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IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728. Volume 5, Issue 6 (Mar. - Apr. 2013), PP 09-13 www.iosrjournals.org

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 3

Department of Mathematics, Osmania University, Hyderabad-500007 India.

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:

đ?’™+đ?’‚ đ?’• đ?’™+đ?’ƒ đ?’• đ?’ˆ đ?’™ +đ?’‰ đ?’™ đ?’•âˆ’đ?’“

=đ?&#x;Ž

(1.1)

and

đ?’™ + đ?’‚ đ?’• đ?’™ + đ?’ƒ đ?’• đ?’ˆ đ?’™ + đ?’‰ đ?’™ đ?’• − đ?’“ = đ?’‘(đ?’•) (1.2) where r is a positive constant, đ?‘Ž(đ?‘Ą), đ?‘?(đ?‘Ą), đ?‘” đ?‘Ľ , đ?‘•(đ?‘Ľ) are real valued functions continuous in their respective arguments; đ?‘” 0 = đ?‘• 0 = 0. The dots indicates differential with respect to đ?‘Ą and all solutions are assumed real. Equations of the forms (1.1) and (1.2) in which đ?‘Ž(đ?‘Ą), đ?‘?(đ?‘Ą) 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

đ?’™+đ?’‚ đ?’• đ?’™+đ?’ƒ đ?’• đ?’™+đ?’‰ đ?’™ đ?’•âˆ’đ?’“

=đ?&#x;Ž

tend to the zero solution as đ?‘Ą → ∞. This result is now extended to equiation (1.1) by considering the semiinvariant 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.

đ?&#x2019;&#x2122; = đ?&#x2019;&#x2021; đ?&#x2019;&#x2022;, đ?&#x2019;&#x2122;đ?&#x2019;&#x2022; , đ?&#x2019;&#x2122;đ?&#x2019;&#x2022; = đ?&#x2019;&#x2122; đ?&#x2019;&#x2022; + đ?&#x153;˝ , â&#x2C6;&#x2019;đ?&#x2019;&#x201C; â&#x2030;¤ đ?&#x153;˝ â&#x2030;¤ đ?&#x;&#x17D;, đ?&#x2019;&#x2022; â&#x2030;Ľ đ?&#x;&#x17D; (2.1) where đ?&#x2018;&#x201C;: đ??ź Ă&#x2014; đ??śđ??ť â&#x2020;&#x2019; đ?&#x2018;&#x2026;đ?&#x2018;&#x203A; is a continuous mapping . đ?&#x2018;&#x201C; đ?&#x2018;Ą, 0 = 0, đ??śđ??ť = {đ?&#x153;&#x2122; â&#x2C6;&#x2C6; đ??ś â&#x2C6;&#x2019;đ?&#x2018;&#x;, 0 , đ?&#x2018;&#x2026;đ?&#x2018;&#x203A; : đ?&#x153;&#x2122; â&#x2030;¤ đ??ť} and for đ??ť1 â&#x2030;¤ đ??ť , there exists đ??ż đ??ť1 > 0 , with đ?&#x2018;&#x201C;(đ?&#x153;&#x2122;) â&#x2030;¤ đ??ż đ??ť1 when đ?&#x153;&#x2122; â&#x2030;¤ đ??ť1 . Definition 2.1 : An element đ?&#x153;&#x201C; â&#x2C6;&#x2C6; đ??ś is in the đ?&#x153;&#x201D;-limit set of đ?&#x153;&#x2122;, say, Ί đ?&#x153;&#x2122; , if đ?&#x2018;Ľ(đ?&#x2018;Ą, 0, đ?&#x153;&#x2122;) is defined on [0, â&#x2C6;&#x17E;) and there is a sequence đ?&#x2018;Ąđ?&#x2018;&#x203A; , đ?&#x2018;Ąđ?&#x2018;&#x203A; â&#x2020;&#x2019; â&#x2C6;&#x17E;, as đ?&#x2018;&#x203A; â&#x2020;&#x2019; â&#x2C6;&#x17E;, with đ?&#x2018;Ľđ?&#x2018;Ą đ?&#x2018;&#x203A; đ?&#x153;&#x2122; â&#x2C6;&#x2019; đ?&#x153;&#x201C; â&#x2020;&#x2019; 0 as đ?&#x2018;&#x203A; â&#x2020;&#x2019; â&#x2C6;&#x17E; where đ?&#x2018;Ľđ?&#x2018;Ą đ?&#x2018;&#x203A; đ?&#x153;&#x2122; = đ?&#x2018;Ľ(đ?&#x2018;Ąđ?&#x2018;&#x203A; + đ?&#x153;&#x192;, 0, đ?&#x153;&#x2122;) for â&#x2C6;&#x2019;đ?&#x2018;&#x; â&#x2030;¤ đ?&#x153;&#x192; â&#x2030;¤ 0. Definition 2.2 : A set đ?&#x2018;&#x201E; â&#x160;&#x201A; đ??śđ??ť is an invariant set if for any đ?&#x153;&#x2122; â&#x2C6;&#x2C6; đ?&#x2018;&#x201E;,the solution of (2.1), đ?&#x2018;Ľ(đ?&#x2018;Ą, 0, đ?&#x153;&#x2122;) is defined on [0, â&#x2C6;&#x17E;) and đ?&#x2018;Ľđ?&#x2018;Ą (đ?&#x153;&#x2122;) â&#x2C6;&#x2C6; đ?&#x2018;&#x201E; for đ?&#x2018;Ą â&#x2C6;&#x2C6; [0, â&#x2C6;&#x17E;) Lemma 2.1. If đ?&#x153;&#x2122; â&#x2C6;&#x2C6; đ??śđ??ť is such that the solution đ?&#x2018;Ľđ?&#x2018;Ą (đ?&#x153;&#x2122;) of (2.1) with đ?&#x2018;Ľ0 đ?&#x153;&#x2122; = đ?&#x153;&#x2122; is defned on [0, â&#x2C6;&#x17E;) and đ?&#x2018;Ľđ?&#x2018;Ą (đ?&#x153;&#x2122;) â&#x2030;¤ đ??ť1 < đ??ť for đ?&#x2018;Ą â&#x2C6;&#x2C6; 0, â&#x2C6;&#x17E; , then Ί(đ?&#x153;&#x2122;) is a non-empty, compact, invariant set and dist(đ?&#x2018;Ľđ?&#x2018;Ą đ?&#x153;&#x2122; , Ί(đ?&#x153;&#x2122;)) â&#x2020;&#x2019; â&#x2C6;&#x17E; as đ?&#x2018;Ą â&#x2020;&#x2019; â&#x2C6;&#x17E;. www.iosrjournals.org

