Research Article | Open Access

Ercan Tunç, "On the Convergence of Solutions of Certain Third-Order Differential Equations", *Discrete Dynamics in Nature and Society*, vol. 2009, Article ID 863178, 12 pages, 2009. https://doi.org/10.1155/2009/863178

# On the Convergence of Solutions of Certain Third-Order Differential Equations

**Academic Editor:**Leonid Shaikhet

#### Abstract

We establish sufficient conditions for the convergence of solutions of a certain third-order nonlinear differential equations. By constructing a Lyapunov function as the basic tool, some results which exist in the relevant literature are generalized.

#### 1. Introduction

As well known, in the investigation of qualitative behaviors of solutions, stability, convergence, boundedness, oscillation, and so forth of solutions are very important problems in theory and applications of differential equations. For example, in applied sciences, some practical problems concerning mechanics, the engineering technique fields, economy, control theory, physics, chemistry, biology, medicine, atomic energy, information theory, and so forth are associated with certain higher-order linear or nonlinear differential equations. Ever since Lyapunov [1] proposed his famous theory on the stability of motion, For some papers published on the qualitative behaviors of solutions of nonlinear second-and third-order differential equations, the readers can referee to the papers of Afuwape and Omeike [2, 3], Ezeilo [4, 5], Meng [6], Tejumola [7, 8], Tunç [9–11], Omeike [12], and the references listed in these papers as well as one can refer to the books of Reissig et al. [13, 14]. The motivation for the present work has been inspired basically by the paper of Afuwape and Omeike [2] and the papers listed above. Our aim here is to extend the results established by Afuwape and Omeike [2] to nonlinear differential equation (1.4) for the convergence of all solutions of this equation. In 2008, Afuwape and Omeike [2] considered third-order nonlinear differential equations of the form and by introducing a Lyapunov function they discussed the convergence of solutions for this equation. During establishment of the results, Afuwape and Omeike [2] defined the following relations with respect to the functions and : for any pair of constants and for any pair of constants , where is a positive constant.

In this paper, we consider nonlinear differential equation of the form where the functions and are continuous in their respective arguments, with the functions , and are not necessarily differentiable. In addition to (1.2) and (1.3) we assume that for any pair of constants .

By convergence of solutions we mean, any two solutions of (1.4) are said to converge to each other if as .

#### 2. Main Results

The following results are established.

Theorem 2.1. *Suppose that , and that*(i)*there are constants such that satisfies inequalities (1.5),*(ii)*there are constants such that satisfies inequalities (1.2),*(iii)*there are constants , such that for any , the incrementary ratio for satisfies
with *(iv)*there is a continuous function such that
**holds for arbitrary , and satisfies
**
for some constant , where is a constant in the range .**Then all solutions of (1.4) converge.*

A very important step in the proof of Theorem 2.1 will be to give estimate for any two solutions of (1.4). This in itself, being of independent interest, is giving as follows.

Theorem 2.2. *Let be any two solutions of (1.4). Suppose that all the conditions of Theorem 2.1 are satisfied, then for each fixed , in the range , there exist constants and such that for ,
**
where
*

We have the following corollaries when and

Corollary 2.3. *Suppose that in (1.4) and suppose further that conditions (i), (ii), and (iii) of Theorem 2.1 hold, then the trivial solution of (1.4) is exponentially stable in the large.*

Also, if we put in (2.1) with arbitrary, we get the following.

Corollary 2.4. *If and hypotheses (i), (ii), and (iii) of Theorem 2.1 hold for arbitrary , and , then there exists a constant such that every solution of (1.4) satisfies
*

#### 3. Preliminary Results

On setting (1.4) can be replaced by an equivalent system Let , , be any two solutions of (3.1) such that where and are finite constants, and will be determined later.

Our investigation rests mainly on the properties of the function, defined by where and are constants.

Following the argument used in [5], we can easily verify the following for

Lemma 3.1. * (i)**(ii) There exist finite positive constants such that
**
where
**
and using the inequality *

If we define the function by and using the fact that the solutions ,, satisfy (3.1), then as defined in (2.5) becomes

Lemma 3.2. *Assume that the conditions (i), (ii), and (iii) of Theorem 2.1 are satisfied. Then, there exist positive finite constants and such that
**
where *

*Proof of Lemma 3.2*

Differentiating the function in (3.3) along the system (3.1) we obtain
in which
with
and and are strictly positive constants such that
Also, let us denote and simply by and , respectively. For strictly positive constants and conveniently chosen later, we get
Thus,
Moreover, in view of (3.2), we obtain for all in
if
and for all in
if
Combining all the inequalities in (3.16) and (3.18), we have for all in
if
Also, for all in
if
for all in
if
for all in
if
and for all in
if
Further
where , on the other hand
where

Bringing together the estimates just obtained for and in (3.10) and using (3.8), we have
This completes the proof of Lemma 3.2.

