Existence of positive solution to a coupled system of singular fractional difference equations via fractional sum boundary value conditions

Abstract

In this article, we study a coupled system of singular fractional difference equations with fractional sum boundary conditions. A sufficient condition of the existence of positive solutions is established by employing the upper and lower solutions of the system and using Schauder's fixed point theorem. Finally, we provide an example to illustrate our results.

Ulam-Hyers stability results for a novel nonlinear Nabla Caputo fractional variable-order difference system

“…In this paper, we are concerned with a class of nonlinear variable-order Nabla Caputo fractional difference system, which is quite different from the related references discussed in the literature [5,9,10,16,18,19,23].…”

“…Such as in [16], Henderson got the existence conditions of solutions by applying Leray-Schauder Nonlinear Alternative method. In [18,23,35], the authors studied fractional difference equations, and the existence of solutions were established by employing Schauder's fixed point theorem. In [10,19], Luo and Chen investigated the uniqueness results for a class of nonlinear fractional difference system with time delay and gave the proof by contradiction and generalized Gronwall inequality.…”

Existence of solutions for a class of fractional difference equations at resonance

“…The following lemma is fundamental for fractional sum and difference (see Refs. 15, 31): Lemma Let , , and . Then where and for .…”

“…Moreover, fractional difference equations, along with the development of fractional differential equations, have made significant improvements, especially, during the past decade. For some recent work, we refer ( 7–16,32–34 ) and the references therein.…”

Existence and multiplicity of positive solutions to a system of fractional difference equations with parameters

“…Basic definitions and properties of fractional difference calculus were presented in [4], and discrete fractional boundary value problems have been found in . However, the studies of a system of fractional boundary value problems are quite rare (see [34][35][36][37][38][39][40][41][42]).…”