9 | Page


Stability and Boundedness of Solutions of Delay Differential Equations of third order Lemma 2.2. Let đ?&#x2018;&#x2030; đ?&#x2018;Ą, đ?&#x153;&#x2122; : đ??ź Ă&#x2014; đ??śđ??ť â&#x2020;&#x2019; đ?&#x2018;&#x2026; be a continuous functional satisfying a local Lipschitz condition. đ?&#x2018;&#x2030; đ?&#x2018;Ą, 0 = 0, and such that: (i) đ?&#x2018;&#x160;1 ( đ?&#x153;&#x2122; 0 ) â&#x2030;¤ đ?&#x2018;&#x2030;(đ?&#x2018;Ą, đ?&#x153;&#x2122;) â&#x2030;¤ đ?&#x2018;&#x160;2 ( đ?&#x153;&#x2122; ) where đ?&#x2018;&#x160;1 đ?&#x2018;&#x; , đ?&#x2018;&#x160;2 đ?&#x2018;&#x; , are wedges (ii) đ?&#x2018;&#x2030;(2.1) đ?&#x2018;Ą, đ?&#x153;&#x2122; â&#x2030;¤ 0, for đ?&#x153;&#x2122; â&#x2C6;&#x2C6; đ??śđ??ť . Then the zero solution of (2.1) is uniformly stable. If we define đ?&#x2018;? = đ?&#x153;&#x2122; â&#x2C6;&#x2C6; đ??śđ??ť â&#x2C6;ś đ?&#x2018;&#x2030;(2.1) đ?&#x2018;Ą, đ?&#x153;&#x2122; = 0 , then the zero solution of (2.1) is asymptotically stable, provided that the largest invariant set in đ?&#x2018;? is đ?&#x2018;&#x201E; = 0 .

3. Main Result The main objective of this paper is to prove the following: Theorem 3.1. Suppose that đ?&#x2018;&#x17D;(đ?&#x2018;Ą), đ?&#x2018;?(đ?&#x2018;Ą) â&#x2C6;&#x2C6; đ??ś â&#x20AC;˛ (đ??ź), đ?&#x2018;&#x2022; â&#x2C6;&#x2C6; đ??ś â&#x20AC;˛ (đ?&#x2018;&#x2026;) and đ?&#x2018;&#x201D; â&#x2C6;&#x2C6; đ??ś(đ?&#x2018;&#x2026;) and that these functions satisfy the following conditions: đ?&#x2018;&#x2022; (đ?&#x2018;Ľ) (i) đ?&#x2018;&#x2022; 0 = 0, â&#x2030;Ľ đ?&#x203A;ż0 > 0, đ?&#x2018;Ľ â&#x2030;  0, đ?&#x2018;Ľ â&#x20AC;˛ (ii) đ?&#x2018;&#x2022; (đ?&#x2018;Ľ) â&#x2030;¤ đ?&#x2018;? đ?&#x2018;&#x201D;(đ?&#x2018;Ś) (iii) đ?&#x2018;&#x201D; 0 = 0, â&#x2030;Ľ đ?&#x2018;? > 0, đ?&#x2018;Ś â&#x2030;  0, đ?&#x2018;Ś

(iv) 0 < đ?&#x203A;ż1 â&#x2030;¤ đ?&#x2018;? đ?&#x2018;Ą , â&#x2C6;&#x2019;đ??ż â&#x2030;¤ đ?&#x2018;? â&#x20AC;˛ (đ?&#x2018;Ą) â&#x2030;¤ 0, đ?&#x2018;Ą â&#x2C6;&#x2C6; đ??ź, (v) 0 < đ?&#x2018;&#x17D; â&#x2030;¤ đ?&#x2018;&#x17D;(đ?&#x2018;Ą) â&#x2030;¤ đ??ż, đ?&#x2018;Ą â&#x2C6;&#x2C6; đ??ź. Then every solution đ?&#x2018;Ľ = đ?&#x2018;Ľ(đ?&#x2018;Ą) of (1.1) is uniform-bounded and satisfies đ?&#x2018;Ľ(đ?&#x2018;Ą) â&#x2020;&#x2019; 0, đ?&#x2018;Ľ (đ?&#x2018;Ą) â&#x2020;&#x2019; 0, đ?&#x2018;Ľ (đ?&#x2018;Ą) â&#x2020;&#x2019; 0 as đ?&#x2018;? 1 đ?&#x2018;Ą â&#x2020;&#x2019; â&#x2C6;&#x17E; provided there exists đ?&#x203A;ź satisfying > đ?&#x203A;ź > such that đ?&#x2018;?

1

đ?&#x2018;&#x17D;

2đ?&#x203A;ż

đ?&#x203A;ż

(vi) đ?&#x2018;&#x17D;â&#x20AC;˛ đ?&#x2018;Ą â&#x2030;¤ đ?&#x203A;ż2 < đ?&#x203A;ż1 đ?&#x2018;? â&#x2C6;&#x2019; đ?&#x203A;źđ?&#x2018;? đ?&#x2018;Ą â&#x2C6;&#x2C6; đ??ź and đ?&#x2018;&#x; < min 5 , 5 . 2 3đ??żđ?&#x2018;? đ??żđ?&#x203A;źđ?&#x2018;? Theorem 3.2. If all the conditions of Theorem 3.1 are satisfied then all solutions of the perturbed equation đ?&#x2018;Ą (1.2) are bounded provided 0 đ?&#x2018;?(đ?&#x2018; ) đ?&#x2018;&#x2018;đ?&#x2018;  < â&#x2C6;&#x17E; , for all đ?&#x2018;Ą â&#x2030;Ľ 0.