#### 4. Proof of Theorem 2.2

This follows directly from [5], on using inequality (3.32). Let be any constant in the range Set , so that We rewrite (3.32) in the form where with , considering the two cases

(i) and(ii)separately. If , then . On the other hand, if then the definition of in (4.2) gives at least and also This implies that Therefore from which together with we obtain where Again due to (4.1) and using the estimate on for , we have where , which follows from In view of the fact that we obtain and on using inequality (3.4), we have for some positive constants and . On integrating (4.10) from to , we have Again, using Lemma 3.1, we obtain (2.4), with , and This completes the proof of Theorem 2.2.

#### 5. Proof of Theorem 2.1

This follows from the estimate (2.4) and the condition (2.3) on Choose in (2.3). From the estimate (2.4), if then the exponential index remains negative for all Then, as we have , and this gives as . This completes the proof of Theorem 2.1.

#### Acknowledgment

The author would like to express sincere thanks to the anonymous referees for their invaluable corrections, comments, and suggestions.

#### References

- A. M. Liapunov,
*Stability of Motion*, Academic Press, New York, NY, USA, 1966. View at: MathSciNet - A. U. Afuwape and M. O. Omeike, “Convergence of solutions of certain non-homogeneous third order ordinary differential equations,”
*Kragujevac Journal of Mathematics*, vol. 31, pp. 5–16, 2008. View at: Google Scholar | MathSciNet - A. U. Afuwape and M. O. Omeike, “Further ultimate boundedness of solutions of some system of third order nonlinear ordinary differential equations,”
*Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica*, vol. 43, pp. 7–20, 2004. View at: Google Scholar | Zentralblatt MATH | MathSciNet - J. O. C. Ezeilo, “A note on the convergence of solutions of certain second order differential equations,”
*Portugaliae Mathematica*, vol. 24, pp. 49–58, 1965. View at: Google Scholar | Zentralblatt MATH | MathSciNet - J. O. C. Ezeilo, “New properties of the equation $\stackrel{\u20db}{x}+a\ddot{x}+b\dot{x}+h(x)=P(t,x,\dot{x},\ddot{x})$ for certain special values of the incrementary ratio ${y}^{-1}\{h(x+y)-h(x)\}$,” in
*Équations différentielles et fonctionnelles non linéaires (Actes Conférence Internat. “Equa-Diff 73”, Brussels/Louvain-la-Neuve, 1973)*, P. Janssens, J. Mawhin, and N. Rouche, Eds., pp. 447–462, Hermann, Paris, France, 1973. View at: Google Scholar | Zentralblatt MATH | MathSciNet - F. W. Meng, “Ultimate boundedness results for a certain system of third order nonlinear differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 177, no. 2, pp. 496–509, 1993. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. O. Tejumola, “On the convergence of solutions of certain third-order differential equations,”
*Annali di Matematica Pura ed Applicata*, vol. 78, no. 1, pp. 377–386, 1968. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. O. Tejumola, “Convergence of solutions of certain ordinary third order differential equations,”
*Annali di Matematica Pura ed Applicata*, vol. 94, no. 1, pp. 247–256, 1972. View at: Publisher Site | Google Scholar | Zentralblatt MATH | 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, pp. 1–6, 2005. View at: Google Scholar | Zentralblatt MATH | MathSciNet - C. Tunç and E. Tunç, “New ultimate boundedness and periodicity results for certain third-order nonlinear vector differential equations,”
*Mathematical Journal of Okayama University*, vol. 48, pp. 159–172, 2006. View at: Google Scholar | Zentralblatt MATH | MathSciNet - C. Tunç and E. Tunç, “On the asymptotic behavior of solutions of certain second-order differential equations,”
*Journal of the Franklin Institute*, vol. 344, no. 5, pp. 391–398, 2007. View at: Google Scholar | MathSciNet - M. O. Omeike, “New result in the ultimate boundedness of solutions of a third-order nonlinear ordinary differential equation,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 9, no. 1, article 15, pp. 1–8, 2008. View at: Google Scholar | MathSciNet - R. Reissig, G. Sansone, and R. Conti,
*Non-Linear Differential Equations of Higher Order*, Noordhoff International, Leyden, The Netherlands, 1974. View at: Zentralblatt MATH | MathSciNet - 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

#### Copyright

Copyright © 2009 Ercan Tunç. 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.