Oscillation Criteria for some Dynamic Equations with Maxima on Time Scales

Abstract

The study of dynamic equations on time scales goes back to its founder Stefan Hilger [28], 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 the time scale equals the set of real numbers, the obtained results yields results of ordinary differential equations, while when the time scale is the set of integers, the obtained results yields 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 = qNo := (q* : t E N0, q > 1} or T = qZ = qZ U {0}) which have important applications in quantum theory (see [31]). 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. So we choose the title of the thesis “ Oscillation Criteria for Some Dynamic Equations 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 first order delay differential equations and dynamic equations on time scales. 3. Establish some new sufficient conditions to ensure that all solutions of first order forced delay dynamic equations with maxima on time scales are oscil¬latory. 4. Give a comparison between the current results and the previous one. Latter, we give some examples to illustrate the importance of the presented results.