Proof of Theorem 3.1. We write the Eq. (1.1) as the following equivalent system: đ?&#x2019;&#x2122;=đ?&#x2019;&#x161; đ?&#x2019;&#x161;=đ?&#x2019;&#x203A; đ?&#x2019;&#x2022; đ?&#x2019;&#x203A; = â&#x2C6;&#x2019;đ?&#x2019;&#x201A; đ?&#x2019;&#x2022; đ?&#x2019;&#x203A; â&#x2C6;&#x2019; đ?&#x2019;&#x192; đ?&#x2019;&#x2022; đ?&#x2019;&#x2C6;(đ?&#x2019;&#x161;) â&#x2C6;&#x2019; đ?&#x2019;&#x2030; đ?&#x2019;&#x2122; + đ?&#x2019;&#x2022;â&#x2C6;&#x2019;đ?&#x2019;&#x201C; đ?&#x2019;&#x2030;â&#x20AC;˛ đ?&#x2019;&#x2122; đ?&#x2019;&#x201D; đ?&#x2019;&#x161; đ?&#x2019;&#x201D; đ?&#x2019;&#x2026;đ?&#x2019;&#x201D;

(3.1)

Define its Lyapunov functional as :

đ?&#x;?đ?&#x2018;˝ đ?&#x2019;&#x2022;, đ?&#x2019;&#x2122;đ?&#x2019;&#x2022; , đ?&#x2019;&#x161;đ?&#x2019;&#x2022; , đ?&#x2019;&#x203A;đ?&#x2019;&#x2022; = đ?&#x;?đ?&#x2018;Ż đ?&#x2019;&#x2122; + đ?&#x;?đ?&#x153;śđ?&#x2019;&#x192; đ?&#x2019;&#x2022; đ?&#x2018;Ž đ?&#x2019;&#x161; + đ?&#x;?đ?&#x153;śđ?&#x2019;&#x2030; đ?&#x2019;&#x2122; đ?&#x2019;&#x161; + đ?&#x2019;&#x201A; đ?&#x2019;&#x2022; đ?&#x2019;&#x161;đ?&#x;? + đ?&#x153;śđ?&#x2019;&#x203A;đ?&#x;? + đ?&#x;?đ?&#x2019;&#x161;đ?&#x2019;&#x203A; đ?&#x;&#x17D; đ?&#x2019;&#x2022; +đ?&#x;?đ??&#x20AC; â&#x2C6;&#x2019;đ?&#x2019;&#x201C; đ?&#x2019;&#x2022;+đ?&#x2019;&#x201D; đ?&#x2019;&#x161;đ?&#x;? đ?&#x153;˝ đ?&#x2019;&#x2026;đ?&#x153;˝đ?&#x2019;&#x2026;đ?&#x2019;&#x201D; (3.2)

đ?&#x2018;Ľ

y

where đ??ť đ?&#x2018;Ľ = 0 đ?&#x2018;&#x2022; đ?&#x2018;  đ?&#x2018;&#x2018;đ?&#x2018;  , G y = 0 g s ds and đ?&#x153;&#x2020; is a positive constant which will be determined later. From (iv) it follows that đ?&#x2018;?(đ?&#x2018;Ą) is non-decreasing functions on 0, â&#x2C6;&#x17E; . Thus, since they are continuous on this interval and bounded below đ?&#x203A;ż1 > 0, they are bounded on 0, â&#x2C6;&#x17E; and the limit of each exists as đ?&#x2018;Ą â&#x2020;&#x2019; â&#x2C6;&#x17E;. Since đ??ż in (iv) and (v) is an arbitrary selected bound, we can also assume that:

đ?&#x;&#x17D; < đ?&#x153;šđ?&#x;? â&#x2030;¤ đ?&#x2019;&#x192; đ?&#x2019;&#x2022; â&#x2030;¤ đ?&#x2018;ł, đ??Ľđ??˘đ??Śđ?&#x2019;&#x2022;â&#x2020;&#x2019;â&#x2C6;&#x17E; đ?&#x2019;&#x192; đ?&#x2019;&#x2022; = đ?&#x2019;&#x192;đ?&#x;&#x17D; , đ?&#x153;šđ?&#x;? â&#x2030;¤ đ?&#x2019;&#x192;đ?&#x;&#x17D; â&#x2030;¤ đ?&#x2018;ł,

(3.3)

Due to (3.1) we write V as

đ?&#x2018;˝ = đ?&#x2018;Ż đ?&#x2019;&#x2122; + đ?&#x153;śđ?&#x2019;&#x192;(đ?&#x2019;&#x2022;)đ?&#x2018;Ž đ?&#x2019;&#x161; + đ?&#x153;śđ?&#x2019;&#x2030; đ?&#x2019;&#x2122; đ?&#x2019;&#x161; + = đ?&#x2018;˝đ?&#x;? +

đ?&#x;? đ?&#x2018;˝ đ?&#x;? đ?&#x;?

+đ??&#x20AC;

đ?&#x;&#x17D; đ?&#x2019;&#x2022; đ?&#x2019;&#x161;đ?&#x;? â&#x2C6;&#x2019;đ?&#x2019;&#x201C; đ?&#x2019;&#x2022;+đ?&#x2019;&#x201D;

đ?&#x;? đ?&#x;?

đ?&#x2019;&#x201A; đ?&#x2019;&#x2022; đ?&#x2019;&#x161;đ?&#x;? + đ?&#x2019;&#x161;đ?&#x2019;&#x203A; + đ?&#x153;śđ?&#x2019;&#x203A;đ?&#x;? + đ??&#x20AC;

đ?&#x;&#x17D; đ?&#x2019;&#x2022; đ?&#x2019;&#x161;đ?&#x;? â&#x2C6;&#x2019;đ?&#x2019;&#x201C; đ?&#x2019;&#x2022;+đ?&#x2019;&#x201D;

đ?&#x153;˝ đ?&#x2019;&#x2026;đ?&#x153;˝đ?&#x2019;&#x2026;đ?&#x2019;&#x201D;

(3.4)

đ?&#x153;˝ đ?&#x2019;&#x2026;đ?&#x153;˝đ?&#x2019;&#x2026;đ?&#x2019;&#x201D;

First, consider

đ?&#x2018;˝đ?&#x;? = đ?&#x2019;&#x201A; đ?&#x2019;&#x2022; đ?&#x2019;&#x161;đ?&#x;? + đ?&#x2019;&#x161;đ?&#x2019;&#x203A; + đ?&#x153;śđ?&#x2019;&#x203A;đ?&#x;? =đ?&#x2019;&#x201A; đ?&#x2019;&#x2022; 1

đ?&#x;?

đ?&#x2019;&#x203A;

đ?&#x2019;&#x161; + đ?&#x;?đ?&#x2019;&#x201A;

đ?&#x2019;&#x2022;

đ?&#x;?

