A Stability Result for the Solutions of a Certain System of Fourth-Order Delay Differential Equation

Abstract

The main purpose of this work is to give sufficient conditions for the uniform stability of the zero solution of a certain fourth-order vector delay differential equation of the following form: By constructing a Lyapunov functional, we obtained the result of stability.

1. Introduction

As is well known, the stability is a very important problem in the theory and applications of delay differential equations. Therefore, in the literature, some methods have been developed to obtain information on the stability behaviour of the delay differential equations when there is no analytical expression for the solutions. One of these methods is known as Lyapunov’s second method; since Lyapunov [1] proposed his famous second method on the stability of motion, the problems related to the investigation of stability of solutions of certain second-, third-, and fourth-order linear and nonlinear, scalar, and vector differential equations have been given great attention in the past five decades due to the importance of the subject.

During this period, stability of solutions for various higher-order linear and nonlinear differential equations has been extensively studied and many results have been obtained in the literature (see, e.g., Krasovskii [2], Yoshizawa [3], Reissig et al. [4], Abou-El-Ela and Sadek [5–7], Bereketoglu and Kart [8], Sadek [9], Tunç [10–13], Abou-El-Ela et al. [14], and the references cited in those works), among which the results performed on asymptotic stability properties of linear and nonlinear scalar and vector differential equations of fourth-order can briefly be summarized as follows.

First in 1990 Abou-El-Ela and Sadek [5] found sufficient conditions for the asymptotic stability of the zero solution of the scalar nonlinear differential equation of the form

Later in 2004 Sadek [9] determined sufficient conditions, under which all solutions of the nonhomogeneous vector differential equation tend to zero as .

Recently in 2012 Abou-El-Ela et al. [14] investigated sufficient conditions for the uniform stability of the zero solution of the real fourth-order vector delay differential equation

In the present paper, we are concerned with the uniform stability of the zero solution of real nonlinear autonomous vector delay differential equation of the fourth-order where ; is an -symmetric matrix; , , and are -vector continuous functions; ; and is a bounded delay and positive constant.

Equation (4) represents a system of real fourth-order differential equation with delay

The Jacobian matrices , , , , , and are given by where , , , , , , , and represent , , , , , , and , respectively. It will also be assumed as basic throughout the paper that the Jacobian matrices , , , , , , and exist and are continuous. The symbol will be used to denote the usual scalar product in for any in ; that is, ; thus . It is well known that the real symmetric matrix , is said to be positive-definite, if and only if the quadratic form is positive-definite, where and denotes the transpose of .

2. Main Result

In order to reach the main result of this paper, we will give some basic information to the stability criteria for a general autonomous delay differential system. We consider where is a continuous mapping, , , and for , there exists an , with when .

Theorem 1 (see [15]). Let be a continuous functional satisfying a local Lipschitz condition, , such that(i), where , are wedges;(ii), for .Then the zero solution of (7) is uniformly stable.

The following theorem will be our main stability result for (4).

Theorem 2. In addition to the essential assumptions imposed on the functions , , , and , suppose the existence of arbitrary positive constants , , , , , and . Suppose also for the following conditions are satisfied.(i), , and are symmetric; , for all .(ii) is symmetric and , for all .(iii)There is a finite constant such that  for all .(iv)One has , for all .(v)One has , for all .(vi) and are negative-definite.(vii)Also , is symmetric, and , for all .(viii) commutes with , for all and , for all , and .(ix)Also , is symmetric, and , for all , where is a positive constant such that  Then the zero solution of (4) is uniformly stable, provided that  where

The following two lemmas are important for proving Theorem 2.

Lemma 3. Let be a real symmetric -matrix and where , are constants. Then

For a proof of the above lemma, see Bellman [16].

Lemma 4. Assume that , , and . Then(1);(2);(3);(4);(5).

Proof. The proof is as follows: The proofs of (2) and (3) are similar to that of (1): since is negative-definite from assumption (vi) and is symmetric from assumption (i). Then Since is negative-definite from assumption (i), we have and then since is negative-definite from assumption (vi).

3. Proof of Theorem 2

For the proof of the main stability theorem, it will be convenient to consider instead of (4) the equivalent system

The proof of Theorem 2 depends on a scalar differentiable function ; now we define the Lyapunov functional as where and are positive constants, which will be determined later. Let Since , for all , it follows that

Further we define and then it follows from (ii) and (iv) that for all , and Since then

Therefore

Also since are nonnegative, consequently we obtain

Then we can find

The matrices and are symmetric because and are symmetric. The eigenvalues of and are positive because of (22) and (24).

Consequently the square roots and exist; these are again symmetric and nonsingular for all .

Therefore we get

From and , because of (22) and (24), we get from Lemma 3 and Cauchy-Schwartz inequality that

From the definitions of , in (11), it follows that where

Since by integrating both sides from to and because of , we obtain

Thus

But from by integrating both sides from to and because of , we find

Therefore by using (11), (24), (vii), (viii), and Lemma 3, we have

To estimate we need since from (11) and (ii) we find that

Now

Thus we obtain from (viii)

From the identity we get from (25) and by Lemma 3

So we have from (9) and (11) since .

To estimate we need by (11), (25), and (45). So from the identity we find

Thus from (9), we obtain since . Then it follows that

Since the coefficients are positive constants from the definitions of , , and in (iv), (v), and (9), then there exists a positive constant such that

To prove that by using the hypotheses of Theorem 2 we find

Since then from (ix) we have and also since then from (iv) we have

Since then from (viii) we get

By using Cauchy-Schwartz inequality and from

Hence there exists a positive constant satisfying

Now from (19), (20), and Lemma 4, we have

Then we get and it follows that where

But

Since is nonnegative by (ix), then from (11) we get since by (9). Also

But is nonnegative by (viii) and from (11), we get

Therefore

We know that and by Lemma 3, we get

Since by (viii) and by using Cauchy-Schwartz inequality, we obtain

Also, since by (iii) and by using Cauchy-Schwartz inequality, we find

Therefore it follows from (11) and (45) that and if we take then we have

Therefore if we obtain for some . Therefore from (54), (64), and (81) the functional satisfies all the conditions of Theorem 1, so that the zero solution of (4) is uniformly stable.

Thus the proof of Theorem 2 is now complete.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.