Bounded and homoclinic-like solutions of second-order singular difference equations

Abstract

We are concerned with the existence of positive solutions for the boundary value problem

$$\begin{aligned} \left\{ \begin{array}{ll} -D^{2}u(n-1)+c(n)Du(n)+a(n)u(n)=\frac{b(n)}{(u(n))^p},&{}\quad n\in \mathbb {Z},\\ \lim _{|n|\rightarrow +\infty }u(n)=0,\\ \end{array}\right. \end{aligned}$$

where \(a,b:\mathbb {Z}\rightarrow \mathbb {R}\), \(c:\mathbb {Z}\rightarrow (0,1)\), \(p>0\), and D is the forward difference operator. The main tools used are fixed point theorems of cone-compressing and cone-condensing type.