+ đ?&#x;&#x2019;đ?&#x2019;&#x201A;

đ?&#x2019;&#x2022;

(đ?&#x;&#x2019;đ?&#x153;śđ?&#x2019;&#x201A; đ?&#x2019;&#x2022; â&#x2C6;&#x2019; đ?&#x;?)đ?&#x2019;&#x203A;đ?&#x;?

By (v), đ?&#x203A;źđ?&#x2018;&#x17D; đ?&#x2018;Ą â&#x2030;Ľ đ?&#x203A;źđ?&#x2018;&#x17D; > 1 since đ?&#x203A;ź > . clearly, 4đ?&#x203A;źđ?&#x2018;&#x17D; đ?&#x2018;Ą â&#x2C6;&#x2019; 1 is positive. Thus , there is a đ?&#x203A;ż3 > 0 such that đ?&#x2018;&#x17D;

đ?&#x;? đ?&#x;?

đ?&#x;? đ?&#x;?

đ?&#x2018;˝đ?&#x;? â&#x2030;Ľ đ?&#x153;šđ?&#x;&#x2018; đ?&#x2019;&#x161;đ?&#x;? + đ?&#x153;šđ?&#x;&#x2018; đ?&#x2019;&#x203A;đ?&#x;? đ?&#x2018;˝đ?&#x;? = đ?&#x2018;Ż đ?&#x2019;&#x2122; + đ?&#x153;śđ?&#x2019;&#x192;(đ?&#x2019;&#x2022;)đ?&#x2018;Ž đ?&#x2019;&#x161; + đ?&#x153;śđ?&#x2019;&#x2030; đ?&#x2019;&#x2122; đ?&#x2019;&#x161; â&#x2030;Ľ đ?&#x153;šđ?&#x;? www.iosrjournals.org

(3.5)

đ?&#x153;ś đ?&#x2018;Ż đ?&#x2019;&#x2122; + đ?&#x2019;&#x192;đ?&#x2019;&#x161;đ?&#x;? + đ?&#x153;śđ?&#x2019;&#x2030; đ?&#x2019;&#x2122; đ?&#x2019;&#x161; đ?&#x;? 10 | Page


Stability and Boundedness of Solutions of Delay Differential Equations of third order Since 𝑏(𝑡) ≥ 1, 𝛿1 > 0 and

𝑔(𝑦) 𝑦

1

≥ 𝑏 > 0 implies that 𝐺(𝑦) ≥ 𝑏𝑦 2 2

The quantity in the brackets above can be written thus, 𝟏 𝟐

Where 𝛿4 = 1 −

𝟏 𝜶 𝜶 (𝒃𝒚 + 𝒉 𝒙 )𝟐 + 𝟐𝑯 𝒙 − 𝒃 𝒉𝟐 (𝒙) 𝒃 𝒙 𝟏𝜶 𝜶𝒉′ 𝒔 = 𝟐 𝒃 (𝒃𝒚 + 𝒉 𝒙 )𝟐 + 𝟎 𝟏 − 𝒃 𝒉 𝒔 𝒅𝒔 𝒙 𝜶𝒄 ≥ 𝟎 𝟏 − 𝒃 𝒉 𝒔 𝒅𝒔 𝒙 ≥ 𝜹𝟒 𝟎 𝒉 𝒔 𝒅𝒔 = 𝜹𝟒 𝑯(𝒙)

𝟐𝑯 𝒙 + 𝜶𝒃𝒚𝟐 + 𝟐𝜶𝒉 𝒙 𝒚 = 𝟐

𝛼𝑐 𝑏

>1−

𝑏 ( )𝑐 𝑐

𝑏

= 0. Thus, since

𝑕(𝑥) 𝑥

𝑽𝟏 ≥

(3.6)

> 𝛿0 > 0, we have

𝜹 𝟎 𝜹𝟏 𝜹𝟒 𝟐 𝒙 𝟐

(3.7)

Combining (3.4),(3.5) and (3.7) we have

𝑽 𝒕, 𝒙𝒕 , 𝒚𝒕 , 𝒛𝒕

𝜹𝟎 𝜹𝟏 𝜹𝟒 𝟐 𝜹𝟑 𝟐 𝜹𝟑 𝟐 ≥ 𝒙 + 𝒚 + 𝒛 +𝝀 𝟐 𝟒 𝟒

𝟎

𝒕

𝒚𝟐 𝜽 𝒅𝜽𝒅𝒔 −𝒓 𝒕+𝒔

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 𝑽 𝒅𝒕

𝟏

𝒕, 𝒙𝒕 , 𝒚𝒕 , 𝒛𝒕 = 𝜶𝒃′ 𝒕 𝑮 𝒚 + 𝟐 𝒂′ 𝒕 𝒚𝟐 − 𝒃 𝒕 𝒈 𝒚 𝒚 − 𝜶𝒉′ 𝒙 𝒚𝟐 − 𝝀𝒓𝒚𝟐 −[𝜶𝒂 𝒕 − 𝟏]𝒛𝟐 + 𝒚 −𝝀

𝒕 𝒚𝟐 𝒕−𝒓

𝒕 𝒉′ 𝒕−𝒓

𝒙 𝒔 𝒚 𝒔 𝒅𝒔 + 𝜶𝒛

𝒕 𝒉′ 𝒕−𝒓

𝒙 𝒔 𝒚 𝒔 𝒅𝒔

𝒔 𝒅𝒔

(3.8)

