الفهرس 
1 PreliminariesOnly 14 pages are availabe for public view
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Abstract
Studing the dynamic equations on time scales was introduced by Stefan Hilger [37]. It is a new area of still fairly theoretical exploration in mathematics. Motivating the subject is a notion that dynamic equations on time scales can build bridges between continuous and discrete mathematics. Further, studying time scales lead to several important applications, e.g., insect population models, neural networks, and heat transfer. 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 yield results of ordinary differential equations, while when the time scale equals the set of integers, the obtained results yield results of difference equations. The new theory of the so - called ”dynamic equation” is not only unify the theories of differential and difference equations, but also extends the classical cases to the so - called q - difference equations (when T=q^(N_0 ):={q^t:t∈N_0, q>1} or T=q^¯Z=q^Z∪{0}) which have important applications in quantum theory (see [38]).
A neutral differential equation with deviating arguments is a differential equation in which the highest order derivative of the unknown function appears with and without deviating arguments. In the last two decades,
This thesis is devoted to
1. Illustrate the new theory of Stefan Hilger by giving an introduction to the theory of dynamic equations on time scales.
2. Summarize some of the recent developments in oscillation of second order neutral differential equations with ”maxima”, oscillation of second-order differential equations with a sublinear or a superlinear neutral terms, oscillation of second order nonlinear integro- dynamic equations on time scales.
3. Establish new sufficient conditions to study the oscillatory and the asymptotic behavior of solutions of second order nonlinear neutral dynamic equations with maxima and second order nonlinear mixed neutral integro-dynamic equations with maxima on time scales, so that the obtained results are more generalized than those obtained in previous studies.
4. Give some examples to illustrate the relevance of our results.
This thesis consists of six chapters :-
Chapter 1, is an introductory chapter that contains the basic concepts of theory of functional differential equations, some previous results of the oscillatory and the asymptotic behavior of solutions of second order neutral differential equations with ”maxima” and oscillation of second-order differential equations with a sublinear or a superlinear neutral terms.
In Chapter 2, we give an introduction to the theory of dynamic equations on time scales, differentiation and integration, and some examples of time scales. Moreover, we present various properties of generalized exponential function on time scales. Additionally, some previous studies for the oscillatory and asymptotic behavior of second order integro- dynamic equations on time scales are presented.
Chapter 3, consists of two sections. In the first section, we present some new oscillation criteria for the second order neutral integro-dynamic equation with damping and distributed deviating arguments:
In Chapter 4, Some new oscillation criteria for the second order neutral dynamic equation with maxima and mixed arguments
Chapter 5, Concerned with the oscillatory and asymptotic behavior for solutions of the second-order mixed nonlinear integro-dynamic equations with ”maxima”
Chapter 6, deals with the oscillatory and the asymptotic behavior of the solutions of the second-order neutral nonlinear integro-dynamic equations with maxima and superlinear or sublinear neutral terms on time scales