Existence of Homoclinic Solutions for a Class of Nonlinear Difference Equations

By using the critical point theory, we establish some existence criteria to guarantee that the nonlinear difference equation. Delta[p(n)(Delta x(n - 1))(delta)] - q(n)(x(n))(delta) = f(n,x(n)) has at least one homoclinic solution, where n is an element of Z, x(n) is an element of R, and f : Z x R -> R is non periodic in n. Our conditions on the nonlinear term f (n, x(n)) are rather relaxed, and we generalize some existing results in the literature.