The document investigates stability and boundedness of solutions for third order nonlinear differential equations with delays by constructing Lyapunov functionals. It presents theorems that establish conditions under which solutions converge to zero and are uniformly bounded, extending existing results in the field. The primary focus is on establishing criteria that ensure the stability and boundedness of solutions under specific conditions related to the functions involved.

1. IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728.Volume 5, Issue 6 (Mar. - Apr. 2013), PP 09-13
www.iosrjournals.org
www.iosrjournals.org 9 | Page
Stability and Boundedness of Solutions of Delay Differential
Equations of third order
P. Shekhar1
, V. Dharmaiah2
and G.Mahadevi3
1
Department of Mathematics,Malla Reddy College of Engineering, Hyderabad-500014, India.
2
Department of Mathematics, Osmania University, Hyderabad-500007 India.
3
Department of Mathematics,St.Martins Engineering College, Hyderabad-500014, India.
Abstract : In this paper we investigate stability and boundedness of solutions for certain third order nonlinear
differential equations with delays by constructing Lyapunov functionals.
1. Introduction
The investigations concerning stability and boundedness of solutions of nonlinear equations of third
order with bounded delay have not been fully developed. Certainly, these results should be obtained to be able
to benefit from the applications of the theory of stability and boundedness of solutions. At the same time, we
should recognize that some significant theoretical results concerning the stability and boundedness of solutions
of third order nonlinear differential equations with delay have been achieved, see for example the papers of
Sadek [6].
We consider the third order non-autonomous nonlinear differential equations with delays:
𝒙 + 𝒂 𝒕 𝒙 + 𝒃 𝒕 𝒈 𝒙 + 𝒉 𝒙 𝒕 − 𝒓 = 𝟎 (1.1)
and
𝒙 + 𝒂 𝒕 𝒙 + 𝒃 𝒕 𝒈 𝒙 + 𝒉 𝒙 𝒕 − 𝒓 = 𝒑(𝒕) (1.2)
where r is a positive constant, 𝑎(𝑡), 𝑏(𝑡), 𝑔 𝑥 , 𝑕(𝑥) are real valued functions continuous in their respective
arguments; 𝑔 0 = 𝑕 0 = 0. The dots indicates differential with respect to 𝑡 and all solutions are assumed real.
Equations of the forms (1.1) and (1.2) in which 𝑎(𝑡), 𝑏(𝑡) are constants have been studied by several authors
namely, Sadek [5] and Zhu[13], to mention a few. They obtained criteria which ensure the stability, uniform
boundedness and uniform ultimate boundedness of solutions.
Recently, in [6], SADEK establishes conditions under which all solutions of the non-autonomous equation
𝒙 + 𝒂 𝒕 𝒙 + 𝒃 𝒕 𝒙 + 𝒉 𝒙 𝒕 − 𝒓 = 𝟎
tend to the zero solution as 𝑡 → ∞. This result is now extended to equiation (1.1) by considering the semi-
invariant set of a related non-autonomous system. Using the same technique, boundedness conditions are then
obtained for (1.2).
2. Preliminaries
Now, we will give the preliminary definitions and the stability criteria for the general non-autonomous delay
differential system.
we consider general non-autonomous delay differential system.
𝒙 = 𝒇 𝒕, 𝒙 𝒕 , 𝒙 𝒕 = 𝒙 𝒕 + 𝜽 , −𝒓 ≤ 𝜽 ≤ 𝟎, 𝒕 ≥ 𝟎 (2.1)
where 𝑓: 𝐼 × 𝐶 𝐻 → 𝑅 𝑛
is a continuous mapping .
𝑓 𝑡, 0 = 0, 𝐶 𝐻 = {𝜙 ∈ 𝐶 −𝑟, 0 , 𝑅 𝑛
: 𝜙 ≤ 𝐻}
and for 𝐻1 ≤ 𝐻 , there exists 𝐿 𝐻1 > 0 , with
𝑓(𝜙) ≤ 𝐿 𝐻1 when 𝜙 ≤ 𝐻1.
