Existence of Homoclinic Orbits for a Class of Asymptotically p-Linear Difference Systems with p-Laplacian

and Applied Analysis 3 W4 W̃ n, x ∇W n, x , x − pW n, x , lim |x|→∞ W̃ n, x ∞ uniformly for n ∈ Z, 1.7 and for any fixed 0 < c1 < c2 < ∞, inf n∈Z, c1≤|x|≤c2 W̃ n, x |x| > 0. 1.8 Then problem 1.1 has at least one nontrivial homoclinic solution. Remark 1.2. The function W n, x in this paper is asymptotically p-linear at infinity. The behavior of the gradient of W n, x at infinity is like that of a function V∞ n |x|p−2x, where V∞ n is a real function but not amatrix function. To the best of our knowledge, similar results of this kind of p-Laplacian difference systems with asymptotically p-linearW n, x at infinity cannot be found in the literature. From this point, our result is new. 2. Preliminaries Let S { {u n }n∈Z : u n ∈ R, n ∈ Z } , E { u ∈ S : ∑ n∈Z [|Δu n − 1 | |u n |p] < ∞ } , 2.1 and for u ∈ E, let ‖u‖ {∑ n∈Z [|Δu n − 1 | |u n |p] < ∞ }1/p . 2.2 Then E is a uniform convex Banach space with this norm. As usual, for 1 ≤ p < ∞, let l ( Z,R ) { u ∈ S : ∑ n∈Z |u n | < ∞ } , l∞ ( Z,R ) { u ∈ S : sup n∈Z |u n | < ∞ } , 2.3 and their norms are given by ‖u‖lp (∑ n∈Z |u n | )1/p , ∀u ∈ l ( Z,R ) , ‖u‖∞ sup{|u n | : n ∈ Z}, ∀u ∈ l∞ ( Z,R ) , 2.4