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 Δ2x(t−1)−L(t)x(t)+Vx′(t,x(t))=0 in the two cases that V(t,⋅)V(t,⋅) is superquadratic and subquadratic. Under the assumptions that L(t)L(t) is positive definite for sufficiently large |t|∈Z|t|∈Z, we show that there exists at least one non-trivial homoclinic orbit of the difference equation. Further, if V(t,x)V(t,x) is superquadratic and even with respect to xx, then it has infinitely many different non-trivial homoclinic orbits. At the end, two illustrative examples are provided.