Multiple Solutions For The Second Order Supperlinear Difference Systems

In this paper, the existence and multiplicity of nontrivial solutions for the following second-order super-linear difference equations systems is studied by using the critical point theory, local linking, homological linking, and the Morse theory, where λ∈R is a parameter,△is the forward difference operator, i.e.△u(t)=△u(t+1)-△u(t), and A2u(t)=△(△u(t)). F∈C2(R2,R) satisfies F(0)=0,(?)F(0)=0, VF=(F1’, F2’). Obvious, u=υ≡0is a trivial solution of (P)λ for any parameter λ∈R. In this paper, we prove that there exists at least one or three nontrivial 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 super linear 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) F(x)=o(|x|2) for|x|�'0. (F∞+) There exist R>0and μ>2such that0<μF(x)≤((?)F(x),x),｜x｜>R.(F∞+) There exist R>0and μ>2such that0>μF(x)≥((?)F(x),x),|x|>R.(F0±) There exist δ>0such that±F(x)≤0,|x|<δ.(F) For p>μ, there is M>0such that｜F(x)｜≤M(1＋｜x｜p),x∈R2.Here,(·,·) and||·||denote the usual inner product and the norm in R2.NoteThe main results are obtained in this paper as follows:Theorem1.2.1(i) Suppose that F satisfies (Fo),(F∞+) and (F). Let i∈Z[1,T-1] be fixed. There exist δ>0such that when M-≤δ, for λ∈(λi-δ,λi),then problem (P)λ has at least three nontrivial solutions.(ii) Suppose that F satisfies (F1),(F∞-) and (F). Let i∈Z[2,T] be fixed. There exist δ>0such that when M+≤δ, for λ∈(λi,λi+δ), then problem (P)λ has at least three nontrivial solutions.Theorem1.2.2(i) Suppose that F satisfies (F0),(F0-),(F∞+) and (F). Let i∈Z[1,T-1] be fixed. There exist δ>0such that when M-≤δ, for λ＝λi,then problem (P)λhas at least three nontrivial solutions.(ii) Suppose that F satisfies (Fo),(Fo+),(F∞-) and (F). Let i∈Z[2,T] be fixed. There exist δ>0such that when M+≤δ, for λ=λi, then problem (P)λhas at least three nontrivial solutions. Theorem1.2.3(i) Suppose that F satisfies (F0),(F∞+),(F) and F≥0. Let i∈Z[1, T-1] be fixed. There exist δ>0such that A∈(λi-δ, λi), then problem (P)λ has at least three nontrivial solutions.(ii) Suppose that F satisfies (Fo),(F∞-),(F) and F≤0. Let i∈Z[2,T] be fixed. There exist δ>0such that λ∈(λi,λi+δ), then problem (P)λhas at least three nontrivial solutions.Theorem1.2.4(i) Suppose that F satisfies (Fo),(F∞+) and (F). Let i∈Z[1,T-1] be fixed. There exist δ>0such that when0<M-≤δ, for λ∈(λi,λi+δ), then problem (-P)λ has at least one nontrivial solutions.(ii) Suppose that F satisfies (F0),(F∞-) and (F). Let i∈Z[2,T] be fixed. There exist δ>0such that when0<M+≤δ, for λ∈(λi-δ,λi), then problem (P)λ has at least one nontrivial solutions.