Abstract
Abstract In this paper, we study the existence of multiple periodic solutions for nonlinear second-order difference equations with resonance at origin. The approach is based on critical point theory, minimax methods, homological linking and Morse theory.
Highlights
1 Introduction In this paper, we consider the existence of multiple periodic solutions for the following nonlinear difference equations: (P) – u(k – ) = λmu(k) + f (k, u(k)), k ∈ Z[, N], u( ) = u(N), u( ) = u(N + ), where N > is a fixed integer, u(k) = u(k + ) – u(k), u(k) = ( u(k)), f (k, ·) : R → R is a differential function satisfying f (k, ) =, k ∈ Z[, N], ( . )
Since ( . ) implies that (P) possesses a trivial periodic solution u ≡, we are interested in finding nontrivial periodic solutions for (P)
Critical point theory has been widely used to study the existence of periodic solutions and solutions for nonlinear difference boundary value problems since the first result was established by using variational methods in
Summary
We consider the existence of multiple periodic solutions for the following nonlinear difference equations:. (P) – u(k – ) = λmu(k) + f (k, u(k)), k ∈ Z[ , N], u( ) = u(N), u( ) = u(N + ), where N > is a fixed integer, u(k) = u(k + ) – u(k), u(k) = ( u(k)), f (k, ·) : R → R is a differential function satisfying f (k, ) = , k ∈ Z[ , N],. ) implies that (P) possesses a trivial periodic solution u ≡ , we are interested in finding nontrivial periodic solutions for (P). It follows from [ ] that all the eigenvalues of (P )
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