Multiplicity Of Solutions For Second Order Discrete Boundary Value Problems With Double Resonance

In this paper, the multiplicity of solutions for second order discrete boundary value problem with double resonance of the form Is discussed by means of variational method, combined with critical point theory, Morse theory and calculation of critical groups, where ,Δdenotes the forward difference operator defined by .Also, f satisfies double resonance.This paper is composed of three chapters.In chapter one, the background and the method are introduced.In chapter two, We state some basic of critical point theory and Morse theory,and we introduce The energy functional J of the problem(1.1.1).Also, Some basic properties of functional J are proved. In chapter three, critical point theory, Morse theory and calculation of critical groups are employed to discuss the multiplicity of solutions for second order discrete boundary value problem with double resonance of the form Under certain hypothesis conditions, we prove (1.1.1) has at least three nontrivial solutions, four nontrivial solutions and six nontrivial solutions.