Journal of Applied Mathematics

Volume 2013 (2013), Article ID 103260, 4 pages

http://dx.doi.org/10.1155/2013/103260

Research Article

On the Global Stability Properties and Boundedness Results of Solutions of Third-Order Nonlinear Differential Equations

Muzaffer Ateş

Department of Electrical and Electronics Engineering, Yüzüncü Yıl University, 65080 Van, Turkey

Received 7 June 2013; Revised 12 September 2013; Accepted 28 September 2013

Academic Editor: Hector Pomares

Copyright © 2013 Muzaffer Ateş. 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 studied the global stability and boundedness results of third-order nonlinear differential equations of the form . Particular cases of this equation have been studied by many authors over years. However, this particular form is a generalization of the earlier ones. A Lyapunov function was used for the proofs of the two main theorems: one with and the other with . The results in this paper generalize those of other authors who have studied particular cases of the differential equations. Finally, a concrete example is given to check our results.

1. Introduction

We will consider here the equations of the form

Now, (1) has an equivalent system

where , , and . We also assume that the real functions , , and depend only on the arguments displayed explicitly. The dots denote differentiation with respect to .

Global stabilities of some special cases of (1) have been studied by a number of authors.

In 1953, Šimanov [1] investigated the global stability of the zero solution of the equation

where and are constants.

Later, Ezeilo [2] and Ogurtsov [3] discussed the global stability of the zero solution of the equation of the form

Then, Goldwyn and Narendra [4] studied on the same subject for the following differential equation:

Recently, Qian [5] and Omeike [6] have discussed the global stability, and in a recent paper, Tunç [7] has investigated the boundedness of solutions of the following differential equations:

Plus, Tunç [8] and Omeike [9, 10] have studies on the global stability of solutions of the differential equation of the form

Moreover, Tunç and Omeike have studies on the asymptotic behavior of the following differential equations:

respectively.

Motivation of this study has been based on recent studies of Qian [5], Tunç [7, 8, 11], and Omeike [6, 9, 10]. Equation (1) is a quite general third-order nonlinear differential equation. Equations (3a), (3b), (3c), (3d), (3e), (3f), and (3g) are some special cases of (1), and our study is reducible to the studies in [1–11], but the inversions are not possible. Thus, the studies which have been done in [1–11] are some special cases of our study. Hence, our results extend and include those results obtained in [1–11].

2. Preliminaries

Before introducing our main results, we state some basic theorems and a Lyapunov function which will be required in future. Consider the autonomous system

where is continuous, with being an open set in containing the origin. Let and for .

Theorem 1. Suppose that there exists a scalar function such that

(i) is positive definite on , and as .(ii) on .

Then, all solutions of (4) are bounded as (i.e., (4) is Lagrange stable).

Proof. See [12].

Theorem 2. In addition to the assumptions of Theorem 1, if the origin is the only invariant subset of , then the zero solution of (4) is globally asymptotically stable.

Proof. is the set where , and assume that is to be the only invariant subset of ; then solution of (4) is globally asymptotically stable.

Theorem 3. In Theorem 1, if (ii) is replaced by the condition that is negative definite at all points , then the zero solution of (4) is globally asymptotically stable.

Proof. See [12].

It is well known that the stability is a very important problem in the theory and applications of differential equations. So far, the most effective method to study the stability of nonlinear differential equations is still Lyapunov’s second method. The major advantage of this method is that stability in large can be obtained without any prior knowledge of solutions. Today, this method is widely recognized as an excellent tool not only in the study of differential equations but also in the theory of control systems, dynamical systems, systems with time lag, power system analysis, time-varying nonlinear feedback systems, and so on. Its chief characteristic is the construction of a scalar function, namely, the Lyapunov function. Unfortunately, it is sometimes very difficult to find a proper Lyapunov function for a given system. Therefore, in this work, we construct a suitable Lyapunov function which is an excellent tool in the proof of the main theorems. Here, this function, , is defined by

Rewrite the function as follows:

where

3. Main Result

In the case , we have the following.

Theorem 4. Let , , , and be positive constants such that is sufficiently small, and , and assume that the following conditions are satisfied:

(i),(ii),(iii) for all , , and ,(iv) for all , , and ,(v), for all , and ,(vi).