Since 𝑕′ (𝑥) ≤ 𝑐 and Using 2𝑢𝑣 ≤ 𝑢2 + 𝑣 2 , we have

𝜶𝒛

𝒕 𝒉′ 𝒕−𝒓

𝟏

𝟏

𝒕 𝒚𝟐 𝒔 𝒅𝒔 𝒕−𝒓 𝒕 𝟏 𝑳𝒄 𝒕−𝒓 𝒚𝟐 𝒔 𝒅𝒔 𝟐

𝒙 𝒔 𝒚 𝒔 𝒅𝒔 ≤ 𝟐 𝜶𝒄𝒓𝒛𝟐 + 𝟐 𝒄 𝟏 𝟐

≤ 𝑳𝜶𝒄𝒓𝒛𝟐 + and

𝒚

𝒕 𝒉′ 𝒕−𝒓

𝒙 𝒔 𝒚 𝒔 𝒅𝒔 ≤ 𝒄

𝒕 𝒚𝟐 𝒕−𝒓

𝒔 𝒅𝒔 ≤ 𝑳𝒄

𝒕 𝒚𝟐 𝒕−𝒓

𝒔 𝒅𝒔

Thus

𝒅 𝟏 𝑽 𝒕, 𝒙𝒕 , 𝒚𝒕 , 𝒛𝒕 ≤ 𝜶𝒃′ 𝒕 𝑮 𝒚 + 𝒂′ 𝒕 𝒚𝟐 − 𝒃 𝒕 𝒈 𝒚 𝒚 − 𝜶𝒉′ 𝒙 𝒚𝟐 − 𝝀𝒓𝒚𝟐 𝒅𝒕 𝟐 𝟏

𝟑

− 𝟐 [𝟐 𝜶𝒂 𝒕 − 𝟏 − 𝑳𝜶𝒓𝒄 𝒛𝟐 + [𝟐 𝑳𝒄 − 𝝀] ′

2

2

𝒕 𝒚𝟐 𝒕−𝒓

𝒔 𝒅𝒔

(3.9)

If 𝑦 = 0, then 𝑏 𝑡 𝑔 𝑦 𝑦 − 𝛼𝑕 𝑥 𝑦 − 𝜆𝑟𝑦 = 0. if 𝑦 ≠ 0, we can rewrite the term as

𝒃 𝒕 𝒈 𝒚 𝒚 − 𝜶𝒉′ 𝒙 𝒚𝟐 − 𝝀𝒓𝒚𝟐 = 𝒃 𝒕

𝒈 𝒚 𝒚

− 𝜶𝒉′ 𝒙 − 𝝀𝒓 𝒚𝟐

≥ [𝒃𝒃 𝒕 − 𝜶𝒄 − 𝝀𝒓]𝒚𝟐 = 𝒃 𝒕 𝒃 − 𝜶𝒄 𝒚𝟐 − 𝝀𝒓𝒚𝟐 𝟐 ≥ 𝜹𝟏 [𝒃 − 𝜶𝒄 − 𝜹−𝟏 𝟏 𝝀𝒓]𝒚 Thus,

𝟏 ′ 𝟏 𝟐 𝒂 𝒕 𝒚𝟐 − [𝒃 𝒕 𝒈 𝒚 𝒚 − 𝜶𝒉′ 𝒙 𝒚𝟐 − 𝝀𝒓𝒚𝟐 ] ≤ 𝒂′ 𝒕 − 𝜹𝟏 (𝒃 − 𝜶𝒄 − 𝜹−𝟏 𝟏 𝝀𝒓) 𝒚 𝟐 𝟐 ≤ 𝜹𝟐 − 𝜹𝟏 𝒃 − 𝜶𝒄 + 𝝀𝒓 𝒚𝟐 ≤ − 𝜹𝟓 − 𝝀𝒓 𝒚𝟐

(3.10)

Where 𝛿5 = 𝛿1 𝑏 − 𝛼𝑐 − 𝛿2 > 0 by (vi) According to (v), 𝑎 𝑡 ≥ 𝛼𝑎 > 1 , thus

𝟐 𝜶𝒂 𝒕 − 𝟏 − 𝑳𝜶𝒓𝒄 𝒛𝟐 ≥ (𝜹𝟔 − 𝑳𝜶𝒓𝒄)𝒛𝟐

Where 𝛿6 = 𝛼𝑎 − 1 > 0. Substituting (3.10) and (3.11) 𝒅 𝑽 𝒅𝒕

(3.11)

3

into (3.9) and also choose 𝜆 = 𝐿𝑐, we have that 2

𝟑

𝒕, 𝒙𝒕 , 𝒚𝒕 , 𝒛𝒕 ≤ 𝜶𝒃′ 𝒕 𝑮 𝒚 − 𝜹𝟓 − 𝟐 𝑳𝒄𝒓 𝒚𝟐 − 𝜹𝟔 − 𝑳𝜶𝒄𝒓 𝒛𝟐 www.iosrjournals.org

(3.12)

11 | Page


Stability and Boundedness of Solutions of Delay Differential Equations of third order Since đ?&#x2018;? â&#x20AC;˛ đ?&#x2018;Ą â&#x2030;¤ 0 and đ??ş đ?&#x2018;Ś â&#x2030;Ľ 0. đ?&#x203A;źđ?&#x2018;? â&#x20AC;˛ đ?&#x2018;Ą đ??ş đ?&#x2018;Ś â&#x2030;¤ 0 for all đ?&#x2018;Ľ, đ?&#x2018;Ś and đ?&#x2018;Ą â&#x2030;Ľ 0. since in (3.6) it was shown that

đ?&#x2018;Ż đ?&#x2019;&#x2122; + đ?&#x153;śđ?&#x2018;Ž đ?&#x2019;&#x161; + đ?&#x153;śđ?&#x2019;&#x2030; đ?&#x2019;&#x2122; đ?&#x2019;&#x161; â&#x2030;Ľ đ?&#x153;šđ?&#x;&#x2019; đ?&#x2018;Ż(đ?&#x2019;&#x2122;) â&#x2030;Ľ đ?&#x;&#x17D;. Thus,

đ?&#x2019;&#x2026; đ?&#x;&#x2018; đ?&#x2018;˝ đ?&#x2019;&#x2022;, đ?&#x2019;&#x2122;đ?&#x2019;&#x2022; , đ?&#x2019;&#x161;đ?&#x2019;&#x2022; , đ?&#x2019;&#x203A;đ?&#x2019;&#x2022; â&#x2030;¤ â&#x2C6;&#x2019; đ?&#x153;šđ?&#x;&#x201C; â&#x2C6;&#x2019; đ?&#x2018;łđ?&#x2019;&#x201E;đ?&#x2019;&#x201C; đ?&#x2019;&#x161;đ?&#x;? â&#x2C6;&#x2019; đ?&#x153;šđ?&#x;&#x201D; â&#x2C6;&#x2019; đ?&#x2018;łđ?&#x153;śđ?&#x2019;&#x201E;đ?&#x2019;&#x201C; đ?&#x2019;&#x203A;đ?&#x;? đ?&#x2019;&#x2026;đ?&#x2019;&#x2022; đ?&#x;? Therefore, if

đ?&#x2019;&#x201C; < đ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;

