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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.

We shall be concerned here, with stability and boundedness of solutions of the third order, non-linear, non- autonomous differential equation of the form:

where a(t), b(t) are positive continuously differentiable functions and

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 [

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. [

solutions of Equation (1.1) for which

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.

Our main results are the following theorems.

Theorem 1 Suppose

(i)

(ii)

(iii)

(iv)

Then, every solution

Theorem 2 Let all the conditions of Theorem 1 be satisfied, and in addition we assume that there exist a finite constant

(i)

where

Then every solution

for all sufficiently large t, while D is a finite constant.

Remark 2.1 Our results develop Qian [

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

and

Lemma 1 Subject to the conditions of Theorem 1 there are positive constants

Furthermore, there are finite constants

provided that

Proof: To verify (2.6) observe first that the expressions

By conditions (ii) of Theorem 1 and

in the re-arrangement of 2V becomes

Since

for all x, y and z. Since

we have

Next, we prove the inequality (2.7). Along any solution

We easily see that by hypothesis (ii) of Theorem 1,

and

By hypothesis (iii)

Also,

Thus,

that is,

where

Using the inequality (2.6) for all

Thus,

let

Just as in (2.7), we obtain

Proof of Theorem 1: It follows that

Thus, in view of (2.9) and (2.10) and the last discussion, it shows that the trivial solution of (1.1) is asymptotically stable.

Hence, the proof of Theorem 1 is complete.

Proof of Theorem 2: The proof of Theorem 2 depends on the scalar differentiable Lyapunov function

For

Since

Hence, it follows that

for a constant

Making use of the inequalities

by (2.6), we have

Hence,

or

We integrate both sides of this inequality from 0 to t and using Gronwall-Bellman inequality, we obtain

where

Now, since the right-hand side is a constant and since

From the system (1.1), this implies that

The proof of Theorem 2 is now complete.

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) is satisfied.

Example 2.1 We consider a certain third order non-autonomous scalar differential equation of the form

Choosing

Thus,

and finally,

and

Thus, all conditions of the Theorems are satisfied. Therefore, all solutions of (3.1) are asymptotically stable and bounded.

Akinwale L.Olutimo,Folashade O.Akinwole, (2016) Stability and Boundedness of Solutions of Certain Non-Autonomous Third Order Nonlinear Differential Equations. Journal of Applied Mathematics and Physics,04,149-155. doi: 10.4236/jamp.2016.41018