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.

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 (1.1) 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 (1.2) for any pair of constants and (1.3) for any pair of constants , where is a positive constant.

In this paper, we consider nonlinear differential equation of the form (1.4) 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 (1.5) 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 (1.6) 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 (2.1) with (iv) there is a continuous function such that (2.2) holds for arbitrary , and satisfies (2.3) 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 (2.4) where (2.5)

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 (2.6)

3. Preliminary Results

On setting (1.4) can be replaced by an equivalent system (3.1) Let , be any two solutions of (3.1) such that (3.2) where and are finite constants, and will be determined later.

Our investigation rests mainly on the properties of the function, defined by (3.3) 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 (3.4) where (3.5) and using the inequality (3.6)

If we define the function by (3.7) and using the fact that the solutions ,, satisfy (3.1), then as defined in (2.5) becomes (3.8)

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 (3.9) where

Proof of Lemma 3.2

Differentiating the function in (3.3) along the system (3.1) we obtain (3.10) in which (3.11) with (3.12) and and are strictly positive constants such that (3.13) Also, let us denote and simply by and , respectively. For strictly positive constants and conveniently chosen later, we get (3.14) Thus, (3.15) Moreover, in view of (3.2), we obtain for all in (3.16) if (3.17) and for all in (3.18) if (3.19) Combining all the inequalities in (3.16) and (3.18), we have for all in (3.20) if (3.21) Also, for all in (3.22) if (3.23) for all in (3.24) if (3.25) for all in (3.26) if (3.27) and for all in (3.28) if (3.29) Further (3.30) where , on the other hand (3.31) where

Bringing together the estimates just obtained for and in (3.10) and using (3.8), we have (3.32) 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 (4.1) where (4.2) 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 (4.3) and also This implies that (4.4) Therefore (4.5) from which together with we obtain (4.6) where Again due to (4.1) and using the estimate on for , we have (4.7) where , which follows from (4.8) In view of the fact that we obtain (4.9) and on using inequality (3.4), we have (4.10) for some positive constants and . On integrating (4.10) from to , we have (4.11) 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 (5.1) then the exponential index remains negative for all Then, as we have , and this gives (5.2) 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.

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