Definition 2.1 : An element 𝜓 ∈ 𝐶 is in the 𝜔-limit set of 𝜙, say, Ω 𝜙 , if 𝑥(𝑡, 0, 𝜙) is defined on [0, ∞) and
there is a sequence 𝑡 𝑛 , 𝑡 𝑛 → ∞, as 𝑛 → ∞, with 𝑥𝑡 𝑛
𝜙 − 𝜓 → 0 as 𝑛 → ∞ where
𝑥𝑡 𝑛
𝜙 = 𝑥(𝑡 𝑛 + 𝜃, 0, 𝜙) for −𝑟 ≤ 𝜃 ≤ 0.
Definition 2.2 : A set 𝑄 ⊂ 𝐶 𝐻 is an invariant set if for any 𝜙 ∈ 𝑄,the solution of (2.1), 𝑥(𝑡, 0, 𝜙) is defined on
[0, ∞) and 𝑥𝑡(𝜙) ∈ 𝑄 for 𝑡 ∈ [0, ∞)
Lemma 2.1. If 𝜙 ∈ 𝐶 𝐻 is such that the solution 𝑥𝑡 (𝜙) of (2.1) with 𝑥0 𝜙 = 𝜙 is defned on [0, ∞) and
𝑥𝑡(𝜙) ≤ 𝐻1 < 𝐻 for 𝑡 ∈ 0, ∞ , then Ω(𝜙) is a non-empty, compact, invariant set and
dist(𝑥𝑡 𝜙 , Ω(𝜙)) → ∞ as 𝑡 → ∞.

2. Stability and Boundedness of Solutions of Delay Differential Equations of third order
www.iosrjournals.org 10 | Page
Lemma 2.2. Let 𝑉 𝑡, 𝜙 : 𝐼 × 𝐶 𝐻 → 𝑅 be a continuous functional satisfying a local Lipschitz condition.
𝑉 𝑡, 0 = 0, and such that:
(i) 𝑊1( 𝜙 0 ) ≤ 𝑉(𝑡, 𝜙) ≤ 𝑊2( 𝜙 ) where 𝑊1 𝑟 , 𝑊2 𝑟 , are wedges
(ii) 𝑉(2.1) 𝑡, 𝜙 ≤ 0, for 𝜙 ∈ 𝐶 𝐻.
Then the zero solution of (2.1) is uniformly stable. If we define
𝑍 = 𝜙 ∈ 𝐶 𝐻 ∶ 𝑉(2.1) 𝑡, 𝜙 = 0 ,
then the zero solution of (2.1) is asymptotically stable, provided that the largest invariant set in 𝑍 is 𝑄 = 0 .
3. Main Result
The main objective of this paper is to prove the following:
Theorem 3.1. Suppose that 𝑎(𝑡), 𝑏(𝑡) ∈ 𝐶′
(𝐼), 𝑕 ∈ 𝐶′
(𝑅) and 𝑔 ∈ 𝐶(𝑅) and that these functions satisfy
the following conditions:
(i) 𝑕 0 = 0,
𝑕(𝑥)
𝑥
≥ 𝛿0 > 0, 𝑥 ≠ 0,
(ii) 𝑕′
(𝑥) ≤ 𝑐
(iii) 𝑔 0 = 0,
𝑔(𝑦)
𝑦
≥ 𝑏 > 0, 𝑦 ≠ 0,
(iv) 0 < 𝛿1 ≤ 𝑏 𝑡 , −𝐿 ≤ 𝑏′
(𝑡) ≤ 0, 𝑡 ∈ 𝐼,
(v) 0 < 𝑎 ≤ 𝑎(𝑡) ≤ 𝐿, 𝑡 ∈ 𝐼.
Then every solution 𝑥 = 𝑥(𝑡) of (1.1) is uniform-bounded and satisfies 𝑥(𝑡) → 0, 𝑥(𝑡) → 0, 𝑥(𝑡) → 0 as
𝑡 → ∞ provided there exists 𝛼 satisfying
𝑏
𝑐
> 𝛼 >
1
𝑎
such that
(vi)
1
2
𝑎′
𝑡 ≤ 𝛿2 < 𝛿1 𝑏 − 𝛼𝑐 𝑡 ∈ 𝐼 and 𝑟 < min
2𝛿5
3𝐿𝑐
,
𝛿5
𝐿𝛼𝑐
.
