Multiple Periodic Solutions Of Nonlinear Differential Equations And Homoclinic Solution

The Z₂-group index theory and the symmetric mountain pass lemma are respectively used to explore the existence of multiple periodic solutions and homoclinic solutions for several kinds of nonlinear difference equations.Sufficient conditions are obtained for the existence of periodic solutions and homoclinic orbits for these difference equations.In chapter 1,we introduce the backgroud of this study and propose the problems to be dealed with.In chapter 2,the critical point theory is briefly introduced and some concepts and symbols are presented.In chapter 3,the Z₂-group index theory is used to explore the existence of multiple periodic solutions for a class of nonlinear second-order difference equations, and the above results for the second-order case are generalized to the corresponding high-order difference equations.That is to say,we obtain the sufficient condition of the existence of multiple periodic Solutions for the corresponding 2qth order difference equations.In chapter 4,we consider the existence of homoclinic solutions for a class of second-order self-adjoint nonlinear difference equations by the symmetric mountain pass lemma.A sufficient condition is obtained for the existence of multiple homoclinic solutions.In chapter 5,we present the summary and prospect the future works.