Multiplicity Of Solutions For Plane Discrete Systems At Resonance

In this paper, multiplicity of solutions for boundary value problem of plane resonance difference equation systems is discussed by the variational method of nonlinear functional analysis and the critical point theory, where Z[1,N]={1,2,…,N}, F∈C2(R2,R1) andΔdenotes the forward difference operator defined byΔu(k)=u(k+1)-u(k),Δ2u(k)=Δ(Δu(k)).This paper is composed of four chapters.In Chapter one, the background and the methods of the study for difference equation systems are introduced. Furthermore, the work and main results of this paper are presented (Four theorems about the existence and multiplicity of solutions are obtained)In Chapter two, some basic knowledge of the critical point theory and Morse theory are introduced.In Chapter three, the matrix form of the problem (1.2.1) is derived and the corresponding energy functional of the problem (1.2.1) is constructed, several relevant properties of it are introduced at the same time. Meanwhile,the eigenvalue of the corresponding to the linear eigenvalue problem of (1.2.1) are obtained, and some related basic conclusions are given.In Chapter four, the demonstration of the main results in this paper is obtained with the calculation of critical groups, using the mountain pass theorem, the saddle point theorem, and Morse theory.