Theorem 3.2. If all the conditions of Theorem 3.1 are satisfied then all solutions of the perturbed equation
(1.2) are bounded provided 𝑝(𝑠) 𝑑𝑠 < ∞
𝑡
0
, for all 𝑡 ≥ 0.
Proof of Theorem 3.1. We write the Eq. (1.1) as the following equivalent system:
𝒙 = 𝒚
𝒚 = 𝒛 (3.1)
𝒛 = −𝒂 𝒕 𝒛 − 𝒃 𝒕 𝒈(𝒚) − 𝒉 𝒙 + 𝒉′
𝒙 𝒔 𝒚 𝒔 𝒅𝒔
𝒕
𝒕−𝒓
Define its Lyapunov functional as :
𝟐𝑽 𝒕, 𝒙 𝒕, 𝒚 𝒕, 𝒛 𝒕 = 𝟐𝑯 𝒙 + 𝟐𝜶𝒃 𝒕 𝑮 𝒚 + 𝟐𝜶𝒉 𝒙 𝒚 + 𝒂 𝒕 𝒚 𝟐
+ 𝜶𝒛 𝟐
+ 𝟐𝒚𝒛
+𝟐𝝀 𝒚 𝟐
𝜽 𝒅𝜽𝒅𝒔
𝒕
𝒕+𝒔
𝟎
−𝒓
(3.2)
where 𝐻 𝑥 = 𝑕 𝑠 𝑑𝑠
𝑥
0
, G y = g s ds
y
0
and 𝜆 is a positive constant which will be determined later. From
(iv) it follows that 𝑏(𝑡) is non-decreasing functions on 0, ∞ . Thus, since they are continuous on this interval
and bounded below 𝛿1 > 0, they are bounded on 0, ∞ and the limit of each exists as 𝑡 → ∞. Since 𝐿 in (iv)
and (v) is an arbitrary selected bound, we can also assume that:
𝟎 < 𝜹 𝟏 ≤ 𝒃 𝒕 ≤ 𝑳,
𝐥𝐢𝐦 𝒕→∞ 𝒃 𝒕 = 𝒃 𝟎, (3.3)
𝜹 𝟏 ≤ 𝒃 𝟎 ≤ 𝑳,
Due to (3.1) we write V as
𝑽 = 𝑯 𝒙 + 𝜶𝒃(𝒕)𝑮 𝒚 + 𝜶𝒉 𝒙 𝒚 +
𝟏
𝟐
𝒂 𝒕 𝒚 𝟐
+ 𝒚𝒛 + 𝜶𝒛 𝟐
+ 𝝀 𝒚 𝟐
𝜽 𝒅𝜽𝒅𝒔
𝒕
𝒕+𝒔
𝟎
−𝒓
(3.4)
= 𝑽 𝟏 +
𝟏
𝟐
𝑽 𝟐 + 𝝀 𝒚 𝟐
𝜽 𝒅𝜽𝒅𝒔
𝒕
𝒕+𝒔
𝟎
