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 Δ 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 | t | ∈ 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.