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Abstract and Applied Analysis

Volume 2013 (2013), Article ID 248717, 7 pages

http://dx.doi.org/10.1155/2013/248717

## Stability and Uniform Boundedness in Multidelay Functional Differential Equations of Third Order

Department of Mathematics, Faculty of Sciences, Yüzüncü Yıl University, 65080 Van, Turkey

Received 11 February 2013; Accepted 18 April 2013

Academic Editor: Tonghua Zhang

Copyright © 2013 Cemil Tunç and Melek Gözen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider a nonautonomous functional differential equation of the third order with multiple deviating arguments. Using the Lyapunov-Krasovskiì functional approach, we give certain sufficient conditions to guarantee the asymptotic stability and uniform boundedness of the solutions.

#### 1. Introduction

Differential equations of third order are valuable tools in the modeling of many phenomena in various fields of science and engineering (Chlouverakis and Sprott [1], Cronin-Scanlon [2], Eichhorn et al. [3], Friedrichs [4], Linz [5], and Rauch [6]). In reality, the stability and boundedness of solutions of certain nonlinear differential equations of the third order have been received intensive attentions by authors (Ademola et al. [7], Afuwape and Castellanos [8], Chukwu [9], Ezeilo ([10, 11]), Hara [12], Mehri and Shadman [13], Ogundare and Okecha [14], Omeike [15], Reissig et al. [16], Swick [17], Tejumola ([18, 19]), Tunç [20–33], and Yoshizawa [34]).

In 2009, Omeike [15] considered the nonlinear differential equation of the third order with the constant delay : and he discussed the stability and boundedness of solutions of this equation.

In this paper, instead of the above equation, we consider the nonautonomous differential equation of the third order with multiple deviating arguments: where are certain positive constants, and and are real valued and continuous functions in their respective arguments with . The existence and uniqueness of the solutions of (2) are also assumed.

The motivation for this paper is a result of the researches mentioned regarding ordinary differential equations of the third order. It follows that the equation discussed in [15] is a special case of (2). Our aim is to improve the results established in [15] from one deviating argument to the multiple deviating arguments for the asymptotic stability and uniform boundedness of solutions. This work contributes to and complements previously known results on the topic in the literature, and it may be useful for researchers working on the qualitative behaviors of solutions. It should be noted that in recent years scores of papers have been published on the qualitative behaviors of solutions (stability of solutions, boundedness of the solutions, existence of the periodic solutions, etc.) of the functional differential equations of the second order with multiple deviating arguments. However, very little attention was given to stability and boundedness of functional differential equations of the third order with multiple deviating arguments ([32]). Therefore, it is worth investigating the qualitative behaviors of solutions in multidelay functional differential equations of the third order. This case is the novelty of the present paper. It should also be noted that the results to be established here are different from those in Tunç [20–33] and the literature.

#### 2. Main Results

Let .

Theorem 1. *One assumes that there exist positive constants , and such that the following conditions hold:*(i)*,
* *,
*(ii)*. *

If

then every solution of (2) is uniform bounded and satisfies

*Remark 2. *It should be noted that it follows from (ii) that and are nonincreasing functions on . Therefore, since these functions are continuous on this interval and bounded below by , they are bounded on and the limit of each exists as . Since in (ii) is an arbitrary selected bound, we can also assume the following estimates:

*Proof. *We write (2) in the system form as follows:

Define a Lyapunov-Krasovskiì functional ([35]) by
where and are certain positive constants, which will be determined later in the proof.

This functional can be arranged as follows:
where

Using the assumptions of Theorem 1, it follows that since and .

Thus, there exist constants and such that
since . Further, using the assumptions of Theorem 1 and , it follows that
so that
where

In view of the previous discussion, we can get

Using a basic calculation, the time derivative of along solutions of (6) results in

Using , and the estimate , we have
where

Noting the previous discussion, it follows that

If , then

If , then it follows that

Since , and , then
where and .

