We give some sufficient conditions to guarantee convergence of solutions to a nonlinear vector differential equation of third order. We prove a new result on the convergence of solutions. An example is given to illustrate the theoretical analysis made in this paper. Our result improves and generalizes some earlier results in the literature.

1. Introduction

This paper is concerned with the following nonlinear vector differential equation of third order: where   and and are continuous functions in their respective arguments.

It should be noted that, in 2005, Afuwape and Omeike [1] considered the following nonlinear vector differential equation of third order: where is real symmetric -matrix. The author established a new result on the convergence of solutions of (2) under different conditions on the function . For some related papers on the convergence of solutions to certain vector differential equations of third order, the readers can referee to the papers of Afuwape [2], Afuwape and Omeike [3], and Olutimo [4]. Further, it is worth mentioning that in a sequence of results Afuwape [2, 5, 6], Afuwape and Omeike [3], Afuwape and Ukpera [7], Ezeilo [8], Ezeilo and Tejumola [9, 10], Meng [11], Olutimo [4], Reissig et al. [12], Tiryaki [13], Tunç [1416], Tunç and Ateş [17], C. Tunç and E. Tunç [18], and Tunç and Karakas [19] investigated the qualitative behaviors of solutions, stability, boundedness, uniform boundedness and existence of periodic solutions, and so on, except convergence of solutions, for some kind of vector differential equations of third order.

The Lyapunov direct method was used with the aid of suitable differentiable auxiliary functions throughout the mentioned papers. However, to the best of our knowledge, till now, the convergence of the solutions to (1) has not been discussed in the literature. Thus, it is worthwhile to study the topic for (1). It should be noted that the result to be established here is different from that in Afuwape [2], Afuwape and Omeike [1, 3], Olutimo [4], and the above mentioned papers. This paper is an extension and generalization of the result of Afuwape and Omeike [3]. It may be useful for the researchers working on the qualitative behaviors of solutions (see, also, Tunç and Gözen [20]).

It should be noted that throughout the paper will denote the real Euclidean space of -vectors and will denote the norm of the vector in .

Definition 1. Any two solutions ,   of (1) in will be said to converge to each other if as .

2. Main Result

The main result of this paper is the following theorem.

Theorem 2. We assume that there are positive constants , , , , , , and such that the following conditions hold: (i)the Jacobian matrices , , and exist and are symmetric and their eigenvalues satisfy for all , , in ;(ii) satisfies for any ,  ,  ,  , in .
If then any two solutions , of (1) necessarily converge, where , , , are some positive constants with and ,

Remark 3. The mentioned theorem itself still holds valid with (5) replaced by the much weaker condition for arbitrary any ,  ,  ,  , in , where it is assumed that for .
The following lemma is needed in our later analysis.

Lemma 4. Let be a real symmetric -matrix and where and are constants.

Proof (see Afuwape [5]). Our main tool in the proof of our result is the continuous function defined for any triple vectors ,  ,   in , by This function can be rearranged as where and
The following result is immediate from the estimate (11).

Lemma 5. Assume that all the conditions on the vectors ,  , and in the theorem hold. Then, there exist positive constants and such that for arbitrary in .

Proof. Let Then the proof can be easily completed by using Lemma 4. Therefore, we omit the details of the proof.

Proof of the Theorem. Let in be any solution of (1). For such a solution, let and be denoted, respectively, by and . Then, we can rewrite (1) in the following equivalent system form:
Let in be any solution of (1), define by where is the function defined in (11) with ,  ,   replaced by ,   and , respectively.
By Lemma 5, it follows that there exist and such that
When we differentiate the function with respect to along the system (15), it follows, after simplification, that where
Note that the existence of the following estimates is clear (see Afuwape and Omeike [1]): where , , , , , .
Subject to the assumptions, it can be easily obtained that
In view of the assumptions of the theorem, it is also clear that Hence, Using the estimate , it follows that Then if with .
Further, since then where .
Moreover, it is obvious that where .
Hence, Using the assumption (5), we get so that There exists a constant such that provided that , where is a sufficiently small positive constant.
In view of (17), the last estimate implies that for some positive constant .
The conclusion of the theorem is immediate if, provided that  , on integrating in (33) between and , we have which implies that By (17), this shows that This completes the proof of the theorem.

Example 6. Let us consider (1), with where are bounded continuous functions on .
Then, it can be easily seen that Thus, , , , , , and .
Let us choose Then, Since , then all the conditions of Theorem 2 hold. Therefore, all solutions of the equation considered converge (see, also, [1]).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.