Existence Of Solutions For 2-dimensional Discrete Systems At Resonance

In this paper, the existence of solutions for boundary value problems of 2-dimensional resonant difference equation systems under the mixed boundary conditions of the form is discussed by means of the variational method of nonlinear functional analysis and the critical point theory, where Z[1, N]={1,2,…, N}, F∈C2(R2,R1) satisfies the sublinear growth condition where (?)F=(Fu,Fv), and△denotes the forward difference operator defined by△u(k)=u(k+1)-u(k),△2u(k)=△(△u(k)), a,b,c>0, and ac>b2.This paper is composed of three chapters. In Chapter one, the background and the methods of the study for difference equation systems, the work and main results of this paper are presented.In Chapter two, some basic knowledge of the critical point theory are introduced. Then the matrix form of the problem (1.2.1) is derived 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:λ(1),λ(2),…,λ(2N) corresponding to the linear eigenvalue problem of (1.2.1), and some related basic conclusions are given.In Chapter three, the existence and multiplicity of solutions for the problem (1.2.1) are obtained by using Ekeland's variational principle, the mountain pass theorem and the saddle point theorem etc.The main results obtained in this paper are as follows:Theorem 1.2.1 Suppose that F satisfies the condition (1.2.2) and (?). Then the problem (1.2.1) has at least three solutions forλ<λ(1) which is sufficiently close toλ(1).Theorem 1.2.2 Suppose that F satisfies the condition (1.2.2). Then the problem (1.2.1) has at least one solution forλ(k)<λ<λ(k+1).Theorem 1.2.3 Suppose that F satisfies the condition (1.2.2) and (?). Then forλ=λ(k), the problem (1.2.1) has at least one solution.Theorem 1.2.4 Suppose that F satisfies the condition (1.2.2) and(?).If(?),▽F(0)=0 and there is a positive numberδsuch that F(s)≤0 for |s|≤δ,then forλ=λ(k)(k≥2),the problem(1.2.1)has at least one nontrivial solution.