Thus, we get

Let . Hence,

If , then since and . For those such that , we have

Thus,

Therefore, if
then we have

The proof for Theorem 1 is complete.

Let .

Theorem 3. *One assumes that all the assumptions of Theorem 1 and the assumption
**
hold. If
**
then all solutions of (2) are bounded. *

*Proof. *Equation (2) is equivalent to the system

Along any solution of (6), we have

Since , then it follows that
where . Noting that , we get
recalling that .

Let , then
or

Multiplying each side of this estimate by the integrating factor , we get

Integrating each side of this estimate from 0 to *t*, we obtain
or
where .

Since for all , this implies

Since the right-hand side of the last estimate is a constant and when , it follows that there exists a positive constant such that

From the system (30) this implies that

The proof for Theorem 3 is complete.

#### References

- K. E. Chlouverakis and J. C. Sprott, “Chaotic hyperjerk systems,”
*Chaos, Solitons & Fractals*, vol. 28, no. 3, pp. 739–746, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Cronin-Scanlon, “Some mathematics of biological oscillations,”
*SIAM Review*, vol. 19, no. 1, pp. 100–138, 1977. View at Publisher · View at Google Scholar · View at MathSciNet - R. Eichhorn, S. J. Linz, and P. Hänggi, “Transformations of nonlinear dynamical systems to jerky motion and its application to minimal chaotic flows,”
*Physical Review E*, vol. 58, no. 6, pp. 7151–7164, 1998. View at Publisher · View at Google Scholar · View at MathSciNet - K. O. Friedrichs, “On nonlinear vibrations of third order,” in
*Studies in Nonlinear Vibration Theory*, pp. 65–103, Institute for Mathematics and Mechanics, New York University, 1946. View at Google Scholar · View at MathSciNet - S. J. Linz, “On hyperjerky systems,”
*Chaos, Solitons & Fractals*, vol. 37, no. 3, pp. 741–747, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. L. Rauch, “Oscillation of a third order nonlinear autonomous system,” in
*Contributions to the Theory of Nonlinear Oscillations*, Annals of Mathematics Studies, no. 20, pp. 39–88, Princeton University Press, Princeton, NJ, USA, 1950. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. A. Ademola, M. O. Ogundiran, P. O. Arawomo, and O. A. Adesina, “Boundedness results for a certain third order nonlinear differential equation,”
*Applied Mathematics and Computation*, vol. 216, no. 10, pp. 3044–3049, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. U. Afuwape and J. E. Castellanos, “Asymptotic and exponential stability of certain third-order non-linear delayed differential equations: frequency domain method,”
*Applied Mathematics and Computation*, vol. 216, no. 3, pp. 940–950, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. N. Chukwu, “On the boundedness and the existence of a periodic solution of some nonlinear third order delay differential equation,”
*Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali 8*, vol. 64, no. 5, pp. 440–447, 1978. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. O. C. Ezeilo, “A stability result for a certain third order differential equation,”
*Annali di Matematica Pura ed Applicata 4*, vol. 72, pp. 1–9, 1966. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. O. C. Ezeilo and H. O. Tejumola, “Boundedness theorems for certain third order differential equations,”
*Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali 8*, vol. 55, pp. 194–201, 1973/1974. View at Google Scholar · View at MathSciNet - T. Hara, “On the uniform ultimate boundedness of the solutions of certain third order differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 80, no. 2, pp. 533–544, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Mehri and D. Shadman, “Boundedness of solutions of certain third order differential equation,”
*Mathematical Inequalities & Applications*, vol. 2, no. 4, pp. 545–549, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. S. Ogundare and G. E. Okecha, “On the boundedness and the stability of solution to third order non-linear differential equations,”
*Annals of Differential Equations*, vol. 24, no. 1, pp. 1–8, 2008. View at Google Scholar · View at MathSciNet - M. O. Omeike, “Stability and boundedness of solutions of some non-autonomous delay differential equation of the third order,”
*Analele Ştiinţifice ale Universităţii Al. I. Cuza din Iaşi. Serie Nouă. Matematică*, vol. 55, supplement 1, pp. 49–58, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Reissig, G. Sansone, and R. Conti,
