differential

In this article, the author studies the stability and boundedness of solutions for the non-autonomous third order differential equation with a deviating argument, r: x ′′′ (t) + a(t)x ′′ (t) + b(t)g1(x ′ (t − r)) + g2(x ′ (t)) + h(x(t − r)) = p(t,x(t),x(t − r),x ′ (t),x ′ (t − r),x ′′ (t)), where r > 0 is a constant. Sufficient conditions are obtained; a stability result in the literature is improved and extended to the preceding equation for the case p(t,x(t),x(t − r),x ′ (t),x ′ (t −r),x ′′ (t)) = 0, and a new boundedness result is also established for the case p(t,x(t),x(t − r),x ′ (t),x ′ (t − r),x ′′ (t)) 6 0.

For the preceding equation, he constructed a positive definite Liapunov function with negative semi-definite time derivative.This established the stability of the null solution.
The motivation of this paper has come by the result of Ponzo [10,Theorem 2].Our purpose here is to extend and improve the result established by Ponzo [10,Theorem 2] to the preceding non-autonomous differential equation with the deviating argument r for the asymptotic stability of null solution and the boundedness of all solutions, whenever p ≡ 0 and p = 0 in Eq.(1), respectively.
At the same time, it is worth mentioning that one can recognize that by now many significant theoretical results dealt with the stability and boundedness of solutions of nonlinear differential equations of third order without delay: in which a 1 , a 2 and a 3 are not necessarily constants.In particular, one can refer to the EJQTDE, 2010 No. 1, p. 2 book of Reissig et al. [11] as a survey and the papers of Ezeilo [4,5], Ezeilo and Tejumola [6], Ponzo [10], Swick [14], Tunç [16,17,18,21], Tunç and Ateş [27] and the references cited in these works for some publications performed on the topic.Besides, with respect our observation from the literature, it can be seen some papers on the stability and boundedness of solutions of nonlinear differential equations of third order with delay (see, for example, the papers of Afuwape and Omeike [2], Omeike [9], Sadek [12], Sinha [13], Tejumola and Tchegnani [15], Tunç ([19,20], [22][23][24][25][26]), Zhu [28]) and the references thereof).
It should be noted that, to the best of our knowledge, we did not find any work based on the result of Ponzo [10,Theorem 2] in the literature.That is to say that, this work is the first attempt carrying the result of Ponzo [10,Theorem 2] to certain non-autonomous differential equations with deviating arguments.The assumptions will be established here are different from that in the papers mentioned above.

Main Results
Let p(t, x, x(t − r), y, y(t − r), z) = 0. We establish the following theorem Theorem 1.In addition to the basic assumptions imposed on the functions a(t), b(t), g 1 , g 2 and h appearing in Eq. ( 1), we assume that there are positive constants a, α, β, b 1 , b 2 , B, c, c 1 and L such that the following conditions hold: Then the null solution of Eq. ( 1) is stable, provided Proof .To prove Theorem 1, we define a Lyapunov functional V (t, x t , y t , z t ) : where λ 1 and λ 2 are some positive constants which will be specified later in the proof.Now, from the assumptions g 1 (y) The preceding inequalities lead to the following: The preceding inequality allows the existence of some positive constants D i , (i = 1, 2, 3), such that where Now, along a trajectory of (2) we find In view of the assumptions of Theorem 1 and the inequality 2 |mn| ≤ m 2 + n 2 , we find the following inequalities: The substituting of the preceding inequalities into (5) gives . Hence we can write Now, the last inequality implies for some positive constants λ 3 and λ 4 , provided This completes the proof of Theorem 1 (see also Burton [3], Hale [7], Krasovskii [8]).
For the case p(t, x, x(t − r), y, y(t − r), z) = 0, we establish the following theorem.
Theorem 2. Suppose that assumptions (i)-(ii) of Theorem 1 and the following condition hold: where q ∈ L 1 (0, ∞).Then, there exists a finite positive constant K such that the solution x(t) of Eq. ( 1) defined by the initial functions Proof.It is clear that under the assumptions of Theorem 2, the time derivative of functional V (t, x t , y t , z t ) satisfies the following: where D 5 = max{1, α}.
Example.Consider nonlinear delay differential equation of third order: Delay differential Eq. ( 9) may be expressed as the following system: Clearly, Eq. ( 9) is special case of Eq. ( 1), and we have the following: Thus all the assumptions of Theorems 1 and 2 hold.This shows that the null solution of Eq. ( 9) is stable and all solutions of the same equation are bounded, when p(t, x, x(t − r)y, y(t − r), z) = 0 and = 0, respectively.EJQTDE, 2010 No. 1, p. 9