đ?&#x;?đ?&#x153;šđ?&#x;&#x201C; đ?&#x153;šđ?&#x;&#x201D; , đ?&#x;&#x2018;đ?&#x2018;łđ?&#x2019;&#x201E; đ?&#x2018;łđ?&#x153;śđ?&#x2019;&#x201E;

,

we have đ?&#x2019;&#x2026; đ?&#x2018;˝ đ?&#x2019;&#x2026;đ?&#x2019;&#x2022;

đ?&#x2018;&#x2018;

đ?&#x2019;&#x2022;, đ?&#x2019;&#x2122;đ?&#x2019;&#x2022; , đ?&#x2019;&#x161;đ?&#x2019;&#x2022; , đ?&#x2019;&#x203A;đ?&#x2019;&#x2022; â&#x2030;¤ â&#x2C6;&#x2019;đ?&#x153;ˇ đ?&#x2019;&#x161;đ?&#x;? + đ?&#x2019;&#x203A;đ?&#x;? , for some đ?&#x203A;˝ > 0.

By đ?&#x2018;&#x2030; đ?&#x2018;Ą, đ?&#x2018;Ľđ?&#x2018;Ą , đ?&#x2018;Śđ?&#x2018;Ą , đ?&#x2018;§đ?&#x2018;Ą = 0 and system (3.1) , we can easily obtain :đ?&#x2018;Ľ = đ?&#x2018;Ś = đ?&#x2018;§ = 0. Thus, the condition of lemma đ?&#x2018;&#x2018;đ?&#x2018;Ą 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 đ?&#x2018;&#x2030; đ?&#x2018;Ą, đ?&#x2018;Ľđ?&#x2018;Ą , đ?&#x2018;Śđ?&#x2018;Ą , đ?&#x2018;§đ?&#x2018;Ą as used in the proof of theorem 3.1. It was shown in the proof of theorem 3.1 that there is a đ?&#x2018;&#x2018; positive constant đ?&#x203A;ż7 such that đ?&#x203A;ż7 đ?&#x2018;&#x2039; 2 â&#x2030;¤ đ?&#x2018;&#x2030; đ?&#x2018;Ą, đ?&#x2018;Ľđ?&#x2018;Ą , đ?&#x2018;Śđ?&#x2018;Ą , đ?&#x2018;§đ?&#x2018;Ą , where đ?&#x2018;&#x2039; = (đ?&#x2018;Ľ, đ?&#x2018;Ś, đ?&#x2018;§) and thus that đ?&#x2018;&#x2030; đ?&#x2018;Ą, đ?&#x2018;Ľđ?&#x2018;Ą , đ?&#x2018;Śđ?&#x2018;Ą , đ?&#x2018;§đ?&#x2018;Ą â&#x2020;&#x2019; â&#x2C6;&#x17E; đ?&#x2018;&#x2018;đ?&#x2018;Ą as đ?&#x2018;Ľ 2 + đ?&#x2018;Ś 2 + đ?&#x2018;§ 2 â&#x2020;&#x2019; â&#x2C6;&#x17E;. 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 đ??ž > 0 such that đ?&#x2018;&#x2030; â&#x2030;¤ đ??ž for all đ?&#x2018;Ą, đ?&#x2018;Ľ, đ?&#x2018;Ś and đ?&#x2018;§ their since đ?&#x2018;&#x2030; đ?&#x2018;Ą, đ?&#x2018;Ľđ?&#x2018;Ą , đ?&#x2018;Śđ?&#x2018;Ą , đ?&#x2018;§đ?&#x2018;Ą â&#x2020;&#x2019; â&#x2C6;&#x17E; as đ?&#x2018;Ľ 2 + đ?&#x2018;Ś 2 + đ?&#x2018;§ 2 â&#x2020;&#x2019; â&#x2C6;&#x17E;, thus there exists a đ??ˇ > 0 such that if đ?&#x2018;Ľ = đ?&#x2018;Ľ(đ?&#x2018;Ą) is a solution of (1.2) then đ?&#x2018;Ľ(đ?&#x2018;Ą) â&#x2030;¤ đ??ˇ, đ?&#x2018;Ľ (đ?&#x2018;Ą) â&#x2030;¤ đ??ˇ, đ?&#x2018;Ľ (đ?&#x2018;Ą) â&#x2030;¤ đ??ˇ for all đ?&#x2018;Ą â&#x2030;Ľ 0. Equation (1.2) is equivalent to the system

đ?&#x2019;&#x2122;=đ?&#x2019;&#x161; đ?&#x2019;&#x161;=đ?&#x2019;&#x203A; đ?&#x2019;&#x203A; = â&#x2C6;&#x2019;đ?&#x2019;&#x201A; đ?&#x2019;&#x2022; đ?&#x2019;&#x203A; â&#x2C6;&#x2019; đ?&#x2019;&#x192; đ?&#x2019;&#x2022; đ?&#x2019;&#x2C6;(đ?&#x2019;&#x161;) â&#x2C6;&#x2019; đ?&#x2019;&#x2030; đ?&#x2019;&#x2122; + Along any solution đ?&#x2018;Ľ đ?&#x2018;Ą , đ?&#x2018;Ś đ?&#x2018;Ą , đ?&#x2018;§ đ?&#x2018;Ą

(3.13) đ?&#x2019;&#x2022; đ?&#x2019;&#x2030;â&#x20AC;˛ đ?&#x2019;&#x2022;â&#x2C6;&#x2019;đ?&#x2019;&#x201C;

đ?&#x2019;&#x2122; đ?&#x2019;&#x201D; đ?&#x2019;&#x161; đ?&#x2019;&#x201D; đ?&#x2019;&#x2026;đ?&#x2019;&#x201D; + đ?&#x2019;&#x2018;(đ?&#x2019;&#x2022;)

we have

đ?&#x2018;˝(đ?&#x;&#x2018;.đ?&#x;?đ?&#x;&#x2018;) đ?&#x2019;&#x2022;, đ?&#x2019;&#x2122;đ?&#x2019;&#x2022; , đ?&#x2019;&#x161;đ?&#x2019;&#x2022; , đ?&#x2019;&#x203A;đ?&#x2019;&#x2022; = đ?&#x2018;˝(đ?&#x;&#x2018;.đ?&#x;?) đ?&#x2019;&#x2022;, đ?&#x2019;&#x2122;đ?&#x2019;&#x2022; , đ?&#x2019;&#x161;đ?&#x2019;&#x2022; , đ?&#x2019;&#x203A;đ?&#x2019;&#x2022; + đ?&#x2019;&#x161; + đ?&#x153;śđ?&#x2019;&#x203A; đ?&#x2019;&#x2018; đ?&#x2019;&#x2022; Since đ?&#x2018;&#x2030;(3.1) â&#x2030;¤ 0 for all đ?&#x2018;Ą, đ?&#x2018;Ľ, đ?&#x2018;Ś, đ?&#x2018;§, thus

