On the Stability of Solutions of Nonlinear Functional Differential Equation of the Fifth-Order

The main purpose of this paper is to investigate global asymptotic stability of the zero solution of the fifth-order nonlinear delay differential equation on the following form ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) x t x t r x t r x t r x t r x t r x t f x t r x t r g x t h x t       5 4 4 , , , , 0. ψ φ + − − − − − + − + − + + = By constructing a Lyapunov functional, sufficient conditions for the stability of the zero solution of this equation are established.


Introduction
As is well-known, the area of differential equations is an old but durable subject, that remains alive and useful to a wide variety of engineers, scientists and mathematicians.Now the subject of differential equations represents a huge body of knowledge including many subfields and a vast array of applications in many disciplines.It should be noted that principles of differential equations are largely related to the qualitative theory of ordinary differential equations.Qualitative theory refers to the study of behaviour of solutions, for example, the investigation of stability, instability, boundedness of solutions and etc., without determining explicit formulas for the solutions.
In particular one can refer that many authors have dealt with delay differential equations and its problems, and many excellent results have been obtained on the behaviour of solutions for various higher-order: second-, third-, fourth-, and fifth-order nonlinear differential equations with delay, for example, [1]- [27], and references quoted therein, which contain the differential equations without delay or with delay.In many of these references, the authors dealt with the problems by using Lyapunov's second method [28].By considering Lyapunov functionals we obtained the conditions which ensured the stability of the problem.It is worth-mentioning that construction of these Lyapunov functionals remains a general problem.We know that a similar problem exists for ordinary differential equations for higher-order [12].Clearly, it is even more difficult to construct Lyapunov functionals for delay differential equations of higher-order.Up to this moment the investigations concerning the stability of solutions of nonlinear equations of fifth-order with delay have not been fully developed.
In particular in 2010 Tunç [29] obtained sufficient conditions, which ensure the stability of the zero solution of a nonlinear delay differential equation of fifth-order x t x t r x t r x t r x t r x t r x t where ψ and f are continuous functions; 3 α , 4 α and 5 α are positive constants, r is a bounded delay and positive constant; the derivatives ( ) ( )

, , ,
z w f z w f z w exist and are continuous for all z, w and ( ) Later in 2011 Abou-El-Ela, Sadek and Mahmoud [30] obtained the sufficient conditions for the uniform stability of the zero solution of a nonlinear fifth-order delay differential equation of the following form

0, x t x t x t x t f x t g x t h x t r
where r is a positive constant; ( ) ( ) ( ) ( ) In the present paper, we are concerned with the stability of the zero solution of the fifth-order nonlinear delay differential equation on the form x t x t r x t r x t r x t r x t r x t or its equivalent system form where ψ , f , φ , g and h are continuous functions for the arguments displayed explicitly in (1.1) with ( ) ( ) ( ) ( ) r is a bounded delay and positive constant; the derivatives ( ) ( ) z φ′ exist and are continuous for all , w z .

Preliminaries and Stability Results
In order to reach the main result of this paper, we will give some basic information to the stability criteria for the general autonomous delay differential system.We consider where : and for 1 H H < , there exists ( ) The following are the classical theorems on uniform stability and global asymptotic stability for the solution of (2.1).Theorem 2.1.[31].Let ( ) : H V ϕ →   be a continuous functional satisfying a local Lipschitz condition ( ) Then the zero solution of (2.1) is uniformly stable.
Theorem 2.2.[32].Suppose ( ) and let ( ) u s be non-negative and continuous function for 0 s ≤ < ∞ , ( ) Then all solutions of (2.1) approach zero as t → ∞ and the origin is globally asymptotically stable.The following will be our main stability result for (1.1).Theorem 2.3.In addition to the basic assumptions imposed on the functions , , , f g ψ φ and h.Suppose that the following conditions are satisfied, where 1 and the following two inequalities for all y and all , t ( ) 21 ( ) Then the zero solution of (1.1) is globally asymptotically stable, provided that ( ) Proof.We define the Lyapunov functional ( ) , , , , where ρ and λ are two positive constants, which will be determined later and δ is a positive constant de- fined by ( ) Then it is convenient to rewrite the expression for the Lyapunov functional defined in (2.5) in the following form This is due to the fact that the integral on the right-hand side is non-negative by vi), therefore we get and by using v) we find ( ) From iv) we find ( ) Clearly, it follows from the first six terms included in (2.8) that there exist sufficiently small positive constants Now we consider the terms  which are contained in (2.9) and by using the inequality ( )   , , Making use of the assumptions ii)-vi), ( ,

∫ ∫
Replacing the last equality and the preceding inequalities into (2.12),we obtain  Making use of the basic information on the positive semi-definite of a quadratic form, one can easily conclude that 6 0 v ≥ , 7 0 v ≥ , 8 0 v ≥ , 9 0 v ≥ , 10 0 v ≥ , 11 0 v ≥ , 12 0 v ≥ , 13 0 v ≥ , 14 0 v ≥ and 15 0 v ≥ provided that ( ) By constructing a Lyapunov functional, sufficient conditions for the stability of the zero solution of this equation are established.
λ ρ ρ and L are sufficiently small positive constants i)

For the component 1 v
, by using (2.6) and the definition of γ of the above discussion and inequality (2.13), it follows that

14 )
So we can choose the constants λ and ρ as the following , )