper, to make this paper more smooth and convenient, we have investigated the existence and uniqueness of solutions for the local boundary value problem of fractional (p, q)-difference equation with a new function obtained g ∈ C ([0, b] × R, R), given by the following:…”