الفهرس 
Chapter 1 . PreliminariesOnly 14 pages are availabe for public view
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Abstract
In this thesis, we are concerned with the oscillatory behavior of solutions
of several classes of differential and dynamic equations. We first start to
discuss the second order nonlinear damped differential equations
(𝑟(𝑡)𝜓(𝑥(𝑡))(𝑥′(𝑡))𝛼)′ + 𝑝(𝑡)(𝑥′(𝑡))𝛼 + 𝑞(𝑡)𝑓 (𝑥(𝜏(𝑡))) = 0, 𝑡 ≥ 𝑡0
> 0, where 𝛼 is a positive ratio of two odd integers with the hypotheses:-
𝑟 ∈ 𝐶1([𝑡0, ∞), (0, ∞)), 𝑝, 𝑞, 𝜏 ∈ 𝐶([𝑡0, ∞), ℝ), and 𝑞(𝑡) ≥ 0, where 𝑞 does
not vanish eventually; 𝑓 ∈ 𝐶(ℝ, ℝ), 𝑓(𝑥)
𝑥𝛼 ≥ 𝜇 > 0 for all 𝑥 ≠ 0 ; 𝜓 ∈
𝐶1(ℝ, ℝ), 𝜓(𝑥(𝑡)) > 0 , and there exist two real positive continuous
functions 𝑔1, 𝑔2 such that 𝑔1(𝑡) < 𝜓(𝑥(𝑡)) < 𝑔2(𝑡) for all 𝑥 ≠ 0.
Secondly, we study the oscillatory behavior of the second order nonlinear
neutral delay differential equations of the type
(𝑟(𝑡)|𝑧′(𝑡)|𝛼−1𝑧′(𝑡))′ + 𝑓 (𝑡, 𝑥(𝛿(𝑡))) = 0, 𝑡 ≥ 𝑡0,
where
𝑧(𝑡) = 𝑥(𝑡) ± 𝑝(𝑡)𝑥(𝜏(𝑡)) and 𝛼 > 0 with the hypotheses 𝑟, 𝑝 ∈
𝐶([𝑡0, ∞), ℝ), 𝑟(𝑡) > 0, and 0 ≤ 𝑝(𝑡) ≤ 1, 𝜏(𝑡), 𝛿(𝑡) ∈ 𝐶1([𝑡0, ∞), ℝ),
𝜏(𝑡) ≤ 𝑡, 𝛿(𝑡) ≤ 𝑡, and lim𝑡→∞ 𝜏(𝑡) = lim𝑡→∞ 𝛿(𝑡) = ∞; 𝑓 ∈ 𝐶([𝑡0, ∞) ×
ℝ, ℝ), 𝑢𝑓(𝑢) ≥ 0 for all 𝑢 ≠ 0, and there exists a ratio of odd positive
integers 𝛽 and a function 𝑞(𝑡) ∈ 𝐶([𝑡0, ∞), (0, ∞)) such that 𝑓(𝑡, 𝑢)/𝑢𝛽 ≥ 𝑞(𝑡), for all 𝑢 ≠ 0. In our study of the above equations, we consider the
cases of condition respectively
∫ 𝑡∞ 𝑟1/ 𝑑𝑡 𝛼(𝑡) = ∞
0
or
∫ 𝑡∞ 𝑟1/ 𝑑𝑡 𝛼(𝑡) < ∞,
0
and the more general equations
(𝑟(𝑡)𝜓(𝑥(𝑡))|𝑧′(𝑡)|𝛼−1𝑧′(𝑡))′ + 𝑓 (𝑡, 𝑥(𝛿(𝑡))) = 0, 𝑡 ≥ 𝑡0,
where 𝑧(𝑡) = 𝑥(𝑡) ± 𝑝(𝑡)𝑥(𝜏(𝑡)) and 𝛼 is a positive constant, with the
same hypotheses of the above equation with 𝜓 ∈ 𝐶1(ℝ, ℝ), 𝜓(𝑥(𝑡)) > 0,
and there exist two real continuous functions 𝑔1, 𝑔2 such that 𝑔1(𝑡) <
𝜓(𝑥(𝑡)) < 𝑔2(𝑡) for all 𝑢 ≠ 0, and there exists a ratio of odd positive
integers 𝛽 and a function 𝑞(𝑡) ∈ 𝐶([𝑡0, ∞), (0, ∞)) such that 𝑓(𝑡, 𝑢)/𝑢𝛽 ≥
𝑞(𝑡), for all 𝑢 ≠ 0 where in the existence of 𝜓(𝑥(𝑡)), we consider the
more general conditions ∫ 𝑑𝑡
(𝑟(𝑡)𝑔2(𝑡))
1𝛼
∞
𝑡
0
= ∞, or ∫ 𝑑𝑡
(𝑟(𝑡)𝑔2(𝑡))
1𝛼
∞
𝑡
0
< ∞.
