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

In this paper, we analyze the solvability of the discrete nonlinear Schrödinger equation

$$i\beta ({\mathrm{\Delta }}_t+{\mathrm{\nabla }}_t)\phi (t,k)+\gamma |\phi (t,k){|}^2\phi (t,k)+\epsilon {\mathrm{\Delta }}_k^2\phi (t,k-1)=g(t,\phi (t,k)),$$

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

$${\mathrm{\Delta }}_k^2\phi (t,k-1)=\phi (t,k+1)-2\phi (t,k)+\phi (t,k-1)$$

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.