đ?&#x2018;˝(đ?&#x;&#x2018;.đ?&#x;?đ?&#x;&#x2018;) â&#x2030;¤ đ?&#x2019;&#x161; + đ?&#x153;śđ?&#x2019;&#x203A; đ?&#x2019;&#x2018; đ?&#x2019;&#x2022; â&#x2030;¤ ( đ?&#x2019;&#x161; + đ?&#x153;ś đ?&#x2019;&#x203A; ) đ?&#x2019;&#x2018; đ?&#x2019;&#x2022; â&#x2030;¤ đ?&#x153;šđ?&#x;&#x2013; ( đ?&#x2019;&#x161; + đ?&#x2019;&#x203A; đ?&#x2019;&#x2018;(đ?&#x2019;&#x2022;) Where đ?&#x203A;ż8 =max 1, đ?&#x203A;ź . Noting that đ?&#x2018;Ľ < 1 + đ?&#x2018;Ľ 2 we get

đ?&#x2018;˝(đ?&#x;&#x2018;.đ?&#x;?đ?&#x;&#x2018;) â&#x2030;¤ đ?&#x153;šđ?&#x;&#x2013; (đ?&#x;? + đ?&#x2019;&#x161;đ?&#x;? + đ?&#x2019;&#x203A;đ?&#x;? ) đ?&#x2019;&#x2018;(đ?&#x2019;&#x2022;) â&#x2030;¤ đ?&#x;?đ?&#x153;šđ?&#x;&#x2013; đ?&#x2019;&#x2018;(đ?&#x2019;&#x2022;) + đ?&#x153;šđ?&#x;&#x2013; đ?&#x2018;ż đ?&#x;? đ?&#x2019;&#x2018;(đ?&#x2019;&#x2022;) đ?&#x153;š â&#x2030;¤ đ?&#x;?đ?&#x153;šđ?&#x;&#x2013; đ?&#x2019;&#x2018;(đ?&#x2019;&#x2022;) + đ?&#x;&#x2013; đ?&#x2018;˝ đ?&#x2019;&#x2022;, đ?&#x2019;&#x2122;đ?&#x2019;&#x2022; , đ?&#x2019;&#x161;đ?&#x2019;&#x2022; , đ?&#x2019;&#x203A;đ?&#x2019;&#x2022; đ?&#x2019;&#x2018;(đ?&#x2019;&#x2022;) đ?&#x153;šđ?&#x;&#x2022;

Recalling that đ?&#x203A;ż7 đ?&#x2018;&#x2039; 2 â&#x2030;¤ đ?&#x2018;&#x2030; đ?&#x2018;Ą, đ?&#x2018;Ľđ?&#x2018;Ą , đ?&#x2018;Śđ?&#x2018;Ą , đ?&#x2018;§đ?&#x2018;Ą . đ?&#x203A;ż Let đ?&#x153;&#x201A; = maxâ Ą (2đ?&#x203A;ż8 , 8 ) then đ?&#x203A;ż7

đ?&#x2018;&#x2030;(3.13) â&#x2030;¤ đ?&#x153;&#x201A; đ?&#x2018;?(đ?&#x2018;Ą) + đ?&#x153;&#x201A;đ?&#x2018;&#x2030; đ?&#x2018;?(đ?&#x2018;Ą) đ?&#x2018;&#x2030;(3.13) â&#x2C6;&#x2019; đ?&#x153;&#x201A;đ?&#x2018;&#x2030; đ?&#x2018;?(đ?&#x2018;Ą) â&#x2030;¤ đ?&#x153;&#x201A; đ?&#x2018;?(đ?&#x2018;Ą) . Multiplying each side of this inequality by the integrating factor exp â&#x2C6;&#x2019;đ?&#x153;&#x201A;

đ?&#x2018;Ą 0

đ?&#x2018;?(đ?&#x2018; ) đ?&#x2018;&#x2018;đ?&#x2018;  , we get

đ?&#x2018;˝(đ?&#x;&#x2018;.đ?&#x;?đ?&#x;&#x2018;) exp â&#x2C6;&#x2019;đ?&#x153;ź

đ?&#x2019;&#x2022; đ?&#x;&#x17D;

đ?&#x2019;&#x2018;(đ?&#x2019;&#x201D;) đ?&#x2019;&#x2026;đ?&#x2019;&#x201D; â&#x2C6;&#x2019; đ?&#x153;źđ?&#x2018;˝ đ?&#x2019;&#x2018;(đ?&#x2019;&#x2022;) đ??&#x17E;đ??ąđ??Š â&#x2C6;&#x2019;đ?&#x153;ź

đ?&#x2019;&#x2022; đ?&#x;&#x17D;

đ?&#x2019;&#x2018;(đ?&#x2019;&#x201D;) đ?&#x2019;&#x2026;đ?&#x2019;&#x201D; ,

We get

đ?&#x2018;˝(đ?&#x;&#x2018;.đ?&#x;?đ?&#x;&#x2018;) exp â&#x2C6;&#x2019;đ?&#x153;ź

đ?&#x2019;&#x2022; đ?&#x;&#x17D;

đ?&#x2019;&#x2018;(đ?&#x2019;&#x201D;) đ?&#x2019;&#x2026;đ?&#x2019;&#x201D; â&#x2C6;&#x2019; đ?&#x153;źđ?&#x2018;˝ đ?&#x2019;&#x2018;(đ?&#x2019;&#x2022;) đ??&#x17E;đ??ąđ??Š â&#x2C6;&#x2019;đ?&#x153;ź

đ?&#x2019;&#x2022; đ?&#x;&#x17D;

đ?&#x2019;&#x2018;(đ?&#x2019;&#x201D;) đ?&#x2019;&#x2026;đ?&#x2019;&#x201D; â&#x2030;¤ đ?&#x153;ź đ?&#x2019;&#x2018;(đ?&#x2019;&#x2022;) đ??&#x17E;đ??ąđ??Š â&#x2C6;&#x2019;đ?&#x153;ź

