부산대학교 도서관

This paper deals with higher-order nonlinear difference equations containing both advance and retardation. In particular, the $2n$th-order difference equation $$ \Delta^n\left(\gamma_{k-n}\Delta^nu_{k-n}\right)+(-1)^n\chi_ku_k=(-1)^nf\left(k,u_{k+1},u_k,u_{k-1}\right),\quad k\in{\Bbb Z}, \tag1 $$ where $n$ is a fixed positive integer, is investigated. Here, $\Delta$ is the forward difference operator, $\Delta^n=\Delta(\Delta^{n-1})$, $\gamma\:{\Bbb Z}\to (0,\infty)$ and $\chi\:{\Bbb Z}\to (0,\infty)$ are $T$-periodic, and $f\:{{\Bbb Z}\times{\Bbb R}^3}\to{\Bbb R}$ is continuous and $T$-periodic in its first variable for a given positive integer $T$. \par The authors provide conditions which guarantee the existence of a nontrivial homoclinic solution to Equation (1). The proof is based on the Mountain Pass Theorem and periodic approximations. One illustrative example is given.