Multiplicity Results For Some Discrete Systems Near Resonance With A Nonprincipal Eigenvalue

In this paper, the multiplicity of solutions for discrete systems of the form is discussed by means of variational method of nonlinear functional analysis, especially the critical point theory, whereλis close to the nonprincipal eigenvalue of linear eigenvalue problem(1.2.3),Z[1,N]={1,2,…,N},△denotes the forward difference operator defined by△u(k)= u(k+1)-u(k),△2u(k)=△(△u(k)). F∈C2(R2,R1),k∈Z[1,N]and satisfies the sublinear growth condition: where▽F= (Fu, Fv) denotes the gradient of F with respect to (u, v)∈R2.This paper is composed of three chapters.In Chapter one, the background and the method of the study for discrete systems, the significance of study and main results of this paper are presented.In Chapter two, some basic knowledge of the critical point theory are given and the corresponding energy functional of the problem (1.2.1) is constructed. Also, we employ the Kronecker product properties of the matrix to solve the eigenvalues corresponding to the linear eigenvalue problem of (1.2.1), and some basic properties possessed by the functional J are also presented.In Chapter three, the main results are proved by saddle point theorem and linking theorem.The main results obtained in this paper are as follows:Theorem 1.2.1 Letλk(k≥2)be an eigenvalue of problem (1.2.3), suppose that F satisfies (1.2.2) and▽F(0)= 0. Assume that Then existsδ00>0 such that forλ∈(λk-δ0,λk), problem (1.2.1) has at least two solutions.Theorem 1.2.2 Letλk(k≥2)be an eigenvalue of problem (1.2.3), suppose that F satisfies (1.2.2) and▽F(0)= 0. Assume that and Then existsδ1>0 such that forλ∈(λk-δ1,λk), problem (1.2.1) has at least two solutions. Theorem 1.2.3 Letλk(k≥2)be an eigenvalue of problem (1.2.3), suppose that F satisfies (1.2.2) and▽F(0)= 0. Assume that Then existsδ2>0 such that forλ∈(λk,λk+δ2),problem (1.2.1) has at least two solutions.Theorem 1.2.4 Letλk(k≥2)be an eigenvalue of problem (1.2.3), suppose that F satisfies (1.2.2) and▽F(0)= 0. Assume that and Then existsδ3>0 such that forλ∈(λk,λk+δ3),problem (1.2.1) has at least two solutions.