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Convergence behaviors of solutions arising from certain system of third-order nonlinear differential equations are studied. Such convergence of solutions corresponding to extreme stability of solutions when relates a pair of solutions of the system considered. Using suitable Lyapunov functionals, we prove that the solutions of the nonlinear differential equation are convergent. Result obtained generalizes and improves some known results in the literature. Example is included to illustrate the result.

We shall consider here systems of real differential equations of the form

which is equivalent to the system

where

So far in the literature, much attention has been drawn to the boundedness of solutions of ordinary scalar and vector nonlinear differential equations of third order. The book of Reissig et al. [

by extending the result of [

a variant of (1), where c is a positive constant and obtained some results which guarantee the convergence of the solutions. With respect to our observation in the literature, no work based on (1) was found. The result to be obtained here is different from that in Olutimo [

Definition 1.1. Any two solutions

If the relations above are true of each other (arbitrary) pair of solutions of (1), we shall describe this saying that all solutions of (1) converge.

We shall state for completeness, some standard results needed in the proofs of our results.

Lemma 1. Let D be a real symmetric

where

Proof of Lemma 1. See [

Lemma 2. Let

1) The eigenvalues

2) The eigenvalues

where

Proof of Lemma 2. See [

Lemma 3. Subject to earlier conditions on

where

Proof of Lemma 3. See [

Lemma 4. Subject to earlier conditions on

1)

2)

Proof of Lemma 4. See [

Lemma 5. Subject to earlier conditions on

1)

2)

Proof of Lemma 5. See [

Throughout the sequel

functions

Our main result which gives an estimate for the solutions of (1) is the following:

Theorem 1. Assume that

Suppose that

1) The

2) P satisfies

for any

Our main tool in the proof of the result is the function

where

and

The following result is immediate from (4).

Lemma 6. Assume that, all the hypotheses on matrix

Proof of Lemma 6. In the proof of the lemma, the main tool is the function

This function, after re-arrangement, can be re-written as

Since

And

we have that

Since matrix

where

Thus,

From (9), the term

Since

by integrating both sides from

But from

integrating both sides from

Hence, (10) becomes

combining the estimate for

By hypothesis (1) of Theorem 1 and lemmas 1 and 2, we have

where

Similarly,

It is obvious that

also,

and

Combining all the estimates of

Now, combining

that is,

Thus, it is evident from the terms contained in (12) that there exists sufficiently small positive constants

where

The right half inequality in lemma 6 follows from lemma 1 and 2.

Thus,

where

Hence,

Let

By

where V is the function defined in (4) with

By lemma 6, (13) becomes

for

The derivative of

where

Using the fact that

and

where

Following (8),

and

Thus,

Note that

and

We have;

On applying Lemma 1 and 2, we have

If we choose

where

Thus,

with

There exists a constants

In view of (14), the above inequality implies

Let

which implies that

Thus, by (14), it shows that

From system (1) this implies that

This completes the proof of Theorem 1.

Analysis of nonlinear systems literary shows that Lyapunov’s theory in convergence of solutions is rarely scarce. The second Lyapunov’s method allows predicting the convergence behavior of solutions of sufficiently complicated nonlinear physical system.

Example 4.0.1. As a special case of system (2), let us take for

Thus,

Clearly,

and

Then, by easy calculation, we obtain eigenvalues of the matrices

It is obvious that

If we choose

Thus, all the conditions of Theorem 1 are satisfied. Therefore, all solutions of (1) converge since (5) and (6) hold.

Akinwale L.Olutimo, (2016) Result on the Convergence Behavior of Solutions of Certain System of Third-Order Nonlinear Differential Equations. International Journal of Modern Nonlinear Theory and Application,05,48-58. doi: 10.4236/ijmnta.2016.51005