Stability and boundedness in delay system of differential equations of third order

Peer review under responsibility of University of Bahrain.

Abstract

In this paper, a class of non-linear vector differential equations of third order with delay is considered. The stability, boundedness and ultimately boundedness of solutions are studied. The technique of proofs involves defining an appropriate Lyapunov functional. The obtained results include and improve the results in the literature.

Keywords:

• Asymptotic stability
• Boundedness
• Lyapunov functional
• Delay
• Vector differential system
• Third order

1 Introduction

During the last years, many good results have been obtained on the qualitative behaviors in ordinary and functional differential equations of third order without and with delay. In particular, for some works on the stability and boundedness in scalar ordinary and functional differential equations of third order without and with delay, we referee the interested reader to the papers of Ademola et al. (2015), Ademola and Arawomo (2011), Afuwape and Castellanos (2010), Graef et al. (2015), Graef and Tunc (2015), Mehri and Shadman (1999), Meng (1993), Omeike (2014), Omeike and Afuwape (2010), Qian (2000), Remili and Oudjedi (2014), Tunc (2004, 2005a,b,c, 2007, 2009a,b, 2010a,b, 2013a,b, 2014, 2015), Tunc and Mohammed (2014), Tunc and Ateş (2006), Zhang and Yu (2013) and their references. However, to the best of our knowledge from the literature, by this time, little attention was given to the investigation into the stability/boundedness/ultimately boundedness in vector functional differential equations of third order with delay (see Tunc and Mohammed (2014)).

It should be noted any investigation into the stability and boundedness in vector functional differential equations of third order, using the Lyapunov functional method, first requires the definition or construction of a suitable Lyapunov functional, which gives meaningful results. In reality, this case can be an arduous task. The situation becomes more difficult when we replace an ordinary differential equation with a functional vector differential equation. However, once a viable Lyapunov functional has been defined or constructed, researchers may end up with working with it for a long time, deriving more information about stability. To arrive at the objective of this paper, we define a new suitable Lyapunov functional.

Recently, the authors in Tunc and Mohammed (2014) discussed the stability and boundedness in non-linear vector differential equation of third order with constant delay :(1) In this paper, we consider vector differential equation of third order of the form(2) where is the fixed constant delay, c is a positive constant; is a continuous differentiable function with and is an continuous differentiable symmetric matrix function such that the Jacobian matrices and exist and are symmetric and continuous, that is,exist and are symmetric and continuous, where , and are components of , and , respectively; is a continuous function, , and the primes in Eq. (2)(2) indicate differentiation with respect to , .

It should be stated that the continuity of the functions , and is a sufficient condition for existence of the solution of Eq. (2)(2) . In addition, we assume that the functions , and satisfy a Lipschitz condition with respect to their respective arguments, like , and . In this case, the uniqueness of solutions of Eq. (2)(2) is guaranteed.

It will be convenient here to consider not Eq. (2)(2) itself, but rather the system(3) derived from it by setting , , .

Along this paper, we assume that the existence and the uniqueness of the solutions of Eq. (2)(2) hold.

The motivation of this paper comes from the results established in Datko (1994), De la Sen (1988a,b), De la Sen and Luo (2004), Omeike and Afuwape (2010), Qian (2000), Tunc and Mohammed (2014), Zhang and Yu (2013)", the mentioned papers and their references. The main purpose of this paper is to get some new stability/boundedness/ultimately boundedness results in Eq. (1)(1) using the Lyapunov-functional approach. By this paper, we will extend and improve the results of Omeike (2014), Tunc (2009b), Tunc and Mohammed (2014), Zhang and Yu (2013)".

This is the novelty of this work. Besides, the results to be established here may be useful for researchers working on the qualitative behaviors of solutions.

One basic tool to be used here is LaSalle’s invariance principle. Let us consider delay differential systemWe take to be the space of continuous function from into and ask that be continuous. We say that is a Lyapunov function on a set relative to if is continuous on , the closure of , is defined on , and on .

