A Study On Stability Of Periodic Solutions Of Dynamic Equations On Time Scales Based On Fixed Point Theorems

The dynamic equations on time scales are a new research field, which aims tointegrate and unify the study of differential and difference equations. The study can betraced back to 1988. A German mathematician named Stenfan Hilger first puts forwardthe notion of time scales and calculus theory on time scales, and then establishes thetheory of time scales and the basic theory of dynamic equations on time scales. Themain purpose of this theory is to unify and extend the existing differential anddifference, and ordinary differential equation and difference equation theory. From thebeginning of this century, this theory has drawn great attention in mathematical filed,and develops rapidly. On one hand, it unifies and extends the classical theory ofdifferential and difference; on the other hand, as a new field in mathematics, it hasbroad application prospects. In recent years, this area has had many research results,especially in stability, oscillation, initial-boundary value problem, which has made greatprogress, the study of dynamic equations on time scales also brings severalimportant applications of real phenomena and the process of mathematical models, forinstance, population dynamics on time scales, epidemiology model, financialconsumption process mathematical model and so on. At present, the vast majority ofresearches of the dynamic equations on time scales are confined to boundary valueproblems and oscillation, while the researches about the problems of periodic solution,impulse, time-delay of the dynamic equations on time scales are comparatively rare.And many researches on these problems are at its initial stage, which are highlysignificant not only in theory but also in practice. Therefore, this thesis mainly exploresthe existence and stability of the dynamic equations on time scales.This thesis deals with the periodic solutions of Nonlinear Neutral DynamicEquations on time scales, the existence of positive periodic solutions of nonlinearfirst-order delay dynamic equations on time scales, and the periodic solutions ofsemilinear dynamic equations on time scales. It proves the existence and uniqueness ofperiodic solutions of Nonlinear Neutral Dynamic Equations on time scales byKrasnosel’skii fixed point theorem and the contraction mapping principle, the existence of positive periodic solution of nonlinear first-order delay dynamic equations on timescales by the fixed point index theorem, and the stability of the periodic solution ofsemilinear dynamic equations on time scales by contraction mapping principle.The thesis consists of six chapters:The first chapter reviews previous studies about the dynamic equations on timescales both in China and abroad, clarifies research purpose and significance, and finallyputs forward research questions.The second chapter mainly elaborates on the relevant theoretical knowledge of thetime scales. Both the concept and the theory content in this chapter are from StefanHilger and other mathematicians, which lays a foundation for the continuous study.The third chapter focuses on the periodic solutions of nonlinear neutral dynamicequations on time scales. It obtains the existence and uniqueness of periodic solutions ofNonlinear Neutral Dynamic Equations on time scales by Krasnosel’skii fixed pointtheorem and the contraction mapping principle as follows:x~Δ( t ) = a (t ) x (σ( t )) + c ( t ) x (t- L ) + q ( t , x ( t ), x ( t- L )), t∈T.The fourth chapter deals with the existence of positive periodic solutions ofnonlinear first-order delay dynamic equations on time scales. It proves the existence ofpositive periodic solution of nonlinear first-order delay dynamic equations on timescales by the fixed point index theorem:x~Δ( t ) = a ( t ) g ( x ( t )) x (σ( t ))λb ( t ) f ( x (t -τ( t ))), t∈T.The fifth chapter considers the periodic solutions of semilinear dynamic equationson time scales. This chapter studies the stability of the periodic solution of semilineardynamic equations on time scales by contraction mapping principle:x~Δ( t ) = a ( t ) x (σ( t )) + f ( t , ( x (t ))), t∈T.The sixth chapter summarizes what have been done in this thesis, and providessuggestions for further research.