On the Convergence of Solutions of Certain Third-Order Differential Equations
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.
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  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 , Tejumola [7, 8], Tunç [9–11], Omeike , 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  and the papers listed above. Our aim here is to extend the results established by Afuwape and Omeike  to nonlinear differential equation (1.4) for the convergence of all solutions of this equation. In 2008, Afuwape and Omeike  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  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.
We have the following corollaries when and
Also, if we put in (2.1) with arbitrary, we get the following.
3. Preliminary Results
Our investigation rests mainly on the properties of the function, defined by where and are constants.
Following the argument used in , we can easily verify the following for
Lemma 3.1. (i)
(ii) There exist finite positive constants such that where and using the inequality
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(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.
The author would like to express sincere thanks to the anonymous referees for their invaluable corrections, comments, and suggestions.
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