Rotation Numbers,Fixed Points And Periodic Solutions For Time-dependent Planar Hamiltonian Systems

In this paper,we consider the existence,multiplicity of periodic solutions for time-dependent planar Hamiltonian systems and the related problems via rotation numbers and fixed point theorems.The thesis consists of the following three parts.1.The existence and multiplicity of periodic solutions for time-dependent planar Hamiltonian systems.2.The existence of periodic solutions for nonconservative perturbation of Hamilto-nian systems.3.Necessary and sufficient conditions for a sequence strong convergence to the fixed point of nonexpansive mappings.In the first part of the thesis,we introduce the concept of spiral curves and use the variations of polar radius to estimate the changes of solution angles.We establish a unified approach for applying Poincare-Birkhoff twist theorem for planar Hamiltoni-an systems without the global existence of the solutions.Thus,the classical work of Jacobowitz and Hartman on the second order time-dependent Hamiltonian equations is extended to the planar time-dependent Hamiltonian systems.Furthermore,we give the definition of rotation numbers of planar linear periodic systems,and describe the twist properties for solutions of nonlinear systems via rotation numbers.When we describe the twist properties of solutions by calculating the revolution numbers,we require the sign condition.Comparing with the linear systems in positively homogeneous sense,we require an additional condition.Our method doesn't require this restriction.Moreover,we also discuss the specific estimates of the rotation numbers for some typical planar linear systems using the phase-plane analysis.The framework of spiral curves and rota-tion numbers can be applied to some important models.This is a new progress in the application of Poincare-Birkhoff twist theorem.As two typical applications of our general approach,we give the sufficient condi-tions for the existence of spiral curves of second order time-dependent Hamiltonian e-quations,and discuss the rotation numbers and the spiral properties of one-dimensional p-Laplacian equations in p-polar coordinate transformation.Thus,we prove the exis-tence and multiplicity of periodic solutions for the equations via Poincare-Birkhoff twist theorem.The results extend the recent work by Fonda,Torres,Boscaggin,Ortega,Zano-lin,Ping Yan and Meirong Zhang.In the second part of the thesis,we investigate the existence of periodic solution-s for nonconservative perturbation Hamiltonian systems without the global existence of solutions.Using the theory of topological degree,we establish a new twist theorem for non area-preserving continuous mappings.This is the first fixed point theorem with an-gle description for non area-preserving continuous mappings.By the technique of spiral curves,we prove the existence of periodic solutions for two typical nonconservative per-turbation of second order superlinear Hamiltonian equations.Thus,the classical work of Jacobowitz and Hartman is extended to the nonconservative perturbation superlin-ear equations.Moreover,it extends the existence of periodic solutions for second order superlinear equations by Capietto,Mawhin and Zanolin via continuation theorems.In the third part of the thesis,we find a necessary and sufficient condition of the sequence generated by a composite iterative algorithm strong convergence to its fixed point,when the fixed point of the nonexpansive mappings exists.