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

We are concerned with the existence of positive solutions for the boundary value problem \(\begin{aligned} \left\{ \begin{array}{ll} -D^{2}u(n-1)+c(n)Du(n)+a(n)u(n)=\frac{b(n)}{(u(n))^p},&{}\quad n\in \mathbb {Z},\\ \lim _{|n|\rightarrow +\infty }u(n)=0,\\ \end{array}\right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-D^{2}u(n-1)+c(n)Du(n)+a(n)u(n)=\frac{b(n)}{{(u(n))}^p},\hfill & n\in \mathbb{Z},\hfill \\ {lim}_{|n|\to +\infty }u(n)=0,\hfill \end{array}\right)\end{array}$$ where \(a,b:\mathbb {Z}\rightarrow \mathbb {R}\) $a,b:\mathbb{Z}\to \mathbb{R}$ , \(c:\mathbb {Z}\rightarrow (0,1)\) $c:\mathbb{Z}\to (0,1)$ , \(p>0\) $p>0$ , and D is the forward difference operator. The main tools used are fixed point theorems of cone-compressing and cone-condensing type.