−𝒓
First, consider
𝑽 𝟐 = 𝒂 𝒕 𝒚 𝟐
+ 𝒚𝒛 + 𝜶𝒛 𝟐
= 𝒂 𝒕 𝒚 +
𝒛
𝟐𝒂 𝒕
𝟐
+
𝟏
𝟒𝒂 𝒕
(𝟒𝜶𝒂 𝒕 − 𝟏)𝒛 𝟐
By (v), 𝛼𝑎 𝑡 ≥ 𝛼𝑎 > 1 since 𝛼 >
1
𝑎
. clearly, 4𝛼𝑎 𝑡 − 1 is positive. Thus , there is a 𝛿3 > 0 such that
𝑽 𝟐 ≥
𝟏
𝟐
𝜹 𝟑 𝒚 𝟐
+
𝟏
𝟐
𝜹 𝟑 𝒛 𝟐
(3.5)
𝑽 𝟏 = 𝑯 𝒙 + 𝜶𝒃(𝒕)𝑮 𝒚 + 𝜶𝒉 𝒙 𝒚 ≥ 𝜹 𝟏 𝑯 𝒙 +
𝜶
𝟐
𝒃𝒚 𝟐
+ 𝜶𝒉 𝒙 𝒚

3. Stability and Boundedness of Solutions of Delay Differential Equations of third order
www.iosrjournals.org 11 | Page
Since 𝑏(𝑡) ≥ 1, 𝛿1 > 0 and
𝑔(𝑦)
𝑦
≥ 𝑏 > 0 implies that 𝐺(𝑦) ≥
1
2
𝑏𝑦2
The quantity in the brackets above can be written thus,
𝟏
𝟐
𝟐𝑯 𝒙 + 𝜶𝒃𝒚 𝟐
+ 𝟐𝜶𝒉 𝒙 𝒚 =
𝟏
𝟐
𝜶
𝒃
(𝒃𝒚 + 𝒉 𝒙 ) 𝟐
+ 𝟐𝑯 𝒙 −
𝜶
𝒃
𝒉 𝟐
(𝒙)
=
𝟏
𝟐
𝜶
𝒃
(𝒃𝒚 + 𝒉 𝒙 ) 𝟐
+ 𝟏 −
𝜶𝒉′ 𝒔
𝒃
𝒉 𝒔 𝒅𝒔
𝒙
𝟎
≥ 𝟏 −
𝜶𝒄
𝒃
𝒉 𝒔 𝒅𝒔
𝒙
𝟎
≥ 𝜹 𝟒 𝒉 𝒔 𝒅𝒔 =
𝒙
𝟎
𝜹 𝟒 𝑯(𝒙) (3.6)
Where 𝛿4 = 1 −
𝛼𝑐
𝑏
> 1 −
(
𝑏
𝑐
)𝑐
𝑏
= 0. Thus, since
𝑕(𝑥)
𝑥
> 𝛿0 > 0, we have
𝑽 𝟏 ≥
𝜹 𝟎 𝜹 𝟏 𝜹 𝟒
𝟐
𝒙 𝟐
(3.7)
Combining (3.4),(3.5) and (3.7) we have
𝑽 𝒕, 𝒙 𝒕, 𝒚 𝒕, 𝒛 𝒕 ≥
𝜹 𝟎 𝜹 𝟏 𝜹 𝟒
𝟐
𝒙 𝟐
+
𝜹 𝟑
𝟒
𝒚 𝟐
+
𝜹 𝟑
𝟒
𝒛 𝟐
+ 𝝀 𝒚 𝟐
𝜽 𝒅𝜽𝒅𝒔
𝒕
𝒕+𝒔
𝟎
−𝒓
Hence we can easily check that 𝑉 𝑡, 𝑥𝑡 , 𝑦𝑡, 𝑧𝑡 satisfies condition (i) of lemma 2.2
From (3.1) and (3.2) we obtain
