Abstract
By using the critical point theory, some existence criteria are established which guarantee that the difference p-Laplacian systems of the form have at least one or infinitely many homoclinic solutions, where , , , , , and are not periodic in n. MSC:34C37, 35A15, 37J45, 47J30.
Highlights
Consider homoclinic solutions of the following p-Laplacian system: u(n – ) p– u(n – ) – a(n) u(n) q–pu(n) + ∇W n, u(n) =, n ∈ Z, ( . )where < p < (q + )/, q >, n ∈ Z, u ∈ RN, a : Z → (, +∞), and W : Z × RN → R are not periodic in n. is the forward difference operator defined by u(n) = u(n + ) – u(n), u(n) = ( u(n))
Where < p < (q + )/, q >, n ∈ Z, u ∈ RN, a : Z → (, +∞), and W : Z × RN → R are not periodic in n. is the forward difference operator defined by u(n) = u(n + ) – u(n)
It is well known that the existence of homoclinic orbits for Hamiltonian systems is a classical problem and its importance in the study of the behavior of dynamical systems has been firstly recognized by Poincaré [ ]
Summary
1 Introduction Consider homoclinic solutions of the following p-Laplacian system: u(n – ) p– u(n – ) – a(n) u(n) q–pu(n) + ∇W n, u(n) = , n ∈ Z, Several authors [ – ] used critical point theory to study the existence of homoclinic orbits for difference equations. Suppose that a and W satisfy the following conditions: (A) Let < p < (q + )/ and q > , a : Z → ( , +∞) is a positive function on Z such that for all n ∈ Z
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