*Non-Linear Differential Equations of Higher Order*, Noordhoff International, Leyden, The Netherlands, 1974. View at MathSciNet - K. E. Swick, “Asymptotic behavior of the solutions of certain third order differential equations,”
*SIAM Journal on Applied Mathematics*, vol. 19, pp. 96–102, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. O. Tejumola, “On the boundedness and periodicity of solutions of certain third-order non-linear differential equations,”
*Annali di Matematica Pura ed Applicata 4*, vol. 83, pp. 195–212, 1969. View at Publisher · View at Google Scholar · View at MathSciNet - H. O. Tejumola, “A note on the boundedness and the stability of solutions of certain third-order differential equations,”
*Annali di Matematica Pura ed Applicata 4*, vol. 92, pp. 65–75, 1972. View at Publisher · View at Google Scholar · View at MathSciNet - C. Tunç, “Uniform ultimate boundedness of the solutions of third-order nonlinear differential equations,”
*Kuwait Journal of Science & Engineering*, vol. 32, no. 1, pp. 39–48, 2005. View at Google Scholar · View at MathSciNet - C. Tunç, “Boundedness of solutions of a third-order nonlinear differential equation,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 6, no. 1, article 3, 6 pages, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Tunç, “On the asymptotic behavior of solutions of certain third-order nonlinear differential equations,”
*Journal of Applied Mathematics and Stochastic Analysis*, no. 1, pp. 29–35, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Tunç, “Stability criteria for certain third order nonlinear delay differential equations,”
*Portugaliae Mathematica*, vol. 66, no. 1, pp. 71–80, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Tunç, “A new result on the stability of solutions of a nonlinear differential equation of third-order with finite lag,”
*Southeast Asian Bulletin of Mathematics*, vol. 33, no. 5, pp. 947–958, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Tunç, “On the stability and boundedness of solutions to third order nonlinear differential equations with retarded argument,”
*Nonlinear Dynamics*, vol. 57, no. 1-2, pp. 97–106, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Tunç, “The boundedness of solutions to nonlinear third order differential equations,”
*Nonlinear Dynamics and Systems Theory*, vol. 10, no. 1, pp. 97–102, 2010. View at Google Scholar · View at MathSciNet - C. Tunç, “On the stability and boundedness of solutions of nonlinear third order differential equations with delay,”
*Filomat*, vol. 24, no. 3, pp. 1–10, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - C. Tunç, “Some stability and boundedness conditions for non-autonomous differential equations with deviating arguments,”
*Electronic Journal of Qualitative Theory of Differential Equations*, no. 1, 12 pages, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Tunç, “Bound of solutions to third-order nonlinear differential equations with bounded delay,”
*Journal of the Franklin Institute*, vol. 347, no. 2, pp. 415–425, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Tunç, “Stability and bounded of solutions to non-autonomous delay differential equations of third order,”
*Nonlinear Dynamics*, vol. 62, no. 4, pp. 945–953, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Tunç, “Existence of periodic solutions to nonlinear differential equations of third order with multiple deviating arguments,”
*International Journal of Differential Equations*, vol. 2012, Article ID 406835, 13 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Tunç, “Qualitative behaviors of functional differential equations of third order with multiple deviating arguments,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 392386, 12 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - C. Tunç, “On the qualitative behaviors of solutions of some differential equations of higher order with multiple deviating arguments,”
*Journal of the Franklin Institute*. In press. - T. Yoshizawa,
*Stability Theory by Liapunov's Second Method*, Publications of the Mathematical Society of Japan, No. 9, The Mathematical Society of Japan, Tokyo, Japan, 1966. View at MathSciNet - N. N. Krasovskiì,
*Stability of Motion. Applications of Lyapunov's Second Method to Differential Systems and Equations with Delay*, Stanford University Press, Stanford, Calif, USA, 1963. View at MathSciNet