Multiplicity Of Solutions For Boundary Value Problems Of Second-order Difference Equations

In this paper, the multiplicity of solutions for difference equations of the form is discussed by using variational methods of the nonlinear functional analysis, combining critical point theory, especially the critical groups and the Morse theory, where N≥3is a fixed integer, the discrete internal Z[1,N]={1,2,...,N}, α <1,β≤1are constant,△denotes the forward difference operator defined by△u(k)=u(k+1)-u(k),△2u(k)=△(△u(k)), f(k,·)∈C1(R1,R1) satisfies the resonance condition, and f(/k,0)=0.This paper is composed of three chapters.In Chapter one, the study background for boundary problems of difference equations and the study methods for the existence of the solutions, the study significance and main results of this paper are introduced. Four theories for multiplicity of solutions are given. In Chapter two, some basic conclutions about the critical point theory that will be used are introduced and the energy functional J corresponded to the problem (1.2.1) is constructed.In Chapter three, the prove for the main results is presented by the conclutions introduced in Chapter two, combining the positive and negtive energy functional and by the computations of the critical groups.