Qualitative Analysis of Solutions of Neutral Dynamic Equations with "Maxima" on Time Scales

Abstract

The study of dynamic equations on time scales goes back to its founder Stefan Hilger [39], in order to unify, extend and generalize ideas from discrete, quantum, and continuous calculus to arbitrary time scale calculus. A time scale T is a nonempty closed subset of the real numbers. When T = R, the general results yield the results of ordinary differential equations. When the time scale is the set of integers, the general results yield the results of difference equations. The new theory of the so called “ dynamic equation” is not only unify the theories of differential equations and difference equations, but also extends these classical cases to the so called q- difference equations (when T = qN0 := {qt : t ∈ N0, q > 1 or T = qZ = qZ 0 ) which have important applications in quantum theory (see [40]). In the last two decades, there has been increasing interest in obtaining sufficient conditions for oscillation (nonoscillation) of the solutions of dynamic equations on time scales. Also, the mathematical importance of various types of differential equations with "Maxima" has grown rapidly but the qualitative theory of these equations is relatively little developed compared to functional differential equa- tions. So we chose the title of the thesis “ Qualitative Analysis of Solutions of Neutral Dynamic Equations with "Maxima" on Time Scales” aiming to use the generalized Riccati transformation and the inequality technique in establishing some new oscillation criteria for second and third order neutral dynamic equa- tions with "Maxima" on time scales.

This thesis is devoted to 1. Illustrate Hilger’s theory by giving a general introduction to the theory of dynamic equations on time scales. 2. Summarize some of the recent developments in oscillation of second and third order neutral delay differential equations with "Maxima".