الفهرس 
Preliminariesيوجد فقط 14 صفحة متاحة للعرض العام
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المستخلص
Summary
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.
This thesis contains three chapters:
Chapter 1 is the introductory chapter and contains the basic concepts and some preliminary results of the oscillation theory of first order delay differential equations with maxima.
Chapter 2 consists of three sections. In the first section, we give an introduction to the theory of dynamic equations on time scales, differentiation, integrations, and some examples of time scales. Also, we present various properties of gen¬eralized exponential function on arbitrary time scale. In the second section, we establish some new oscillation results for the first order dynamic equations with maxima
xA(t) + q(t) max x(s) = 0,
s€[t-S,t]
xA(t) + pi(t) max x(s) — p2(t) max x(s) = 0,
se[ri(t),t] se[T2(t),t]
The results of this section published in:
International Journal of Scientific and Innovative Mathematical Re¬search, 5 (2017), 1-8.
In the third section, we establish some new oscillation criteria for first order forced delay dynamic equation of the form
The result of this section are published in:
International journal of Dynamical Systems and Differential Equations.
In Chapter 3, we establish some new oscillation criteria for first order sublinear delay dynamic equations with and without maxima of the form
xA(t) + p(t)xa(T(t)) = 0,t > to, (1)
and
xA(t) + p(t) max xa(s) = 0,t > t0, (2)
s€[r (t),t]
where p(t) G Crd(\t0, w)T, R+), a is a quotient of odd positive integer. Oscillation behavior of these equations is not studied before. In Section 3.1, we studied the oscillatory of first order superlinear delay dynamic equations with and without maxima on time scales and the obtained results was submitted for publication. Also, in section 3.2, we studied the oscillatory of first order sublinear delay dy¬namic equations with and without maxima on time scales and the obtained results was submitted for publication.