Multiplicity Of Solutions For Second Order Discrete Problems With Resonance

In this paper,multiplicity of solutions of boundary value problems of second order discrete with resonance of form Is discussed by means of variational method of nonlinear functional analysis,especially the critical point theory,critical groups and the Morse theory,where Z [1 ,N]= {1,2,,N}, f∈C1 ( Z[1,N]×R1,R1), f (t ,0)=0,Δis the forward difference operator, i.e.Δu (t )=u(t+1)?u(t),Δ2 u (t)=Δ(Δu(t)). f satisfies resonant conditions at origin and infinity.This paper is composed of three chapters.In chapter one,the background and the method of the study for discrete problem,and main task of this paper are introduced.In chapter two,some basic knowledge of the critical point theory are given.some basic properties possessed by the corresponding energy functional J of the problem(1.2.1)are also presented.In the chapter three, the main results are proved. Respectively write Whereλn,λmare eigenvalues of the(1.2.1)corresponding linear eigenvalues value problem. Assume The main results obtained in this paper are as follows:Theorem 1.2.1 If in each of following cases holds, problem(1.2.1)has at least five nontrivial solutions,...