Research Article | Open Access
A. M. A. Abou-El-Ela, A. I. Sadek, A. M. Mahmoud, R. O. A. Taie, "A Stability Result for the Solutions of a Certain System of Fourth-Order Delay Differential Equation", International Journal of Differential Equations, vol. 2015, Article ID 618359, 11 pages, 2015. https://doi.org/10.1155/2015/618359
A Stability Result for the Solutions of a Certain System of Fourth-Order Delay Differential Equation
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.
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  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 , Yoshizawa , Reissig et al. , Abou-El-Ela and Sadek [5–7], Bereketoglu and Kart , Sadek , Tunç [10–13], Abou-El-Ela et al. , 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  found sufficient conditions for the asymptotic stability of the zero solution of the scalar nonlinear differential equation of the form
Later in 2004 Sadek  determined sufficient conditions, under which all solutions of the nonhomogeneous vector differential equationtend to zero as .
Recently in 2012 Abou-El-Ela et al.  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-orderwhere ; 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 bywhere , , , , , , , 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 considerwhere is a continuous mapping, , , and for , there exists an , with when .
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 andwhere , are constants. Then
For a proof of the above lemma, see Bellman .
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). ThenSince is negative-definite from assumption (i), we haveand thensince 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 aswhere and are positive constants, which will be determined later. LetSince , for all , it follows that
Further we defineand then it follows from (ii) and (iv) thatfor all , andSincethen
Also sinceare nonnegative, consequently we obtain
Then we can find
Consequently the square roots and exist; these are again symmetric and nonsingular for all .
Therefore we get
From the definitions of , in (11), it follows thatwhere
Sinceby integrating both sides from to and because of , we obtain
But fromby integrating both sides from to and because of , we find
To estimate we needsince from (11) and (ii) we find that
Thus we obtain from (viii)
Thus from (9), we obtainsince . 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 thatby using the hypotheses of Theorem 2 we find
Sincethen from (ix) we haveand also sincethen from (iv) we have
Sincethen from (viii) we get
By using Cauchy-Schwartz inequality and from
Hence there exists a positive constant satisfying
Then we getand it follows thatwhere