On homoclinic solutions for a second order difference equation with p-Laplacian

In this paper, we obtain conditions under which the difference equation $-\Delta \left( a(k)\phi {p}(\Delta u(k-1))\right) +b(k)\phi{p}(u(k))=\lambda f(k,u(k)),\quad k\in $\mathbb{Z}$$, has infinitely many homoclinic solutions. A variant of the fountain theorem is utilized in the proof of our theorem. It improves the results in [L.Kong, homoclinic solutions for a second order difference equation with $p-$Laplacian, $\mathit{\text{Appl.}}\mathit{\text{Math.}}\mathit{\text{Comput}}$., $\mathbf{\text{247}}$ (2014), 1113–1121], where the set of conditions imposed on nonlinearity is inconsistent.