On the Stability of Solutions of Nonlinear Functional Differential Equation of the Fifth-Order ()
1. 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
where 
and f are continuous functions;
, 
and 
are positive constants, r is a bounded delay and positive constant; the derivatives 
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
where 
is a positive constant; 
and 
are continuous functions and 
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
(1.1)
or its equivalent system form
(1.2)
where
, 
, 
, 
and 
are continuous functions for the arguments displayed explicitly in (1.1) with 
r is a bounded delay and positive constant; the derivatives 
and 
exist and are continuous for all
.
2. 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
(2.1)
where 
is a continuous mapping, 
and for
, there exists 
with 
when 
The following are the classical theorems on uniform stability and global asymptotic stability for the solution of (2.1).
Theorem 2.1. [31] . Let 
be a continuous functional satisfying a local Lipschitz condition 
and the functions 
are wedges such that i) 
and ii)
.
Then the zero solution of (2.1) is uniformly stable.
Theorem 2.2. [32] . Suppose
, let V be a continuous functional defined on 
with
, and let 
be non-negative and continuous function for
, 
as 
such that for all 
i)
, 
and ii) 
for
.
Then all solutions of (2.1) approach zero as 
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 
and h. Suppose that the following conditions are satisfied, where 
are arbitrary positive constants and 
and L are sufficiently small positive constants i) 
(2.2)
and the following two inequalities
(2.3)
(2.4)
for all 
and all 
where
ii) 
iii) 
and
iv) 
and
v) 
and
vi) 
and
Then the zero solution of (1.1) is globally asymptotically stable, provided that
.
Proof. We define the Lyapunov functional 
as:
(2.5)
where 
and 
are two positive constants, which will be determined later and 
is a positive constant defined by
(2.6)
Then it is convenient to rewrite the expression for the Lyapunov functional defined in (2.5) in the following form
(2.7)
where
For the component
, by using (2.6) and the definition of 
since 
by v), thus we obtain
This is due to the fact that the integral on the right-hand side is non-negative by vi), therefore we get
From the identity
therefore
and by using v) we find
provided that
From iv) we find
Summing up the four inequalities obtained from 
into (2.7), we have
(2.8)
Clearly, it follows from the first six terms included in (2.8) that there exist sufficiently small positive constants 
such that
(2.9)
Now we consider the terms
which are contained in (2.9) and by using the inequality
, we obtain
for some
, if
By using the previous inequality, we get from (2.9) that
(2.10)
As a result, since the integrals
are non-negative, it is obvious that there exists a positive constant 
which satisfies the following inequality
(2.11)
where
.
Now by a direct calculation from (1.2) and (2.5) one finds
(2.12)
Making use of the assumptions ii)-vi), (2.3) and (2.6), we get
and
By v), vi) and (2.6).
By using the assumptions 
and 
of the theorem and inequality
, we obtain the following inequalities
and
Replacing the last equality and the preceding inequalities into (2.12), we obtain
(2.13)
where
and
It is clear that the expressions given by 
and 
represent certain specific quadratic forms, respectively.
Making use of the basic information on the positive semi-definite of a quadratic form, one can easily conclude that
, 
, 
, 
, 
, 
, 
, 
, 
and 
provided that
and
respectively.
Thus in view of the above discussion and inequality (2.13), it follows that
(2.14)
So we can choose the constants 
and 
as the following
and
then the inequality in (2.14) implies that
(2.15)
Hence one can easily get from (2.15) that
(2.16)
for some positive constants
, provided that
Finally, it follows that 
if and only if 
for 
and 
.
Thus all the conditions of Theorem 2.2 are satisfied. This shows that the zero solution of (1.1) is globally asymptotically stable.
Then the proof of Theorem 2.3 is completed.