Oscillation Criteria for Neutral Dynamic Equations on Time Scales

Abstract

The study of dynamic equations on time scales goes back to its founder Stefan Hilger [19], 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 result yields a result of an ordinary di erential equations. When the time scale is the set of integers, the general result yields a result for di erence equations. The new theory of the so - called \ dynamic equation" is not only unify the theories of di erential equations and di erence equations, but also extends these classical cases to the so - called q- di erence equations (when T = qN0 := fqt : t 2 N0; q > 1g or T = qZ = qZ [ f0g) which have important applications in quantum theory (see [22]). In the last two decades, there has been increasing interest in obtaining su cient conditions for oscillation (nonoscillation) of the solutions of dynamic equations on time scales. So we chose the title of the thesis \ Oscillation Criteria for Neutral Dynamic Equations on Time Scales" aiming to use the generalized Riccati transformation and the inequality technique in establishing some new oscillation criteria for the neutral dynamic equations. 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 order neutral delay di erential equations and neutral dynamic equations on time scales. 3. Establish some new su cient conditions to ensure that all solutions of second order neutral delay dynamic equations on unbounded time scales are oscillatory. iii SUMMARY 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. This thesis contains four chapters: Chapter 1 contains the basic concepts of the theory of functional di erential equations and some preliminary results of the theory of second order neutral delay di erential equations. In Chapter 2, we give an introduction to the theory of dynamic equations on time scales, di erentiation and integration on arbitrary time scale. Additionally, the most important studies for the oscillation theory of second order neutral delay dynamic equations on time scales are presented. In Chapter 3, we establish some new oscillation criteria for the second-order nonlinear functional dynamic equation with neutral term (r(t)((m(t)y(t) + p(t)y( (t))) ) ) + f(t; y( (t))) = 0; on a time scale T by using the generalized Riccati technique. The present results not only improve, generalize and extend some of the previous results [21, 27, 39, 45, 48] but also can be applied to some oscillation problems that are not covered before. At the end of this Chapter, a counter example is given to illustrate the main theorem of E. Thandapani et al. [41]. The correct formula for this theorem and related results in their work are given. The results of this chapter are published (see [4] and [5]). In Chapter 4, we introduce some new oscillation criteria for the second-order nonlinear functional dynamic equation with non positive neutral term (r(t)((m(t)y(t) 􀀀 p(t)y( (t))) ) ) + f(t; y( (t)))