الفهرس 
Preliminariesيوجد فقط 14 صفحة متاحة للعرض العام
المستخلص
Studing the dynamic equations on time scales was introduced by Stefan Hilger
[28]. 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 di erential equations, while when the time scale
equals the set of integers, the obtained results yield results of di erence equations.
The new theory of the so - called ”dynamic equation” is not only unify the theories
of di erential and di erence equations, but also extends the 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
[31]).
A neutral di erential equation with deviating arguments is a di erential equation
in which the highest order derivative of the unknown function appears with
and without deviating arguments. In recent years, there has been an increasing
interest in studying oscillation and nonoscillation of solutions of neutral dynamic
equations on time scales which seek to harmonize the oscillation of continuous
and discrete mathematics, however the study was restricted to speci c equations
under certain conditions. The oscillation conditions of their equations are not applicable
when these conditions change. For this reason we aim to generalize these
equations. So we select the title of our thesis to be ”Oscillation of Second Order
Dynamic Equations with Mixed Arguments on Time Scales” using the generalized
Riccati transformation, the inequality technique and the generalized exponential
function in establishing some new oscillation criteria for second order neutral dynamic
equations with mixed arguments on time scales.
iii
SUMMARY
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 di erential equations, oscillation of second order nonlinear neutral
dynamic equations and oscillation of second order dynamic equations with
damping on time scales.
3. Establish new sucient conditions to ensure that all solutions of second order
nonlinear neutral dynamic equations with mixed arguments and second
order nonlinear mixed neutral dynamic equations with nonpositive neutral
term on time scales are oscillatory or almost oscillatory or tend to zero,
so that the obtained results are more generalized than those obtained in
previous studies.
4. Trying to nd some new oscillation criteria for second-order mixed nonlinear
neutral dynamic equations with damping on time scales.
5. Give some examples to illustrate the importance of our results.
This thesis consists of ve chapters :-
Chapter 1, is an introductory chapter that contains the basic concepts of
studying the oscillation of solutions for functional di erential equations as well as
some previous results in studying the oscillation of second order neutral di erential
equations.
In Chapter 2, we give an introduction to the theory of dynamic equations
on time scales, di erentiation 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 oscillation theory of second
order neutral dynamic equations and second order dynamic equations with
damping on time scales are presented.
In Chapter 3, we establish some new oscillation criteria for the second-order
nonlinear neutral dynamic equation with mixed arguments on a time scale T
r(t)[
x(t)+p1(t)x(1(t))+p2(t)x(2(t))
]
+f(t; x( 1(t)))+g(t; x( 2(t))) = 0;
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SUMMARY
The results of this chapter generalize and extend the results of Tao Ji et al. [29],
and published in: Journal of Basic and Applied Research International
17(1)(2016) 49-66. [5].
In Chapter 4, we present some new oscillation results for the second-order
nonlinear mixed neutral dynamic equation with non positive neutral term on a
time scale T
(r(t)[(x(t)p1(t)x(1(t))+p2(t)x(2(t)))]
)+f(t; x( 1(t)))+g(t; x( 2(t))) = 0;
Our results not only generalize some existing results in [22], but also can be
applied to some oscillation problems that do not covered before. Also, we give
some examples to explain our results. The results of this chapter published in:
Academic Journal of Applied Mathematical Sciences 3 (2)(2017)8-20
[6]
Chapter 5 is concerned with the oscillatory behavior of all solutions of the
second-order mixed nonlinear neutral dynamic equation with damping on a time
scale T
(r(t) (z(t))) + p(t) (z(t)) + f(t; x( 1(t))) + g(t; x( 2(t))) = 0;
where (s) = jsj
1s and z(t) = x(t) + p1(t)x(1(t)) + p2(t)x(2(t)). Our results
generalize the results of [22] and [30] which are considered as special cases
of our results when taking = =
; p(t) = p2(t) = 0 and considering either
g(t; x( 2(t))) = 0 or f(t; x( 1(t))) = 0. Also, we introduce an illustrated example
to explain our results.