𝒅
𝒅𝒕
𝑽 𝒕, 𝒙 𝒕, 𝒚 𝒕, 𝒛 𝒕 = 𝜶𝒃′
𝒕 𝑮 𝒚 +
𝟏
𝟐
𝒂′
𝒕 𝒚 𝟐
− 𝒃 𝒕 𝒈 𝒚 𝒚 − 𝜶𝒉′
𝒙 𝒚 𝟐
− 𝝀𝒓𝒚 𝟐
−[𝜶𝒂 𝒕 − 𝟏]𝒛 𝟐
+ 𝒚 𝒉′
𝒙 𝒔 𝒚 𝒔 𝒅𝒔 + 𝜶𝒛
𝒕
𝒕−𝒓
𝒉′
𝒙 𝒔 𝒚 𝒔 𝒅𝒔
𝒕
𝒕−𝒓
− 𝝀 𝒚 𝟐 𝒔 𝒅𝒔
𝒕
𝒕−𝒓
(3.8)
Since 𝑕′
(𝑥) ≤ 𝑐 and Using 2𝑢𝑣 ≤ 𝑢2
+ 𝑣2
, we have
𝜶𝒛 𝒉′
𝒙 𝒔 𝒚 𝒔 𝒅𝒔 ≤
𝒕
𝒕−𝒓
𝟏
𝟐
𝜶𝒄𝒓𝒛 𝟐
+
𝟏
𝟐
𝒄 𝒚 𝟐
𝒔 𝒅𝒔
𝒕
𝒕−𝒓
≤
𝟏
𝟐
𝑳𝜶𝒄𝒓𝒛 𝟐
+
𝟏
𝟐
𝑳𝒄 𝒚 𝟐
𝒔 𝒅𝒔
𝒕
𝒕−𝒓
and
𝒚 𝒉′
𝒙 𝒔 𝒚 𝒔 𝒅𝒔 ≤ 𝒄 𝒚 𝟐
𝒔 𝒅𝒔
𝒕
𝒕−𝒓
≤ 𝑳𝒄 𝒚 𝟐
𝒔 𝒅𝒔
𝒕
𝒕−𝒓
𝒕
𝒕−𝒓
Thus
𝒅
𝒅𝒕
𝑽 𝒕, 𝒙 𝒕, 𝒚 𝒕, 𝒛 𝒕 ≤ 𝜶𝒃′
𝒕 𝑮 𝒚 +
𝟏
𝟐
𝒂′
𝒕 𝒚 𝟐
− 𝒃 𝒕 𝒈 𝒚 𝒚 − 𝜶𝒉′
𝒙 𝒚 𝟐
− 𝝀𝒓𝒚 𝟐
−
𝟏
𝟐
[𝟐 𝜶𝒂 𝒕 − 𝟏 − 𝑳𝜶𝒓𝒄 𝒛 𝟐
+ [
𝟑
𝟐
𝑳𝒄 − 𝝀] 𝒚 𝟐
𝒔 𝒅𝒔
𝒕
𝒕−𝒓
(3.9)
If 𝑦 = 0, then 𝑏 𝑡 𝑔 𝑦 𝑦 − 𝛼𝑕′
𝑥 𝑦2
− 𝜆𝑟𝑦2
= 0. if 𝑦 ≠ 0, we can rewrite the term as
𝒃 𝒕 𝒈 𝒚 𝒚 − 𝜶𝒉′
𝒙 𝒚 𝟐
− 𝝀𝒓𝒚 𝟐
= 𝒃 𝒕
𝒈 𝒚
𝒚
− 𝜶𝒉′
𝒙 − 𝝀𝒓 𝒚 𝟐
≥ [𝒃𝒃 𝒕 − 𝜶𝒄 − 𝝀𝒓]𝒚 𝟐
= 𝒃 𝒕 𝒃 − 𝜶𝒄 𝒚 𝟐
− 𝝀𝒓𝒚 𝟐
≥ 𝜹 𝟏[𝒃 − 𝜶𝒄 − 𝜹 𝟏
−𝟏
𝝀𝒓]𝒚 𝟐
Thus,
𝟏
𝟐
𝒂′
𝒕 𝒚 𝟐
− [𝒃 𝒕 𝒈 𝒚 𝒚 − 𝜶𝒉′
𝒙 𝒚 𝟐
− 𝝀𝒓𝒚 𝟐
] ≤
𝟏
𝟐
𝒂′
𝒕 − 𝜹 𝟏(𝒃 − 𝜶𝒄 − 𝜹 𝟏
−𝟏
𝝀𝒓) 𝒚 𝟐
≤ 𝜹 𝟐 − 𝜹 𝟏 𝒃 − 𝜶𝒄 + 𝝀𝒓 𝒚 𝟐
≤ − 𝜹 𝟓 − 𝝀𝒓 𝒚 𝟐
(3.10)
Where 𝛿5 = 𝛿1 𝑏 − 𝛼𝑐 − 𝛿2 > 0 by (vi)
According to (v), 𝑎 𝑡 ≥ 𝛼𝑎 > 1 , thus
𝟐 𝜶𝒂 𝒕 − 𝟏 − 𝑳𝜶𝒓𝒄 𝒛 𝟐
≥ (𝜹 𝟔 − 𝑳𝜶𝒓𝒄)𝒛 𝟐
(3.11)
Where 𝛿6 = 𝛼𝑎 − 1 > 0.
