Multiple Solutions to Fractional Difference Boundary Value Problems

Abstract

The following fractional difference boundary value problems ${▵}^{\nu }y\left(t\right)=-f\left(t+\nu -1,y\left(t+\nu -1\right)\right)$, $y(\nu -2)=y(\nu +b+1)=0$ are considered, where is a real number and ${▵}^{\nu }y(t)$ is the standard Riemann-Liouville fractional difference. Based on the Krasnosel’skiǐ theorem and the Schauder fixed point theorem, we establish some conditions on $f$ which are able to guarantee that this FBVP has at least two positive solutions and one solution, respectively. Our results significantly improve and generalize those in the literature. A number of examples are given to illustrate our main results.