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In this paper, we study certain non-autonomous third order delay differential equations with continuous deviating argument and established sufficient conditions for the stability and boundedness of solutions of the equations. The conditions stated complement previously known results. Example is also given to illustrate the correctness and significance of the result obtained.

This paper considers the third order non-autonomous nonlinear delay differential

or its equivalent system

where

In applied science, some practical problems are associated with Equation (1.1) such as after effect, nonlinear oscillations, biological systems and equations with deviating arguments (see [

Equation of the form (1.1) in which

and established conditions for the stability and boundedness of solution when

and obtains the conditions for its boundedness of solution.

Results obtained are now extended to non-autonomous delay differential Equation (1.1). Results obtained in this work are comparable in generality to the results of Sadek [

Now, we will state the stability criteria for the general non-autonomous delay differential system. We consider:

where

and for

Definition 1.0.1 ( [

Definition 1.0.2 ( [

Lemma 1 ([8,13]) An element

Lemma 2 ( [

1)

2)

Then the zero solution of (1.3) is uniformly stable. If we define

The following will be our main stability result (when

Theorem 1 In addition to the basic assumptions imposed on the functions a(t), b(t), c(t),

1)

2)

3)

4)

Then, the zero solution of system (1.2) is asymptotically stable, provided that

and

Our main tool is the following Lyapunov functional

where

We also assume that

where

By the assumption

The Lyapunov functional (2.4) can be arranged in the form

From Theorem 1,

Thus, there is a

By (2) and (3) of Theorem 1, we have that the third term on the right in (2.5)

and next two terms give

Using (2.6), (2.7) and (2.8) in (2.5), we have

where

and integrals

Thus, for some positive constants

For the time derivative of the Lyapunov functional (2.3), along a trajectory of the system (1.2), we have

From (4) of Theorem 1,

Similarly, we obtain

Thus,

If

where by (3) of Theorem 1,

And by (1) and (2) of Theorem 1,

as

According to (2) of Theorem 1,

and by (3) and (4) of Theorem 1, we have that

for all

Thus, from (2.11), (2.12), (2.13), (2.14) and (2.15), we have

If we choose

and

and using

Choosing

we have

for some

Finally, it follows that

Thus, (2.10) and (2.16) and the last statement agreed with Lemma 2. This shows that the trivial solution of (1.1) is asymptotically stable.

Hence, the proof of the Theorem 1 is now complete.

Remark 2.1 If

Remark 2.2 If

Theorem 2 We assume that all the assumptions of Theorem 1 and

hold, where

Then, there exists a finite positive constant K such that the solution

satisfies the inequalities

for all

As in Theorem 1, the proof of Theorem 2 depends on the scalar differentiable Lyapunov function

Since

In view of (2.16),

Since

Hence, it follows that

for a constant

Making use of the inequalities

By (2.10), we have

Hence,

or

where

Multiplying each side of this inequality by the integrating factor

Integrating each side of this inequality from 0 to t, we get, where

or

Since

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

From the Equation (1.1) this implies

The proof of Theorem 2 is now complete.

Remark 3.1 If

The solutions of the third-order non-autonomous delay system are asymptotically stable and bounded according to the Lyapunov’s theory if the inequalities (2.1) and (2.2) are satisfied.

Example 3.1 We consider non-autonomous third-order delay differential equation

with equivalent system of (3.1) as:

comparing (1.2) with (3.2), it is easy to see that

The function

The function

also,

Since

we have

It follows that

tually by a period-doubling cascade leading to chaos.

Finally,

and

Thus, all assumptions of Theorem 1 and Theorem 2 are held. That is, zero solution of Equation (1.1) is asymptotically stable and all the solutions of the same equation are bounded.

Akinwale L.Olutimo,Daniel O.Adams, (2016) On the Stability and Boundedness of Solutions of Certain Non-Autonomous Delay Differential Equation of Third Order. Applied Mathematics,07,457-467. doi: 10.4236/am.2016.76041