On the solvability of the discrete nonlinear Schrodinger equation with subcubic potential

In this paper, we analyze the solvability of the discrete nonlinear Schrödinger equation \begin{equation*} iβ(Δ_t+\nabla_t)φ(t,k) +γ|φ(t,k)|^2φ(t,k) +\varepsilon Δ_k^2φ(t,k-1) = g(t,φ(t,k)), \end{equation*} where ${\mathrm{\Delta }}_t$ and ${\mathrm{\Delta }}_k$ denote the standard forward difference operators in the variables $t$ and $k$, respectively, ${\mathrm{\nabla }}_t$ denotes the standard backward difference operator in $t$, and \begin{equation*} Δ_k^2φ(t,k-1) = φ(t,k+1)-2φ(t,k)+φ(t,k-1) \end{equation*} is the discrete Laplacian operator in the spatial variable $k$. Throughout, we will assume the parameters $\beta$ and $\epsilon$ are positive real numbers, the parameter $\gamma$ is a nonzero real number, and the potential function $g:\mathbb{Z}\times \mathbb{C}\to \mathbb{C}$ is continuous.