On sequences of large homoclinic solutions for a difference equations on the integers involving oscillatory nonlinearities

In this paper, we determine a concrete interval of positive parameters $\lambda$, for which we prove the existence of infinitely many homoclinic solutions for a discrete problem $-\Delta \left( a(k)\phi _{p}(\Delta u(k-1))\right) +b(k)\phi_{p}(u(k))=\lambda f(k,u(k))$, $k\in Z$; where the nonlinear term $f : Z \times R \rightarrow R$ has an appropriate oscillatory behavior at infinity, without any symmetry assumptions. The approach is based on critical point theory.