Multiplicity Of Solutions For Boundary Value Problems Of Second Order Resonant Difference Equations

In this paper, the multiplicity of solutions for boundary value problems of second order resonant difference equation of the form is discussed by means of variational method of nonlinear functional analysis, especially the critical point theory and the Morse theory, where denotes the forward difference operator defined byThis paper is composed of three chapters.In the Chapter one, the background and the method of the study for difference equations, the significance of study and main results of this paper are presented. In the Chapter two, some basic knowledge of the critical point theory and the Morse theory are given and the corresponding energy functional of the problem(1.2.1)is constructed. Some basic properties possessed by the functional J are also presented.In the Chapter three, the main results are proved.The main results obtained in this paper are as follows:Theorem 1.2.1 Let ( f 1) hold and lti→m 0 f ( kt , t)<λ1. Then the problem (1.2.1) has at least three nontrivial solutions in each of the following cases:Theorem 1.2.2 Let ( f 1) hold and l =1. Then the problem (1.2.1) has at least two nontrivial solutions in each of the following cases:Theorem 1.2.3 Let ( f 1) and ( f 2) hold and l , m≥2. Then the problem (1.2.1) has at least four nontrivial solutions in each of the following cases: (iii) ( g^-), ( f₀⁺) with m≠l?1; (iv) ( g^-), ( f₀^-) with m≠l.