The Existence of Solutions to a System of Discrete Fractional Boundary Value Problems

Abstract

We study the existence of solutions for the boundary value problem $-{\Delta }^{\nu }{y}_1\left(t\right)=f\left(y_1\right(t+\nu -1),y_2(t+\mu -1\left)\right)$, $-{\Delta }^{\mu }{y}_2\left(t\right)=g\left(y_1\right(t+\nu -1),y_2(t+\mu -1\left)\right)$, $y_1(\nu -2)=\Delta y_1(\nu +b)=0$, $y_2(\mu -2)=\Delta y_2(\mu +b)=0$, where , $f,g:\mathbb{R}\{\backslash times\}\mathbb{R}\to \mathbb{R}$ are continuous functions, $b\in {\mathbb{N}}_0$. The existence of solutions to this problem is established by the Guo-Krasnosel'kii theorem and the Schauder fixed-point theorem, and some examples are given to illustrate the main results.