Convergence of Solutions to a Certain Vector Differential Equation of Third Order

and Applied Analysis 3 By Lemma 5, it follows that there exist δ 3 > 0 and δ 4 > 0

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, Tunc ¸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  1 (),  2 () of (1) in   will be said to converge to each other if then any two solutions  1 (),  2 () of (1) necessarily converge, where , , ,  are some positive constants with 0 <  < 1 and (< 1), Remark 3. The mentioned theorem itself still holds valid with (5) replaced by the much weaker condition for arbitrary  any   ,   ,   , ( = 1, 2), in   , where it is assumed that The following lemma is needed in our later analysis.

Lemma 4.
Let  be a real symmetric  × -matrix and where  and  are constants.Then 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 0 <  < 1 and  > 0 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  1 and  2 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 Ẋ and Ẍ be denoted, respectively, by  and .Then, we can rewrite (1) in the following equivalent system form: Let  1 (),  2 () in   be any solution of (1), define  = () by where  is the function defined in (11)  ( 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 0 <  ℎ ≤   ( ℎ ()) ≤ Δ ℎ , it follows that Then provided that Δ 1 < , where  is a sufficiently small positive constant.