Abstract
In this article, we discuss how to use a standard minimizing argument in critical point theory to study the existence of non-trivial homoclinic solutions of the following second-order non-autonomous discrete systems Δ 2 x n - 1 + A Δ x n - L ( n ) x n + ∇ W ( n , x n ) = 0 , n ∈ Z , without any periodicity assumptions. Adopting some reasonable assumptions for A and L, we establish that two new criterions for guaranteeing above systems have one non-trivial homoclinic solution. Besides that, in some particular case, for the first time the uniqueness of homoclinic solutions is also obtained. MSC: 39A11.
Highlights
The theory of nonlinear discrete systems has widely been used to study discrete models appearing in many fields such as electrical circuit analysis, matrix theory, control theory, discrete variational theory, etc., see for example [1,2]
We give the following definition: if xn is a solution of a discrete system, xn will be called a homoclinic solution emanating from 0 if xn ® 0 as |n| ® +∞
We establish that two new criterions for guaranteeing the above system have one non-trivial homoclinic solution by adopting some reasonable assumptions for A and L
Summary
The theory of nonlinear discrete systems has widely been used to study discrete models appearing in many fields such as electrical circuit analysis, matrix theory, control theory, discrete variational theory, etc., see for example [1,2]. If xn ≠ 0, xn is called a non-trivial homoclinic solution. We consider the existence of non-trivial homoclinic solutions for the following second-order non-autonomous discrete system
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