Substituting (3.10) and (3.11) into (3.9) and also choose 𝜆 =
3
2
𝐿𝑐, we have that
𝒅
𝒅𝒕
𝑽 𝒕, 𝒙 𝒕, 𝒚 𝒕, 𝒛 𝒕 ≤ 𝜶𝒃′
𝒕 𝑮 𝒚 − 𝜹 𝟓 −
𝟑
𝟐
𝑳𝒄𝒓 𝒚 𝟐
− 𝜹 𝟔 − 𝑳𝜶𝒄𝒓 𝒛 𝟐
(3.12)

4. Stability and Boundedness of Solutions of Delay Differential Equations of third order
www.iosrjournals.org 12 | Page
Since 𝑏′
𝑡 ≤ 0 and 𝐺 𝑦 ≥ 0. 𝛼𝑏′
𝑡 𝐺 𝑦 ≤ 0 for all 𝑥, 𝑦 and 𝑡 ≥ 0.
since in (3.6) it was shown that
𝑯 𝒙 + 𝜶𝑮 𝒚 + 𝜶𝒉 𝒙 𝒚 ≥ 𝜹 𝟒 𝑯(𝒙) ≥ 𝟎.
Thus,
𝒅
𝒅𝒕
𝑽 𝒕, 𝒙 𝒕, 𝒚 𝒕, 𝒛 𝒕 ≤ − 𝜹 𝟓 −
𝟑
𝟐
𝑳𝒄𝒓 𝒚 𝟐
− 𝜹 𝟔 − 𝑳𝜶𝒄𝒓 𝒛 𝟐
Therefore, if
𝒓 < 𝑚𝑖𝑛
𝟐𝜹 𝟓
𝟑𝑳𝒄
,
𝜹 𝟔
𝑳𝜶𝒄
,
we have
𝒅
𝒅𝒕
𝑽 𝒕, 𝒙 𝒕, 𝒚 𝒕, 𝒛 𝒕 ≤ −𝜷 𝒚 𝟐
+ 𝒛 𝟐
, for some 𝛽 > 0.
By
𝑑
𝑑𝑡
𝑉 𝑡, 𝑥𝑡, 𝑦𝑡 , 𝑧𝑡 = 0 and system (3.1) , we can easily obtain :𝑥 = 𝑦 = 𝑧 = 0. Thus, the condition of lemma
2.2 are satisfied. Therefore the proof of the theorem 3.1 is now complete.
Proof of theorem 3.2. the proof of theorem 3.2 will depend upon the same scalar valued function
𝑉 𝑡, 𝑥𝑡 , 𝑦𝑡, 𝑧𝑡 as used in the proof of theorem 3.1. It was shown in the proof of theorem 3.1 that there is a
positive constant 𝛿7 such that 𝛿7 𝑋 2
≤
𝑑
𝑑𝑡
𝑉 𝑡, 𝑥𝑡, 𝑦𝑡 , 𝑧𝑡 , where 𝑋 = (𝑥, 𝑦, 𝑧) and thus that 𝑉 𝑡, 𝑥𝑡, 𝑦𝑡 , 𝑧𝑡 → ∞
as 𝑥2
+ 𝑦2
+ 𝑧2
→ ∞.
