(Periodic Solutions And Homoclinic Orbits For Differential Equations And Inclusions)

This doctoral dissertation mainly concerns the applications of variational method and critical point theory to investigating the existence of periodic solutions and homoclinic orbits for differential equations and inclusions. It is composed of two parts. In Part one (Chapter2and Chapter3), we deal with the existence and multiplicity of homoclinic orbits for several smooth dynamical systems. In Part two (Chapter4), we investigate the existence of periodic solutions and homoclinic orbits for several types of non-smooth dynamical systems.In Chapter1, we introduce the historic background and the recent develop-ment of problems to be studied. The status and the up-to-date progress for all the investigated problems are given. Some preliminaries are also introduced in this chapter.In Chapter2, by using the variational methods, we investigate the existence and multiplicity of homoclinic orbits for second-order Hamiltonian systems. Some open problems are resolved. We obtain some new existence results upon critical point theory. Furthermore, some results in the literatures are improved.In Chapter3, we investigate the homoclinic orbits for p(t)-Laplacian systems and obtain new results.In Chapter4, we deal with the existence of periodic solutions and homoclinic orbits for second-order p(t)-Laplacian systems with non-smooth potential, and obtain some new results.