Existence And Multiplicity Of Periodic Solutions For Several Types Of Second-order Differential Equations

The development of ordinary differential equations is closely related to the de-velopment of physics,mechanics,astronomy,and other science and technology.The problem of periodic solution for ordinary differential equations is one of the impor-tant branches of the qualitative theory of differential equations.This kind of problem embodies the regularity and balance in the development,which is an objective phe-nomenon in nature and a persistent hot issue.Many scholars have devoted themselves to investigating this type problem and have obtained abundant and classic results.Nowa-days,with the development of differential equation theory,the methods and theories to study the existence of periodic solutions are increasingly available,such as fixed point theory,critical point theory,topological degree theory,upper and lower solution method,Poincare-Birkhoff theorem,Lyapunov method,Poincare-Bendixson limit cy-cle theory and so on.The existence and multiplicity of periodic solutions for several kinds of differential equations are studied by using the critical point theory and the topological degree theory.Some new results are obtained,which extend and develop the previous results.The whole paper contains five chapters.The introduction part briefly introduces the research status of periodic solutions for second-order differential equations under the framework of Fucik spectral,the bound-ary value problems for second-order impulsive differential equations and the rotating periodic solutions for second-order impulsive Hamiltonian systems.Moreover,the main results are also simply described.The main aim of the second chapter is to investigate the existence of periodic solutions of second-order p-Laplacian equation with variable coefficients across half-eigenvalues.By using the p-Laplacian type continuity theorem,we obtain the exis-tence of periodic solutions for conservative and dissipative p-Laplacian equations with variable coefficients when the asymptotic behavior of nonlinearity at infinity across infinitely many half-eigenvalues.In the third chapter,the multiplicity of the periodic solutions of the second-order quasilinear equations with impulse effects is studied.The impulse effects produced by quasilinear terms is more general than that of linear terms.A new energy functional is given under impulsive effects.By using the symmetric mountain pass lemma and the properties of the genus,several theorems of multiplicity of solutions are obtained when the nonlinearity satisfies superliner,sublinear and concave-convex growth.In the fourth chapter,the existence and multiplicity of the rotating periodic so-lutions for second-order impulsive Hamiltonian systems are obtained.The rotating periodic solutions are much more extensive than the traditional periodic solutions(in-cluding periodic solutions,subharmonic solutions and quasi-periodic solutions).First,under the superlinear conditions,the multiplicity of the rotating periodic solutions is obtained by using the fountain lemma.Moreover,if the nonlinearity satisfies the sub-linear growth,the existence theorem of the rotation ground state periodic solution is also obtained.In the fifth chapter,the existence of the periodic solution of second-order impul-sive p-Laplacian equation is investigated.By using the Nehari manifold method,the existence theorem of the ground state solution is given when the nonlinearity satisfies the suplinear growth condition that is weaker than the classical Ambrosetti-Rabinowitz condition.Moreover,the existence of the periodic solutions for a kind of dissipative problem is also considered when the the primitive nonlinearity satisfies the asymptotic p-order growth.The last part is the summary of this dissertation and the prospect of future research.