The proof that all solutions of (1.2) are bounded is based on the method in ([1], [11]) if it can be shown
that there exists a constant 𝐾 > 0 such that 𝑉 ≤ 𝐾 for all 𝑡, 𝑥, 𝑦 and 𝑧 their since 𝑉 𝑡, 𝑥𝑡, 𝑦𝑡 , 𝑧𝑡 → ∞ as
𝑥2
+ 𝑦2
+ 𝑧2
→ ∞, thus there exists a 𝐷 > 0 such that if 𝑥 = 𝑥(𝑡) is a solution of (1.2) then 𝑥(𝑡) ≤ 𝐷,
𝑥(𝑡) ≤ 𝐷, 𝑥(𝑡) ≤ 𝐷 for all 𝑡 ≥ 0. Equation (1.2) is equivalent to the system
𝒙 = 𝒚
𝒚 = 𝒛 (3.13)
𝒛 = −𝒂 𝒕 𝒛 − 𝒃 𝒕 𝒈(𝒚) − 𝒉 𝒙 + 𝒉′
𝒙 𝒔 𝒚 𝒔 𝒅𝒔 + 𝒑(𝒕)
𝒕
𝒕−𝒓
Along any solution 𝑥 𝑡 , 𝑦 𝑡 , 𝑧 𝑡 we have
𝑽(𝟑.𝟏𝟑) 𝒕, 𝒙 𝒕, 𝒚 𝒕, 𝒛 𝒕 = 𝑽(𝟑.𝟏) 𝒕, 𝒙 𝒕, 𝒚 𝒕, 𝒛 𝒕 + 𝒚 + 𝜶𝒛 𝒑 𝒕
Since 𝑉(3.1) ≤ 0 for all 𝑡, 𝑥, 𝑦, 𝑧, thus
𝑽(𝟑.𝟏𝟑) ≤ 𝒚 + 𝜶𝒛 𝒑 𝒕 ≤ ( 𝒚 + 𝜶 𝒛 ) 𝒑 𝒕 ≤ 𝜹 𝟖( 𝒚 + 𝒛 𝒑(𝒕)
Where 𝛿8 =max 1, 𝛼 . Noting that 𝑥 < 1 + 𝑥2
we get
𝑽(𝟑.𝟏𝟑) ≤ 𝜹 𝟖(𝟐 + 𝒚 𝟐
+ 𝒛 𝟐
) 𝒑(𝒕)
≤ 𝟐𝜹 𝟖 𝒑(𝒕) + 𝜹 𝟖 𝑿 𝟐
𝒑(𝒕)
≤ 𝟐𝜹 𝟖 𝒑(𝒕) +
𝜹 𝟖
𝜹 𝟕
𝑽 𝒕, 𝒙 𝒕, 𝒚 𝒕, 𝒛 𝒕 𝒑(𝒕)
Recalling that 𝛿7 𝑋 2
≤ 𝑉 𝑡, 𝑥𝑡, 𝑦𝑡 , 𝑧𝑡 .
Let 𝜂 = max⁡(2𝛿8,
𝛿8
𝛿7
) then
𝑉(3.13) ≤ 𝜂 𝑝(𝑡) + 𝜂𝑉 𝑝(𝑡)
𝑉(3.13) − 𝜂𝑉 𝑝(𝑡) ≤ 𝜂 𝑝(𝑡) .
Multiplying each side of this inequality by the integrating factor
exp −𝜂 𝑝(𝑠)
𝑡
0
𝑑𝑠 , we get
𝑽(𝟑.𝟏𝟑) exp −𝜼 𝒑(𝒔)
𝒕
𝟎
𝒅𝒔 − 𝜼𝑽 𝒑(𝒕) 𝐞𝐱𝐩 −𝜼 𝒑(𝒔)
𝒕
𝟎
𝒅𝒔 ,
We get
𝑽(𝟑.𝟏𝟑) exp −𝜼 𝒑(𝒔)
𝒕
𝟎
𝒅𝒔 − 𝜼𝑽 𝒑(𝒕) 𝐞𝐱𝐩 −𝜼 𝒑(𝒔)
𝒕
𝟎
𝒅𝒔 ≤ 𝜼 𝒑(𝒕) 𝐞𝐱𝐩 −𝜼 𝒑(𝒔)
𝒕
𝟎
𝒅𝒔
Integrating each side of this inequality from 0 to 𝑡, we get, where 0 = 𝑥 0 , 𝑦 0 , 𝑧 0 ,
𝑽𝐞𝐱𝐩 −𝜼 𝒑(𝒔)
𝒕
𝟎
𝒅𝒔 − 𝑽 𝟎 ≤ 𝟏 − 𝐞𝐱𝐩 −𝜼 𝒑(𝒔)
𝒕
𝟎
𝒅𝒔
Or
𝑽 ≤ 𝑽 𝟎 𝐞𝐱𝐩 𝜼 𝒑 𝒔
𝒕
𝟎
𝒅𝒔 + 𝐞𝐱𝐩 𝜼 𝒑(𝒔)
𝒕
𝟎
𝒅𝒔 − 𝟏
Since 𝑝(𝑠)
𝑡
0
𝑑𝑠 ≤ 𝐴 for all t, this implies

5. Stability and Boundedness of Solutions of Delay Differential Equations of third order
www.iosrjournals.org 13 | Page
𝑽 𝒕, 𝒙 𝒕, 𝒚 𝒕, 𝒛 𝒕 ≤ 𝑽 𝟎 𝒆 𝜼𝑨
+ [𝒆 𝜼𝑨
− 𝟏] for 𝒕 ≥ 𝟎
Now, since the right-hand side is a constant, and since
𝑉 𝑡, 𝑥𝑡 , 𝑦𝑡, 𝑧𝑡 → ∞ as 𝑥2
+ 𝑦2
+ 𝑧2
→ ∞, it follows that there exists a 𝐷 > 0 such that
𝑥(𝑡) ≤ 𝐷,
𝑦(𝑡) ≤ 𝐷
𝑧(𝑡) ≤ 𝐷 for 𝑡 ≥ 0.
From the system (3.13) this implies that
𝑥(𝑡) ≤ 𝐷, 𝑥(𝑡) ≤ 𝐷, 𝑥(𝑡) ≤ 𝐷 for 𝑡 ≥ 0.
I. Conclusions :
In this paper, we considered the third order non-autonomous non-linear differential equations with
delays. The differential equations we have discussed in this paper in which 𝑎(𝑡), 𝑏(𝑡) are constants have been
