Research On Oscillation Behavior Of Delay Dynamic Equations

Oscillation theory is not only an important branch in qualitative theory of dynamicequations, but also has numerous applications in practical issue, especially in control field.The oscillation behavior of solutions of differential equation is one of the importantqualitative properties of differential equations, and all the structures of its emergence havedeep physical background and realistic mathematical models. Recently, with the increasingdevelopment of science and technology, this theory has achieved rapid development andextensive attention in the field of applied mathematics.With the purpose of integrating continuous and discrete analysis, the theory of measurechains was introduced by Hilger, and gives rise to many applications. As a special case ofmeasure chains, time scales are very representative. The theory of calculus on time scales isnot only the needs of mathematical theory development, but also the demands of the practicalproblems. It has great potential in applications, so recently it has received a lot of attention.This paper mostly study some oscillation criteria for delay dynamic equations, and theresults obtained in this thesis are new, or extend and complement some existing results inliteratures. This dissertation is composed of five chapters. Main contents are as follows:In the first chapter of introduction, we introduce the background and development of theoscillation theory of differential equations and dynamic equations on time scales. Somecalculus theory on time scales is recommended. The primary results gained in this thesis arebriefly presented.In the second chapter, we study the oscillation behavior of second order nonlineardynamic equations on time scales. In section2.1, using Riccati transformation technique andinequalities, we consider the oscillation of second order quasi-linear neutral delay dynamicequations on time scales. We obtain some sufficient conditions which guarantee that everysolution of these equations is oscillatory or converges to zero. And we give an example toverify the main result. In section2.2, we examine the interval oscillation criteria for secondorder nonlinear forced dynamic equations with damping. The main results obtained here arenew and improve some known results. In the third chapter, we study the oscillation behavior of third order delay dynamicequations on time scales. In section3.1, applying the theory of time scales, generalizedRiccati transformation technique and inequalities, we study the oscillation of third ordernonlinear delay dynamic equations on time scales and give an example to verify the mainresult. In section3.2, using Riccati transformation technique, the averaging functionstechnique and inequalities, we consider the oscillation of third order nonlinear neutral delaydynamic equations on time scales, and present some new Kamenev-type and Philos-typeoscillation criteria for these equations.In the fourth chapter, we investigate the oscillation behavior of even order differentialequations. In section4.1, basing on the new comparison principles, Riccati transformationtechnique and inequalities, we study the oscillation criteria for even order nonlinear neutraldifferential equations, and give an example to verify the main result. In section4.2, by meansof the generalized Riccati transformation technique and inequalities, we consider theoscillation theorems for even order damped neutral delay differential equations with mixednonlinearities.In the fifth chapter, we summarize the main results in this paper, and point out theinnovations of our work. Finally, we prospect some future research work.