đ?&#x2019;&#x2022; đ?&#x;&#x17D;

đ?&#x2019;&#x2018;(đ?&#x2019;&#x201D;) đ?&#x2019;&#x2026;đ?&#x2019;&#x201D;

Integrating each side of this inequality from 0 to đ?&#x2018;Ą, we get, where 0 = đ?&#x2018;Ľ 0 , đ?&#x2018;Ś 0 , đ?&#x2018;§ 0 , đ?&#x2019;&#x2022;

đ?&#x2018;˝đ??&#x17E;đ??ąđ??Š â&#x2C6;&#x2019;đ?&#x153;ź

đ?&#x2019;&#x2022;

đ?&#x2019;&#x2018;(đ?&#x2019;&#x201D;) đ?&#x2019;&#x2026;đ?&#x2019;&#x201D; â&#x2C6;&#x2019; đ?&#x2018;˝ đ?&#x;&#x17D; â&#x2030;¤ đ?&#x;? â&#x2C6;&#x2019; đ??&#x17E;đ??ąđ??Š â&#x2C6;&#x2019;đ?&#x153;ź đ?&#x;&#x17D;

đ?&#x2019;&#x2018;(đ?&#x2019;&#x201D;) đ?&#x2019;&#x2026;đ?&#x2019;&#x201D; đ?&#x;&#x17D;

Or đ?&#x2019;&#x2022;

đ?&#x2018;˝ â&#x2030;¤ đ?&#x2018;˝ đ?&#x;&#x17D; đ??&#x17E;đ??ąđ??Š đ?&#x153;ź Since

đ?&#x2018;Ą 0

đ?&#x2019;&#x2022;

đ?&#x2019;&#x2018; đ?&#x2019;&#x201D; đ?&#x2019;&#x2026;đ?&#x2019;&#x201D; + đ??&#x17E;đ??ąđ??Š đ?&#x153;ź đ?&#x;&#x17D;

đ?&#x2019;&#x2018;(đ?&#x2019;&#x201D;) đ?&#x2019;&#x2026;đ?&#x2019;&#x201D; â&#x2C6;&#x2019; đ?&#x;? đ?&#x;&#x17D;

đ?&#x2018;?(đ?&#x2018; ) đ?&#x2018;&#x2018;đ?&#x2018;  â&#x2030;¤ đ??´ for all t, this implies www.iosrjournals.org

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Stability and Boundedness of Solutions of Delay Differential Equations of third order đ?&#x2018;˝ đ?&#x2019;&#x2022;, đ?&#x2019;&#x2122;đ?&#x2019;&#x2022; , đ?&#x2019;&#x161;đ?&#x2019;&#x2022; , đ?&#x2019;&#x203A;đ?&#x2019;&#x2022; â&#x2030;¤ đ?&#x2018;˝ đ?&#x;&#x17D; đ?&#x2019;&#x2020;đ?&#x153;źđ?&#x2018;¨ + [đ?&#x2019;&#x2020;đ?&#x153;źđ?&#x2018;¨ â&#x2C6;&#x2019; đ?&#x;?] for đ?&#x2019;&#x2022; â&#x2030;Ľ đ?&#x;&#x17D; Now, since the right-hand side is a constant, and since đ?&#x2018;&#x2030; đ?&#x2018;Ą, đ?&#x2018;Ľđ?&#x2018;Ą , đ?&#x2018;Śđ?&#x2018;Ą , đ?&#x2018;§đ?&#x2018;Ą â&#x2020;&#x2019; â&#x2C6;&#x17E; as đ?&#x2018;Ľ 2 + đ?&#x2018;Ś 2 + đ?&#x2018;§ 2 â&#x2020;&#x2019; â&#x2C6;&#x17E;, it follows that there exists a đ??ˇ > 0 such that đ?&#x2018;Ľ(đ?&#x2018;Ą) â&#x2030;¤ đ??ˇ, đ?&#x2018;Ś(đ?&#x2018;Ą) â&#x2030;¤ đ??ˇ đ?&#x2018;§(đ?&#x2018;Ą) â&#x2030;¤ đ??ˇ for đ?&#x2018;Ą â&#x2030;Ľ 0. From the system (3.13) this implies that đ?&#x2018;Ľ(đ?&#x2018;Ą) â&#x2030;¤ đ??ˇ, đ?&#x2018;Ľ(đ?&#x2018;Ą) â&#x2030;¤ đ??ˇ, đ?&#x2018;Ľ (đ?&#x2018;Ą) â&#x2030;¤ đ??ˇ for đ?&#x2018;Ą â&#x2030;Ľ 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 đ?&#x2018;&#x17D;(đ?&#x2018;Ą), đ?&#x2018;?(đ?&#x2018;Ą) 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 nonautonomous equation tend to the zero solution as đ?&#x2018;Ą â&#x2020;&#x2019; â&#x2C6;&#x17E;. These results are now extended by considering the semi-invariant set of a related non-autonomous system.

References: [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

Antoslewicz, H.A. On nonlinear differential equations of the second order with integrable forcing term, J. London Math. Soc. 30 (1955), 64-67. G. Makay. On the asymptotic stability of the solutions of functional differential equations with infinite delay. J. Diff. Eqs.,108:(1994),139-151. J. K. Hale. Theory of Functional Differential Equations. Springer-Verlag, New York. (1977) Omeike, M.O. Further results on global stability of third-order nonlinear differential equations, Nonlinear Analysis 67 (2007), 3394-3400. Sadek, A.I. Stability and boundedness of a kind of third-order delay differential system, Applied Mathematics Letters 16 (2003), 657662. 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. S. Li, L. Wen.Functional Differential Equations. Human Science and Technology Press. (1987) 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. T. A. Burton. Volterra Integral and Differential Equations. Academic Press New York. (1983) T. A. Burton. Stability and Periodic Solutions of Ordinary and Functional Differential Equations. Academic Press.(1985) Tunc. C. Boundedness of solutions of a third-order nonlinear differential equation, JIPAM. J. Inequal. Pure Appl. Math. 6 (2005), 16. V. Kelmanovskii, A. Myshkis. Introduction to the Theory and Applications of Functional Differential Equations. Kluwer Academic,Dordrecht. (1999) 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.

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