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

Abstract

We study a class of nonlinear fractional difference equations with nonlocal boundary conditions at resonance. The system is inspired by the three‐point boundary value problem for differential equations that have been extensively studied. It is also an extension to a fractional difference equation arising from real‐world applications. Converting the problem to an equivalent system corresponding to the integral operator and Green's function for differential equations, we are able to apply the coincidence degree theory for semilinear operators to obtain sufficient conditions for the existence of solutions. In addition, we prove a new property of the Gamma function and construct a family of examples to illustrate the applications of the results.