Affine-Periodic Solutions For Ordinary Differential Equations

This thesis is concerned with the existence of affine-periodic solutions to ordinary differential equations including first order,second order,functional differential equations and vector ordinary differential equations.In the first section of the first chapter,we briefly introduce the birth,background and wide applications of differential equations.Meanwhile,the structure of solutions to differential equations are illustrated simply together with the concept of regular solutions.Then we give the definition of affine-periodic solution,which is studied in our thesis.Also,some fruitful results on affine-periodic solutions are presented roughly.In the second section,we introduce the Leray-Schauder's continuation method in solving nonlinear e-quations,including algebraic equations and integral equations.In the third section,we introduce the history of the lower and upper solutions method,and some classical results of boundary value probelms.Finally,some basic lemmas related to our research are listed in the fourth section.In the second chapter,we obtain the existence results for dissipative affine-periodic systems.In the first section,we introduce some works on dissipative systems.Then we introduce the main result on dissipative affine-periodic solutions to first order dissipative systems in[0,?).In the third section,some specific examples are given as an application of the existence theorem.In the last section,we firstly turn the affine-periodic problem to the affine-periodic boundary value problem,and then define the solution operator using the equivalent integral equation.Finally,we prove the existence of affine-periodic solution with the help of the continuation method,the lower and upper solutions method and the homotopy invariance of topological degree.In the third chapter,at first,we introduce the Nagumo lemma including its contents,proof and counterexample as the Nagumo condition falls,and we obtain another similar version of Nagumo lemma where the control function h adds some positive constant.In the next section,we introduce the existence of affine-periodic solutions to second-order ordinary differential equations in R.In the third section,some specific examples on oscillators are listed as an application of the main theorem.In the final section,we firstly turn the affine-periodic problem to the affine-periodic boundary value problem,and then define the solution operator using the equivalent integral equation.At last,we prove the existence of affine-periodic solution via the continuation method,the lower and upper solutions method and the homotopy invariance of topological degree.In the fourth chapter,we study the existence of affine-periodic solution to functional differential equations.In the first section,we introduce Schmitt,Hale and Lopes's classical results on existence of periodic solutions to func-tional differential equations.In the next section,we list our main result on the existence of affine-periodic solutions to functional differential equations.In the third section,we take some explicit examples to illustrate the efficiency of our conclusion.In the final section,we prove the main result.At first,we re-duce the existence problem of affine-periodic solution to functional differential equation to one of boundary value problem.Secondly,we use the continuation method,the theory of topological degree and the lower and upper solutions method to show the existence of solution to the boundary value problem.In the final chapter,we explore the existence of affine-periodic solutions to nonlinear systems of vector ordinary differential equations.We firstly reduce the existence problem of affine-periodic solutions to its associated solutions of affine-periodic boundary value problem.Then we apply Opial's linearization method to such nonlinear boundary value problem,which is reduced to its corresponding linear probelm.Finally we apply the Leray-Schauder's fixed point theorem to prove the existence of affine-periodic solution to the linear problem under some suitable asymptotic assumptions.