Gap solitons in periodic difference equations with sign-changing saturable nonlinearity

In this paper, we consider the existence of gap solitons for a class of difference equations:

$$L{u}_n-\omega u_n=f_n(u_n),n\in \mathbb{Z},$$

where $L{u}_n=a_n{u}_{n+1}+a_{n-1}{u}_{n-1}+b_n{u}_n$ is the discrete difference operator in one spatial dimension, $\{a_n\}$ and $\{b_n\}$ are real valued T-periodic sequences, $\omega \in \mathbb{R}$, $f_n(\cdot )\in C(\mathbb{R},\mathbb{R})$ and $f_{n+T}(\cdot )=f_n(\cdot )$ for each $n\in \mathbb{Z}$. Under general asymptotically linear conditions on the nonlinearity $f_n(\cdot )$, we establish the existence of gap solitons for the above equation via variational methods when $t{f}_n(t)$ is allowed to be sign-changing. Our methods further extend and improve the existing results.