Applications Of Variational Methods To Problems Of Differential Equations With Gyroscopic Terms

Variational method is an important method to the boundary value problems for dif-ferential equations with variational structures. In this paper, we study the existence and multiplicity of solutions for Hamilton systems with gyroscopic terms under different condi-tions. Moreover, the homoclinic solutions of second order Hamilton systems with gyroscopic terms are also considered. Finally, we discuss the existence of solutions of differential systems for impulsive problems with mixed boundary value conditions under certain conditions.The thesis contains four chapters.In the first chapter, we introduce the backgrounds, preliminaries and structure of the paper.Chapter2is devoted to the periodic solutions of second order nonautonomous Hamilton systems with gyroscopic terms. We consider the solutions using generalized Ahmad-Lazer-Paul type conditions and don’t need the gyroscopic term being small. We not only obtain the existence of nontrivial solutions but also get multiple solutions of the second order nonau-tonomous Hamilton systems.In chapter3, we study the homoclinic solutions of the non-autonomous Hamilton systems with gyroscopic terms. We obtain infinitely many homoclinic solutions of the system when the gyroscopic term is small.In chapter4. we study the bondary value problems of impulsive differential equations with mixed two-point boundary value conditions. We get some existence results both in the resonant and non resonant case.