Received 23 January 2016; accepted 21 March 2016; published 24 March 2016
This paper considers the third order non-autonomous nonlinear delay differential
or its equivalent system
where, , , β and are some positive constants, will be determined later, , , are real valued functions continuous in their respective arguments on, , , , , and respectively and. Besides, it is supposed that the derivatives, are continuous for all x, y, with. In addition, it is also assumed that the functions, , and satisfy a Lipschitz condition in and z; throughout the paper, and are respectively abbreviated as x, y and z. Then the solutions of (1.1) are unique.
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  -  ). It is well known that the stability of solutions plays a key role in characterizing the behavior of nonlinear delay differential equations. Stability is much more complicated for delay equations. Thus, it is worthwhile to continue to investigate the stability and boundedness of solutions of Equation (1.1) and its various forms.
Equation of the form (1.1) in which, and are constants has been studied by several authors Sadek   , Zhu  , Afuwape and Omeike  , Ademola and Aramowo  , Yao and Meng  , Tunc  and Ademola et al  to mention a few. They obtain the stability, uniform boundedness and uniform ultimate boundedness of solutions. In a sequence of results, Omeike  considers the following nonlinear delay differential equation of the third order, with a constant deviating argument r,
and established conditions for the stability and boundedness of solution when and while Tunc  considers a similar system with a constant deviating argument r of the form
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  on analogous third order differential equation which itself generalizes an analogous third-order results of Zhu  , and also complement existing results on third order delay differential equations. We establish sufficient conditions for the stability (when) and boundedness (when) of solutions of Equation (1.1) which extend and improve the results of Omeike  and Tunc  . An example is given to illustrate the correctness and significance of the result obtained.
Now, we will state the stability criteria for the general non-autonomous delay differential system. We consider:
where is a continuous mapping,
and for, there exists, with
Definition 1.0.1 (  ) An element is in the -limit set of, say, , if is defined on and there is a sequence, as, with as where
Definition 1.0.2 (   ) A set is an invariant set if for any, the solution of (1.2), , is defined on and for.
Lemma 1 ([8,13]) An element is such that the solution of (1.3) with is defined on and for, then is a non-empty, compact, invariant set and
Lemma 2 (   ) Let be a continuous functional satisfying a local Lipschitz con- dition., and such that:
1) where, are wedges;
Then the zero solution of (1.3) is uniformly stable. If we define, then the zero solution of (1.3) is asymptotically stable provided that the largest invariant set in Z is.
The following will be our main stability result (when) for (1.1).
2. Statement of Results
Theorem 1 In addition to the basic assumptions imposed on the functions a(t), b(t), c(t), and p, let us assume that there exist positive constants such that the following conditions are satisfied:
4); and, , for all x, y.
Then, the zero solution of system (1.2) is asymptotically stable, provided that
Our main tool is the following Lyapunov functional defined as
where and are positive constants which will be determined later.
We also assume that
By the assumption and, from (2.3) we have
The Lyapunov functional (2.4) can be arranged in the form
From Theorem 1, and which makes.
Thus, there is a such that
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
Thus, for some positive constants and, where small enough such that
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, , and using, we have that
Similarly, we obtain
If, then. If, we can rewrite the term as
where by (3) of Theorem 1,.
And by (1) and (2) of Theorem 1,
According to (2) of Theorem 1, and by (3), and certainly thus,
and by (3) and (4) of Theorem 1, we have that
for all and.
Thus, from (2.11), (2.12), (2.13), (2.14) and (2.15), we have
If we choose
and using, we obtain
Finally, it follows that if and only if, for and for.
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 is a constant and (1.1) is the constant co-efficient delay differential equation , then conditions (1)-(4) reduce to the Routh-Hurwitz conditions a > 0, c > 0 and. To show this we set and, and.
Remark 2.2 If and in (1.1), the non-autonomous Equation (1.1) reduces to the autonomous equation considered in Sadek  .
3. The Boundedness of Solution
Theorem 2 We assume that all the assumptions of Theorem 1 and
hold, where is a positive constant.
Then, there exists a finite positive constant K such that the solution of Equation (1.1) defined by the initial function
satisfies the inequalities
for all, where provided that
Proof of Theorem 2
As in Theorem 1, the proof of Theorem 2 depends on the scalar differentiable Lyapunov function defined in (2.3).
Since, in (1.1).
In view of (2.16),
Since for all thus
Hence, it follows that
for a constant, where.
Making use of the inequalities and, it is clear that
By (2.10), we have,
Multiplying each side of this inequality by the integrating factor, we get
Integrating each side of this inequality from 0 to t, we get, where,
Since and using the fact that for all t, this implies
Now, since the right-hand side is a constant, and since as, it follows that there exist a such that
From the Equation (1.1) this implies
The proof of Theorem 2 is now complete.
Remark 3.1 If is a constant, , and in (1.1), the result obtained reduces to Omeike  and a result of Tunc  .
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, it is clear from the equation that
The function, it is clear from the equation that
It follows that, if the delay is increased beyond this range a limit cycle appear, followed even-
tually by a period-doubling cascade leading to chaos.
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.