Multiplicity Of Solutions For Sturm-Liouville Difference Equations At Resonance

In this paper the multiplicity of solutions for Sturm-Liouville difference equations at resonance is discussed by means of variational method of nonlinear functional analysis, where is a given positive sequence, f satisfies resonant conditions both at zero and at infinity,Δis the forward difference operator,This paper is composed of three chapters.In chapter one, the background, the method and the latest advances in research of the problem (1.2.1) are introduced.In chapter two, some basic theories of critical point and energy functional J of the problem (1.2.1) are induced. And several basic properties which J satisfies are proved.In chapter three, Morse theory and computations of the critical groups at infinity are employed to discuss the multiplicity of solutions for Sturm-Liouville difference equations at resonance, which implies that the problem (1.2.1) has at least five nontrivial solutions under certain assumptions .