Stability and Boundedness of Solutions of Certain Non-Autonomous Third Order Nonlinear Differential Equations

In this paper, by defining an appropriate Lyapunov functional, we obtain sufficient conditions for which all solutions of certain real non-autonomous third order nonlinear differential equations are asymptotically stable and bounded. The results obtained improve and extend some known results in the literature.


Introduction
We shall be concerned here, with stability and boundedness of solutions of the third order, non-linear, nonautonomous differential equation of the form: where a(t), b(t) are positive continuously differentiable functions and φ , f and p are continuous real-valued functions depending only on the arguments shown, and the dots indicate the differentiation with respect to t.Moreover, the existence and uniqueness of solutions of (1.1) will be assumed.The Lyapunov function or functional approach has been a powerful tool to ascertain the stability and boundedness of solutions of certain differential equations.Up to now, perhaps, the most effective method to determine the stability and boundedness of solutions of non-linear differential equations is still the Lyapunov's direct (or second) method.The major advantage of this method is that stability in the large and boundedness of solutions can be obtained without any prior knowledge of solutions.Today, this method is widely recognized as an excellent tool not only in the study of differential equations but also in the theory of control systems, dynamical systems, systems with time lag, power system analysis, time varying non-linear feedback systems, and so on.Its chief characteristic is the construction of a scalar function or functional, namely, the Lyapunov function or functional.This function or functional and its time derivative along the system under consideration must satisfy some fundamental inequalities.But, finding an appropriate Lyapunov function or functional is in general a difficult task.See [1].
Stability analysis and boundedness of solutions of nonlinear systems are important area of current research and many concept of stability and boundedness of solutions have in the past been studied by several authors.See for instance, a survey book, Ressig et al. [2] and in a sequence of results by [3]- [8].With respect to our observation in the relevant literature, these authors consider stability, asymptotic behavior and boundedness of solutions of Equation (1.1) for which ( ) ≠ have received little attention due to the difficulty in constructing suitable scalar function.For example, see [9]- [12].However, no work based on (1.1) was found.
The result here will be different from those mentioned.The motivation for the present work is derived from the papers of the authors mentioned above.Our aim is to extend their results to the very special case in Equation (1.1) for the boundedness and asymptotic behavior of solutions.

Statement of Results
Our main results are the following theorems.
− > for all x, y; ( ) where  is a small positive constant whose magnitude depends only on the constants appeared in (i)-(iii).Then, every solution ( ) ) is asymptotically stable and satisfies Theorem 2 Let all the conditions of Theorem 1 be satisfied, and in addition we assume that there exist a finite constant 0 o δ > and a non-negative and continuous function ( ) q t such that p satisfies where ( ) Then every solution ( ) for all sufficiently large t, while D is a finite constant.Remark 2.1 Our results develop Qian [13], Omeike [14] and Tunc's [15] results to the non-autonomous of the form (1.1).
It is convenient here to consider, the equivalent system of (1.1); , and show that under the conditions stated in the theorem, every solution for all sufficiently large t, where D is the constant in (2.1).Our proof of (2.3) rests entirely on the lemma stated below and the scalar function By conditions (ii) of Theorem 1 and ( ) are positive.This implies that there exists a constant small enough such that ( ) ( ) ( ) Next, we prove the inequality (2.7).Along any solution 2), we have We easily see that by hypothesis (ii) of Theorem 1, , , , 0 .

Conclusions
The solutions of the third-order non-autonomous nonlinear system are bounded and asymptotically stable according to the Lyapunov's theory if the inequality (2.5)  Thus, all conditions of the Theorems are satisfied.Therefore, all solutions of (3.1) are asymptotically stable and bounded.

Lemma 1
Subject to the conditions of Theorem 1 there are positive constants 1 2 3 4 , , , , D D D D and 5 D depending only on , , , , o o a b c a b , and α such that a b δ and α such that any solution all x, y and z.Since α satisfy (2 the right-hand side is a constant and since ( ), , V x y z → ∞ as 2 2 2 is satisfied.Example 2.1 We consider a certain third order non-autonomous scalar differential equation of the form