Positive solutions for singular discrete boundary value problems

Abstract

We study the existence of zero-convergent solutions for the second-order nonlinear difference equation $\Delta \left(a_n{\Phi }_p\left(\Delta x_n\right)\right)=g\left(n,x_{n+1}\right)$, where ${\Phi }_p\left(u\right)={\left|u\right|}^{p-2}u$, $p>1$, $\left\{a_n\right\}$ is a positive real sequence for $n\ge 1$, and $g$ is a positive continuous function on $\mathbb{N}\times \left(0,u_0\right)$, $0<u_0\le \infty$. The effects of singular nonlinearities and of the forcing term are treated as well.