Positive solutions for nonlinear discrete periodic boundary value problems

Abstract

We are concerned with the nonlinear discrete periodic boundary value problem

(P)$$-\Delta \left[p(t-1)\Delta u(t-1)\right]+q\left(t\right)u\left(t\right)=rg\left(t\right)f\left(u\left(t\right)\right)\text{,}t\in {[1,T]}_{\mathbb{Z}}\text{,}$$$$u\left(0\right)=u\left(T\right)\text{,}p\left(0\right)\Delta u\left(0\right)=p\left(T\right)\Delta u\left(T\right)\text{,}$$

where $r$ is a positive parameter. Optimal interval will be given to the parameter $r$ to ensure that (P) has at least one positive solution.