Then, the zero solution of (1) is globally asymptotically stable.

Proof. From conditions (i)–(iv) of Theorem 4, we obtain

where . It follows that there exists a constant small enough that

Hence, is a positive definite function (see Global Asymptotic Stability on page 223, Theorem 5.2.12 that of Rao [12]).

Now, we show that the derivative of with respect to along the solution path of system (2) is negative semidefinite.

Let

where

First, from condition (v) of Theorem 4, we obtain

Next, from conditions (iii), (v), and (vi) of Theorem 4 we obtain that .

Hence

In addition, we can conclude that as (see Global Asymptotic Stability on page 223, Theorem 5.2.12 of Rao [12]).

The whole discussions (conditions of Theorems 1, 2, 3, and 4) show that the zero solution of system (2) is globally asymptotically stable (also see Theorem 1.5 that of Reissing et al. [13]). Then, the rest of the proof may now follow as in [2]. Thus, the proof of Theorem 4 is completed.

In the case , we have the following.

Theorem 5. Suppose that the following conditions are satisfied:

(i) all the conditions of Theorem 4 hold;(ii), where , is a space of integrable Lebesgue functions.

Then, there is a finite positive constant such that every solution of system (2) satisfies

Proof. Consider the function defined as above. Since , then the total derivative of can be revised as

Let . Then, we have

Using the inequality

we obtain

From (9), we have

where

Integrating (19) from to , we obtain

Setting

then, we have

Using Gronwall-Bellman inequality (see Rao [12]) yields

The proof of Theorem 5 is complete.

4. Example

Consider the equation

Equation (25) is in the form of (1), where

Case 1 (). With , , , from condition (vi) of Theorem 4, we have

Hence, condition (vi) of Theorem 4 is satisfied.

Then, it is easy to check that all the other conditions [(i)–(vi)] of Theorem 4 are satisfied. Hence, the trivial solution of (25) is globally asymptotically stable.

Case 2 (). Assume that , , , , are the same as in Case 1 of the example, and

and that

Then, all the conditions of Theorem 5 are satisfied. Hence, the solutions of (25) are bounded.

References

S. N. Šimanov, “On stability of solution of a nonlinear equation of the third order,” Akademii Nauk SSSR, vol. 17, pp. 369–372, 1953 (Russian).

J. O. C. Ezeilo, “On the stability of solutions of certain differential equations of the third order,” The Quarterly Journal of Mathematics, vol. 11, pp. 64–69, 1960.

A. I. Ogurtsov, “On the stability of the solutions of some non-linear differential equations of third and fourth order,” Izvestiya Vysshikh Uchebnykh Zavedenii, vol. 10, pp. 2000–2009, 1959 (Russian).

M. Goldwyn and S. Narendra, Stability of Certain Nonlinear Differential Equations Using the Second Method of Lyapunov, Craft Laboratory, Harvard University, Cambridge, Mass, USA, 1963.

C. Qian, “On global stability of third-order nonlinear differential equations,” Nonlinear Analysis: Theory, Methods and Applications, vol. 42, no. 4, pp. 651–661, 2000.

M. Omeike, “New results on the asymptotic behavior of a third-order nonlinear differential equation,” Differential Equations & Applications, vol. 2, no. 1, pp. 39–51, 2010.

C. Tunç, “On the asymptotic behavior of solutions of certain third-order nonlinear differential equations,” Journal of Applied Mathematics and Stochastic Analysis, vol. 1, pp. 29–35, 2005.

C. Tunç, “Global stability of solutions of certain third-order nonlinear differential equations,” Panamerican Mathematical Journal, vol. 14, no. 4, pp. 31–35, 2004.

M. O. Omeike, “Further results on global stability of third-order nonlinear differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 12, pp. 3394–3400, 2007.

M. O. Omeike, “Further results on global stability of solutions of certain third-order nonlinear differential equations,” Acta Universitatis Palackianae Olomucensis, vol. 47, pp. 121–127, 2008.

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.

M. R. M. Rao, Ordinary Differential Equations, Affiliated East-West Press, New Delhi, India, 1980.

R. Reissing, G. Sansone, and R. Conti, Nonlinear Differential Equations of Higher Order, Noordhoff International Publishing, Leyden, The Netherlands, 1974.