Homoclinic orbits of a class of second-order difference equations

In this paper, we apply the variational method and the spectral theory of difference operators to investigate the existence of homoclinic orbits of the second-order difference equation Delta(2)x(t - 1) - L(t)x(t) + V-x' (t, x(t)) = 0 in the two cases that V(t, .) is superquadratic and subquadratic. Under the assumptions that L(t) is positive definite for sufficiently large vertical bar t vertical bar is an element of Z, we show that there exists at least one non-trivial homoclinic orbit of the difference equation. Further, if V(t, x) is superquadratic and even with respect to x, then it has infinitely many different non-trivial homoclinic orbits. At the end, two illustrative examples are provided. (C) 2012 Elsevier Inc. All rights reserved.