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

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