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The study of dynamic equations on time scales goes back to its founder Stefan
Hilger , 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 dierential equations. When the time scale is the
set of integers, the general result yields a result for dierence equations. The
new theory of the so - called \ dynamic equation” is not only unify the theories
of dierential equations and dierence equations, but also extends these classical
cases to the so - called q- dierence equations (when T = qN0 := fqt : t 2 N0;
q > 1g or T = qZ = qZ [ f0g) which have important applications in quantum
theory (see ).
In the last two decades, there has been increasing interest in obtaining sucient
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 dierential equations and neutral dynamic equations on time
3. Establish some new sucient conditions to ensure that all solutions of second
order neutral delay dynamic equations on unbounded time scales are
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 dierential
equations and some preliminary results of the theory of second order neutral
delay dierential equations.
In Chapter 2, we give an introduction to the theory of dynamic equations on
time scales, dierentiation 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. . The correct formula for this
theorem and related results in their work are given. The results of this chapter
are published (see  and ).
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))) = 0; t 2 T;
on a time scale T: The current results not only improve and extend results of
[8, 32], but also can be applied to some oscillation problems that are not covered
before. The results of this chapter are submitted for publication.