| In recent years, a large number of mathematical models that are described by dynamicequations are proposed in the field of natural sciences including physics, economics, medicine,and theory of control, the theoretical study of dynamic equation is of great significance.Sturm proposed a second order linear differential equations problem in1836at the fristtime, oscillation theory of differential equations has become a popular study field. With thedevelopment of study field, study methods and study contents, oscillation theory plays animportant role in qualitative theory of dynamical equations.Stenfan Hilger proposed the measure chain analysis at the frist time, a mathematical toolthat unify the theories of continuous and discrete analysis, in his Ph. D. thesis in1988. Thetheory of calculus on time scale not only promote and unify theories of differential equationsand difference equations, but also extend it to more general cases, enrich the study contents ofdynamic equation. Oscillation theory of dynamic equations on time scales has attracted lots ofmathematicians.In this paper, we study oscillation and non-oscillation of the solution of dynamicequations on time scales, extend and improve some results. The main content is as follows:In chapter1, we introduce the theory of calculus on time scales and oscillation ondynamic equation, the historical background and the recent development of dynamicequations.In chapter2, the study focus on oscillation of second order dynamic equations on timescales. In section2.1, Using Riccati transformation and inequality technique, we studyoscillation of a class of second order half linear dynamic equations with damping term (2.1),and get some suffcient conditions that oscillate or converge to zero. In section2.2, usingRiccati transformation and inequality technique, we study oscillation of a class of secondorder nonlinear dynamic equations (2.18), and establish some oscillation theorems.In chapter3, the study focuses on oscillation of third order dynamic equations on timescales. In section3.1, we study oscillation of a class of third order functional dynamicequations (3.1) by means of Riccati transformation and inequalitiy technique, get some suffcient conditions that oscillate or converge to zero.In chapter4, the study focuses on existence of nonoscillation and oscillation solutions ofsecond order dynamic equations. In section4.1, we study existence of nonoscillation solutionsof second order neutral dynamic equations (4.1) by means of Banach contraction mappingtheory, establish some suffcient conditions that exist nonoscillation solutions. In section4.2,we study existence of oscillation solutions of second order functional dynamic equations(4.13) by means of Schauder fixed point theorem, establish a suffcient condition that existoscillation solutions.In chapter5, the main research works are summarized, and we give some ideas for thefuture research work. |
