Multiplicity Of Solutions For Discrete Boundary Value Problems With Multi-resonance

In this paper, multiplicity of solutions for discrete multi-resonance second-order boundary value problem is discussed by means of variational method of nonlinear functional analysis, the critical point theory, especially Morse theory, as well as the calculations of the critical groups. Where positive integer N≥3,[1,N]={1,2,3,……,N},△denotes the forward difference operator, i.e.△u(k-1)=u(k)-u(k-1),△2u(k)=△u(△(k)). The nonlinear term f(k；)∈C1(R1,R1) satisfies multi-resonance at infinity: Where λκ is theKth eigenvalue of the linear boundary value problemThis paper is composed of three chapters. In chapter one, the background and the method of the study for resonance discrete boundary value problems, the significance of study and main results of this paper are presented.In chapter two, the basic lemmas and definitions about critical point theory and the energy functional of the problem (1.1.1) are introduced.In chapter three, the main results are proved by using Morse theory and the calculations of the critical groups.