Multiplicity Of Periodic Solutions For The Second-order Nonlinear Difference Equations With Resonance

In this paper, the existence and multiplicity of nontrivial periodic solutions for the following second-order nonlinear difference equations with resonance at both origin and infinity is studied by using the critical point theory, the minimax methods, homological linking, the Morse theory and the computations of the critical groups, where f satisfies f (k,0)=0for k∈Z[1,N],△is the forward difference operator, i.e.△x(k)=△x(k+1)-△x(k), and△2x(k)△(△x(k)). In this paper, we prove that there exists at least one, two or three nontrivial periodic solutions which based on some given conditions. The paper is organized as follows.In Chapter1, we introduce the background, methods and significance of the study for the resonant difference equations, and the main results in this paper.In Chapter2, we introduce Morse theory and variational structure of discrete system.In Chapter3, we give some lemmas and the proofs of main results in this paper.Throughout this paper, let f satisfy the following conditions: (f0±) There exists δ>0such that when|u|≤δ,(f1+) There exists r>0and μ>2such that(f1-) There exists r>0andμ>2such that(F) For any p≥μ,. there exists C>0such that Here F(k,u)=f0u f(k,s)ds. λm, λl are two eigenvalues of (P0).Problem (P) is called resonant at origin (or at infinity) if f satisfies (f0)(or f∞).The main results are obtained in this paper as follows:Theorem1.2.1Let f satisfy (f∞-),(f∞), and m∈Z[2,N1]. If one of the following conditions:(i) f(k,0)<λ1,k∈Z[1,N](ii)f(k,0)>λN1,k∈Z[1,N](iii)λl<f1(k,0)<λl+1,k:∈Z[1, N] and l≠m-1is satisfied, then (P) has at least one nontrivial periodic solution.Theorem1.2.2Let f satisfy (f∞),(f∞), and m∈Z[1,N1-1]. If one of the following conditions:(i)f1(k,0)<λ1,k∈Z[1,N](ii) f1(k,0)>λN1,k∈Z[1,N] (ⅲ)λl<f’(k,0)λl+1,k∈Z[1;N]and m≠l is satisfied,then(P)has at least one nontrivial periodic solution.Theorem1.2.3Aussme that f satisfy(f0),(f∞),and l,m∈Z[1,N1-1].If one of the following conditions:(ⅰ)(f∞-),(f0-)and l∈Z[2,N1],m≠l(ⅱ)(f∞-),(f0+)and l∈Z[2,N1],m≠l+1(ⅲ)(f∞+),(f0-)and l∈Z[1,N1-1],m≠l-1(ⅳ)(f∞+),(f0+)and l∈Z[1,N1-1],m≠l is satisfied,then(P)has at least one nontrivial periodic solution.Theorem1.2.4Let f satisfy(f0),and k∈Z[1,N].If one of the following conditions:(ⅰ)(f0-),l∈Z[2,N1]andm≠l-1,l∈Z[2,N1](ⅱ)(f0+),l∈Z[1,N1-1] and m≠l,l∈Z[1,N1-1] is satisfied,then(P)has at least two nontrivial periodic solutions.NoteTheorem1.2.5Let f satisfy(f0)and(F).If one of the following conditions:(ⅰ)f satisfies(f1+)and(f0-),there exists τ>0such that M-≤τ,l∈Z[1,N-2](ⅱ)f satisfies(f1-)and(f0+),there exists τ>0such that M+≤τ,l∈Z[3,N1] is satisfied,then(P)has at least three nontrivial periodic solutions.