Further we are concerned with the oscillatory behavior of solutions of
nonlinear second-order neutral delay dynamic equations
(𝑟(𝑡)𝜓(𝑥(𝑡)) (𝑧∆(𝑡))𝛼)∆ + 𝑓 (𝑡, 𝑥(𝛿(𝑡))) = 0,
where 𝑡 ∈ [𝑡0, ∞)𝕋 = [𝑡0, ∞) ∩ 𝕋 with 𝑠𝑢𝑝𝕋 = ∞, 𝑧(𝑡) = 𝑥(𝑡) +
𝑝(𝑡)𝑥(𝜏(𝑡)) and 𝛼 is a quotient of odd positive integers with the hypotheses:- 𝑟 ∈ 𝐶𝑟𝑑([𝑡0, ∞)𝕋, (0, ∞)), and 𝑝 ∈ 𝐶𝑟𝑑([𝑡0, ∞)𝕋, [0, ∞)), and
0 ≤ 𝑝(𝑡) ≤ 𝑝1 < 1; 𝜓 ∈ 𝐶𝑟𝑑 1 (ℝ, ℝ), 𝜓(𝑥(𝑡)) > 0 , and there exist two
functions 𝑔1, 𝑔2 ∈ 𝐶([𝑡0, ∞)𝕋, (0, ∞)) such that 𝑔1(𝑡) < 𝜓(𝑥(𝑡)) < 𝑔2(𝑡)
for all 𝑥 ≠ 0; 𝜏(𝑡), 𝛿(𝑡) ∈ 𝐶𝑟𝑑 1 ([𝑡0, ∞)𝕋, 𝕋), 𝜏(𝑡) ≤ 𝑡, 𝛿(𝑡) ≤ 𝑡 , and
lim
𝑡→∞ 𝜏(𝑡) = lim𝑡→∞ 𝛿(𝑡) = ∞; 𝑓(𝑡, 𝑥) ∈ 𝐶𝑟𝑑([𝑡0, ∞)𝕋 × ℝ, ℝ), and there
exists a positive function 𝑞(𝑡) ∈ 𝐶𝑟𝑑([𝑡0, ∞)𝕋, (0, ∞)) such that 𝑓(𝑡, 𝑥)/
𝑥𝛼 ≥ 𝑞(𝑡) for all 𝑥 ≠ 0. We also study this equation with the canonical
condition ∫ ∆𝑡
(𝑟(𝑡)𝑔2(𝑡))
1𝛼
∞
𝑡
0
= ∞.
Then we further extend the results of above equation but with positive and
negative neutral term and consider the two cases canonical or non-canonical
conditions ( ∫ ∆𝑡
(𝑟(𝑡)𝑔2(𝑡))
1𝛼
∞
𝑡
0
= ∞ or ∫ ∆𝑡
(𝑟(𝑡)𝑔2(𝑡))
1𝛼
∞
𝑡
0
< ∞ ) in the two cases
0 ≤ 𝑝(𝑡) ≤ 𝑝1 < 1,0 ≤ 𝑝(𝑡) ≤ 𝑝0 < ∞.
Finally, we discuss the oscillatory behavior of solutions of the nonlinear
neutral delay dynamic equation on time scales
(𝑟(𝑡)𝜓(𝑥(𝑡))|𝑧∆(𝑡)|𝛼−1𝑧∆(𝑡))∆ + 𝑓 (𝑡, 𝑥(𝛿(𝑡))) = 0,
where 𝑡 ∈ [𝑡0, ∞)𝕋 = [𝑡0, ∞) ∩ 𝕋 with 𝑠𝑢𝑝𝕋 = ∞, 𝑧(𝑡) = 𝑥(𝑡) −
𝑝(𝑡)𝑥(𝜏(𝑡)), 𝛼 is a positive constant with the hypotheses:- 𝑟 ∈
𝐶
𝑟𝑑([𝑡0, ∞)𝕋), and 𝑝(𝑡) is a positive and rd-continuous function on 𝕋, 0 ≤
𝑝(𝑡) ≤ 𝑝1 < 1; 𝜏(𝑡), 𝛿(𝑡) ∈ 𝐶𝑟𝑑 1 ([𝑡0, ∞)𝕋, 𝕋), 𝜏(𝑡) ≤ 𝑡, 𝛿(𝑡) ≤ 𝑡 , and
lim
𝑡→∞ 𝜏(𝑡) = lim𝑡→∞ 𝛿(𝑡) = ∞; 𝑓(𝑡, 𝑥) ∈ 𝐶𝑟𝑑([𝑡0, ∞)𝕋 × ℝ, ℝ), and there
exists a function 𝑞(𝑡) ∈ 𝐶𝑟𝑑([𝑡0, ∞), (0, ∞)) such that 𝑓(𝑡, 𝑥)/𝑥𝛽 ≥ 𝑞(𝑡), for which 𝛽 is a positive constant. With the canonical case ∫ ∆𝑡
(𝑟(𝑡)𝑔2(𝑡))
1𝛼
∞
𝑡
0
=
∞, in all cases 𝛼 > 𝛽, 𝛼 = 𝛽, 𝛼 < 𝛽.
The obtained results improve and extend some known results in the
literature. Finally, we give some examples to justify our results.