The following form of the LaSalle’s invariance principle can be found in Tunc and Mohammed (2014).

Theorem A

If is a Lyapunov function on and is a bounded solution such that for then is contained in the largest invariant subset of , denotes the omega limit set of a solution.

We need the following lemmas in the proofs of main results.

Lemma A

Hale (1965) suppose . Let be a continuous functional defined on with , and let be a function, non-negative and continuous for , as with . If for all , , , , then the zero solution of is stable.

If we define , then the zero solution of is asymptotically stable, provided that the largest invariant set in is .

Lemma B

Let be a real symmetric -matrix. Then for any where and are, respectively, the least and greatest eigenvalues of the matrix .

2 Stability

Let . The stability result of this paper is the following theorem.

Theorem 1

In addition to the basic assumptions imposed on and with , we assume that there exist positive constants , , , , , and such that the following conditions hold:

, exists, -symmetric matrices and commute with each other,and

If

withandthen all solutions of Eq. (2)(2) are bounded and the zero solution of Eq. (2)(2) is asymptotically stable.

Proof

We define a functional given by(4) where(5) , , and and are positive constants which will be determined in the proof.

Sinceit follows that

Then, from (4)(4) , we have clearly(6)

Under the hypotheses of Theorem 1, we have

In summary, in view of (6)(6) , the above estimates imply that

It is clear from the first four terms that there exist sufficiently small positive constants , , such thatLetso that

A straightforward calculation from (3) and (4) gives that

The assumptions of Theorem 1 lead to

On combining the above obtained inequalities into , we have thatLet

HenceSinceandit is clear that

By Lemma B and the assumptions of Theorem 1, we getLetandso thatIfthen, for some positive constants , and , it follows that

In addition, we can easily see thatConsider the set defined by

When we apply LaSalle’s invariance principle, we observe that implies that . Clearly, this fact leads that the largest invariant set contained in is . By Lemma B, we conclude that the zero solution of system (3)(3) is asymptotically stable. Hence, the zero solution of Eq. (2)(2) is asymptotically stable. This completes the proof of Theorem 1.

3 Boundedness

Let . The boundedness result of this paper is the following theorem.

Theorem 2

We assume that all the assumptions of Theorem 1 hold, except . Further, we suppose that there exists a non-negative and continuous function such thatwhere denotes the space of Lebesgue integrable functions.

Ifwithand

then there exists a constant such that any solution of system (3)(3) determined bySatisfiesfor all .

Proof

Let . In the case of , under the assumptions of Theorem 2, we can easily arrive atwhere

Besides, in view of the discussion made, it is clear thatso that

Integrating both sides of the last estimate from to , we have

Let

Then

By noting the Gronwall–Bellman inequality, we can get

By the estimate and the assumption , we can conclude that all solutions of system (2)(2) are bounded. This completes the proof of Theorem 2.

4 Ultimately boundedness

For the case , the ultimately boundedness result of this paper is the following theorem.

Theorem 3

We assume that all assumptions of Theorem 1 hold, except . In addition, we assume that there exists a positive constant such that the conditionholds.

IfWithand

then there exists a constant such that any solution of system (3)(3) determined byultimately satisfies

for all .

Proof

For the case , in the light of the assumptions of Theorem 3, we can conclude that

The rest of the proof can be easily done by following a similar procedure as shown in Meng (1993), Tunc and Mohammed (2014)". Hence, we omit the details of the proof.

5 Conclusion

A kind of nonlinear vector functional differential equations of third order with a constant delay has been considered. Some qualitative behaviors of solutions, stability/boundedness/ultimately boundedness of solutions, have been discussed. The technique of proofs involves defining an appropriate Lyapunov functional. Our results include and improve some recent results in the literature.

Competing interests

The author declares that he has no competing interests.

Notes

Peer review under responsibility of University of Bahrain.