studied by several authors. They obtained criteria which ensure the stability, uniform boundedness and uniform
ultimate boundedness of solutions. Sadek establishes conditions under which all solutions of the non-
autonomous equation tend to the zero solution as 𝑡 → ∞. These results are now extended by considering the
semi-invariant set of a related non-autonomous system.
References:
[1] Antoslewicz, H.A. On nonlinear differential equations of the second order with integrable forcing term, J. London Math. Soc. 30
(1955), 64-67.
[2] G. Makay. On the asymptotic stability of the solutions of functional differential equations with infinite delay. J. Diff.
Eqs.,108:(1994),139-151.
[3] J. K. Hale. Theory of Functional Differential Equations. Springer-Verlag, New York. (1977)
[4] Omeike, M.O. Further results on global stability of third-order nonlinear differential equations, Nonlinear Analysis 67 (2007),
3394-3400.
[5] Sadek, A.I. Stability and boundedness of a kind of third-order delay differential system, Applied Mathematics Letters 16 (2003), 657-
662.
[6] Sadek, A.I. On the stability of solutions of some non-autonomous delay differential equations of the third order, Asymptotic Analysis
43(2005), 1-7.
[7] S. Li, L. Wen.Functional Differential Equations. Human Science and Technology Press. (1987)
[8] Swick, K. On the boundedness and the stability of solutions of some non-autonomous differential equations of the third order, J.
London Math. Soc. 44(1969),347-359.
[9] T. A. Burton. Volterra Integral and Differential Equations. Academic Press New York. (1983)
[10] T. A. Burton. Stability and Periodic Solutions of Ordinary and Functional Differential Equations. Academic Press.(1985)
[11] Tunc. C. Boundedness of solutions of a third-order nonlinear differential equation, JIPAM. J. Inequal. Pure Appl. Math. 6 (2005), 1-
6.
[12] V. Kelmanovskii, A. Myshkis. Introduction to the Theory and Applications of Functional Differential Equations. Kluwer
Academic,Dordrecht. (1999)
[13] Zhu, Y.F. On stability, boundedness and existence of periodic solution of a kind of third-order nonlinear delay differential system,
Ann. Differential Equations, 8(1992),249-259.