الفهرس 
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Abstract
The study of dynamic equations on a time scales goes back to its founder Stefan Hilger [26], in order to unify, extend and generalize ideas from discrete calculus, quantum calculus, and continuous calculus to arbitrary time scale calculus. A time scale is a nonempty closed subset of the real numbers. By choosing the time scale to be the set of real numbers, the general result yields a result of an ordinary differential equations. On the other hand, choosing the time scale to be the set of integers, the same general result yields a result for 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 or ) which have important applications in quantum theory (see [28]).
A delay differential equation (DDE) is an equation for a function of a single variable, usually called time, in which the derivative of the function at a certain time is given in terms of the values of the function at earlier times. In the last two decades, there has been increasing interest in obtaining sufficient conditions for oscillation (nonoscillation) of the solutions of delay dynamic equations on time scales. So we chose the title of the thesis ” Oscillation of Delay Dynamic Differential Equations on Time Scales” aiming to use the generalized Riccati transformation and the inequality technique in establishing some new oscillation criteria for the delay dynamic equations.
This thesis is devoted to
1. Illustrate the new theory of Stefan Hilger by giving a general introduction into the theory of dynamic equations on time scales,
2. Summarize some of the recent developments in the oscillation of first and second order delay differential equations and delay dynamic equations on time scales,
3. Establish new sufficient conditions to ensure that all solutions of second order delay dynamic equations on unbounded time scales are oscillatory.
Chapter 1 contains the basic concepts of the theory of functional differential equation and some preliminary results of the oscillation theory of first and second order delay differential equations.
In Chapter 2, we give an introduction to the theory of dynamic equations on time scales, differentiation, integration, and some examples of time scales. Also, we present various properties of the exponential function on arbitrary time scale, and use it to solve linear dynamic equations on time scales.
In Chapter 3, we present the most important studies for the oscillation theory of first and second order delay dynamic equations on time scales. In section 3.3, we establish some new oscillation criteria for the second order nonlinear delay dynamic equation
on a time scale . where is the quotient of odd positive integers, and are positive right dense continuous (rd-continuous) functions on . Our results improve and extend some results established by Sun et al. [43]. These results are given in [8]. At the end of this Chapter, we establish some new oscillation criteria for the nonautonomous second order delay dynamic equation
on a time scale by using the generalized Riccati technique, and generalized
exponential function.
Finally in Chapter 4, we present some new oscillation criteria for the second order nonlinear delay dynamic equations. In section 4.1, we give some new oscillation criteria for the second order nonlinear delay dynamic equation with damping of the form
on a time scale . Our results are not only unify the oscillation of nonlinear differential and difference equations but also can be applied on different types of time scales such as for and improve most previous results in differential and difference equations. In section 4.2, we establish some new oscillation criteria for the second order nonlinear advanced dynamic equation of the form
on a time scale Our results improve and extend some results established by S. Tang et. al. [44].