Oscillation And Existence Of Solutions And Boundary Value Problems For Several Classes Of Dynamic Equations On Time Scales

Recently, the study of dynamic equations on time scales is an area of mathematics that has received a lot of attention. Theory of dynamic equations on time scales, founded by Stefan Hilger in 1988, is undergoing a rapid development as it provides a powerful tool not only to unify the existing theories of both differential equations and difference equations but also to generalize the discussion to a wide class of equations on time scales. In addition, the study of dynamic equations on time scales has lead to several important applications related to mathematical modeling of real phenomena. For instance, investigation of population dynamics, financial assumption process, and epidemic models have been involved with the concept of dynamic equations on time scales. Clearly, theory of dynamic equations on time scales has been and are being proved to be of great importance and potential application.In this thesis, oscillation properties,existence properties of solutions of dynamic equations and boundary value problems on time scales are researched.Firstly, oscillation properties of solutions for a class of second order nonlinear neutral dynamic equations on time scales are researched, a series of sufficient conditions for oscillation of the solutions are established. These results are illustrated by examples.Secondly, some sufficient conditions for oscillation of a class of second order neutral delay dynamic equations on time scales are obtained. These results are explained by examples.The existence of nonoscillatory solutions for a class of high-order forced neutral delay dynamic equations on time scales are studied by using Banach contraction mapping principle. Some sufficient conditions to satisfy existence of nonoscillatory solutions of these equations are derived.A class of forced high-order nonlinear neutral variable delay dynamic equations with variable coefficient on time scales is studied. The sufficient conditions are given for the forced oscillation and existence of nonoscillatory solutions. These results generalize the corresponding known results.Finally, the existence of positive solutions for a class of second order nonlinear boundary value problems on time scales and nonlinear singular m-point boundary value problems on the half-line on time scales are considered. Some sufficient conditions to satisfy existence of positive solutions of these equations are derived.