Infinitely Many Homoclinic Orbits for 2 n th-Order Nonlinear Functional Difference Equations Involving the p -Laplacian

Abstract

By establishing a new proper variational framework and using the critical pointtheory, we establish some new existence criteria to guarantee that the 2$n$th-order nonlinear difference equation containing both advance and retardation with $p$-Laplacian ${\Delta }^n\left(r\left(t-n\right){\phi }_p\left({\Delta }^{n}u\left(t-1\right)\right)\right)+q\left(t\right){\phi }_p\left(u\left(t\right)\right)=f\left(t,u\left(t+n\right),\dots ,u\left(t\right),\dots ,u\left(t-n\right)\right)$, $n\in \mathbb{Z}\left(3\right)$, $t\in \mathbb{Z}$, has infinitely many homoclinic orbits, where ${\phi }_p\left(s\right)$ is $p$-Laplacian operator; $r$, $q$, $f$ are nonperiodic in $t$. Our conditions on the potential are rather relaxed, and some existing results in the literature are improved.