Stability for Functional Differential Equations with Delay in Banach Spaces ()
1. Introduction
Since 1892, the Lyapunov’s direct method has been used for the study of stability properties of ordinary, functional, differential and partial differential equations. Nevertheless, the applications of this method to problem of stability in differential equations with delay have encountered serious difficulties if the delay is unbounded or if the equation has unbounded terms (see [1] [5] [6] ). Recently, investigators, such as Burton and Furumochi have noticed that some of these difficulties vanish or might be overcome by means of fixed point theory (see [1] - [17] ). The fixed point theory does not only solve the problem on stability but has a significant advantage over Lyapunov’s direct method. Sadovskii’s fixed point theorem (see [18] ) and techniques of the theory of the measure of noncompactness are used to prove the existence and stability of the solution of the problem investigated in this paper.
Let E denote a Banach space. We consider the nonlinear differential equation with variable delay
(1)
with the initial condition
where
,
,
.
Here
denotes the set of all continuous and bounded functions
with the supremum norm
defined
. Throughout this paper, we assume that
,
with
as
,
. We also assume that the function
is uniformly continuous. Moreover, we assume that the function G is bounded, so there exists a constant
, such that
and
.
Special cases of the equation (1) have been investigated by many authors. For example, Burton in [6] and Zhang in [17] have studied boundedness and stability of the linear equation:
In [13] , Burton and Furumochi using the fixed point theorem of Krasnosielski obtained boundedness and asymptotic stability for the equation:
where
is constant;
and
is a quotient with odd positive integer denominator and
.
Next, Jin and Luo (see [4] ), proved the boundedness and stability of solutions of the equation
and generalized the results claimed in [6] [12] [17] . Ardjouni and Djoudi in [3] considered the more general neutral nonlinear differential equation
and obtained the boundedness and stability results.
We mentioned here that the neutral delay differential equations appear in modelling of the networks containing lossless transmission lines (as in high-speed computers where the lossless transmission lines are used to interconnect switching circuits), in the study of vibrating masses attached to an elastic bar, as the Euler equation in some variational problems, theory of automatic control and in neuromechanical systems in which inertia plays an important role, we refer the reader to the papers by Boe and Chang [19] , Brayton and Willoughby [20] and to the books by Driver [21] , Hale [14] and Popov [22] and reference cited therein.
The fundamental tool in this paper is the Kuratowski measure of noncompactness (see [23] ).
For any bounded subset A of E we denote by
the Kuratowski measure of noncompactness of A, i.e. the infimum of all
such that there exists a finite covering of A by sets of diameters smaller than
. The properties of the measure of noncompactness
are:
1) if
then
;
2)
, where
denotes the closure of A;
3)
if and only if A is relatively compact;
4)
;
5)
,
;
6)
7)
, where
denotes the convex hull of A,
8)
, where
.
Lemma 1. [24] . Let
be a family of strongly equicontinuous functions. Let
, for
and
. Then
where
denotes the measure of noncompactness in
and the function
is continuous.
Let us denote by
the set of all nonnegative real sequences. For
,
, we write
if
(i.e.
, for
) and
.
Let X be a closed convex subset of
and let
be a function which assigns to each nonempty subset Z of X a sequence
such that
(2)
(3)
if
(the zero sequence), then
is compact. (4)
In the proof of the main theorem, we will apply the following fixed point theorem
Theorem 1 [18] If
is continuous mapping satisfying
for arbitrary nonempty subset Z of X with
, then F has a fixed point in X.
2. Main Results
A solution of the problem (1) is a continuous function
such that x satisfies (1) on
and
on
. Stability definitions may found in [1] , for example.
In this paper, we extend stability theorem proved in [3] by giving a necessary and sufficient conditions for asymptotic stability of the zero solution of the Equation (1). By using some conditions expressed in terms of the measure of noncompactness which G satisfies, we define a continuous, bounded operator
over the Banach space
, whose fixed points are solutions of (1). The fixed point theorem of Sadovskii is used to prove the existence of a fixed point of the operator
. To construct our mapping, we begin transforming (1) to a more tractable, but equivalent equation, which we invert to obtain an equivalent integral equation for which we derive a fixed point mapping. We need the following lemma in our proof of the main theorem.
Lemma 2 Let
be an arbitrary continuous function and suppose that
(5)
Then x is a solution of (1) if and only if
(6)
where
(7)
Now we will present our main results. We set
(8)
Let
be a fixed. Put
(9)
where
and
, this mean that
Let us recall that for continuous function
, the first order modulus of smoothness for f is the function
defined for any
by
Define
Now, we use (6) to define the operator
by
if
and for
we let
where
where
(10)
Theorem 2 Assume that the function
is uniformly continuous, satisfies the condition
and there exists a constant
such that
(11)
for each
and for each
, where
denotes the Kuratowski measure of noncompactness. Moreover, we assume that there exists a constant
satisfies, for
:
(12)
where
and
, as
.
Then there exists at least one solution of the problem (1), which tends to zero if
(13)
Proof. We divide our proof into parts. In the first part, we show that
complete with is well defined. Next, using Sadovskii’s fixed point theorem and techniques of the theory of the measure of noncompactness, we prove, that there exists a fixed point of the operator
, which is a solution of the problem (1). In the last part of our proof, we will show, that the solution of (1) tends to zero as
if
, as
.
Part I. It is clear that
is continuous. We will show the boundedness of this operator. Notice that
, so there exists a constant
such that
,
.
For
, the boundedness of the operator
is clear. For
we have
where
Therefore, because
satisfies
, we have, by (9), that
.
Now we will show that for each
we obtain
Let’s note that
where
Because the exponential function is continuous and using assumptions about functions
and properties of Bochner integral we have
By equicontinuity of function G and equicontinuity of family
we have that
So we obtain that
.
In the same way, we estimate the remaining ingredients and we get
Now we will show, that
as
. It is obvious that the first term
tends to zero as
, by the condition (13). Because
and
as
and b is a bounded function so the second term
tends to zero as
. Because functions
are bounded functions, so
,
. Analogously
and
tends to zero if
. Moreover,
In conclusion
as
. Hence
maps
into
.
Part II. Suppose that
and
denotes the Kuratowski measure of noncompactness. Let
,
. Then,
So, using (12) and properties of
we have
Because
, we get
for each
.
Define
for any nonempty subset V of
, where
,
. Evidently
. By the properties of
, the function
satisfies conditions (2)-(4). From (10)
, whenever
. If
then for each
we have
. Hence Arzela-Ascoli’s theorem proves that the set V is relatively compact. Consequently, by Theorem 1, the operator
has a fixed point which is a solution of the problem (1) with
on
and
as
, if
, for
. The proof is complete.
Remark 1. It is clear that we can assume about the function G other types of continuity and other conditions on measures of noncompactness. When we investigate the existence of solutions of (1) with non-continuous right-hand side, it is natural to consider the so-called Carathéodory-type solutions.