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Fixed Points and Stability in Nonlinear Equations with Variable Delays
Fixed Point Theory and Applications volume 2010, Article number: 195916 (2010)
Abstract
We consider two nonlinear scalar delay differential equations with variable delays and give some new conditions for the boundedness and stability by means of the contraction mapping principle. We obtain the differences of the two equations about the stability of the zero solution. Previous results are improved and generalized. An example is given to illustrate our theory.
1. Introduction
Fixed point theory has been used to deal with stability problems for several years. It has conquered many difficulties which Liapunov method cannot. While Liapunov's direct method usually requires pointwise conditions, fixed point theory needs average conditions.
In this paper, we consider the nonlinear delay differential equations
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ1_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ2_HTML.gif)
where ,
,
,
are continuous functions. We assume the following:
(A1) is differentiable,
(A2) the functions is strictly increasing,
(A3) as
.
Many authors have investigated the special cases of (1.1) and (1.2). Since Burton [1] used fixed point theory to investigate the stability of the zero solution of the equation , many scholars continued his idea. For example, Zhang [2] has studied the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ3_HTML.gif)
Becker and Burton [3] have studied the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ4_HTML.gif)
Jin and Luo [4] have studied the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ5_HTML.gif)
Burton [5] and Zhang [6] have also studied similar problems. Their main results are the following.
Theorem 1.1 (Burton [1]).
Suppose that , a constant, and there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ6_HTML.gif)
for all and
. Then, for every continuous initial function
, the solution
of (1.3) is bounded and tends to zero as
.
Theorem 1.2 (Zhang [2]).
Suppose that is differentiable, the inverse function
of
exists, and there exists a constant
such that for
-
(i)
(1.7)
-
(ii)
(1.8)
where . Then, the zero solution of (1.3) is asymptotically stable if and only if
-
(iii)
(1.9)
Theorem 1.3 (Burton [7]).
Suppose that , a constant. Let
be odd, increasing on
, and satisfies a Lipschitz condition, and let
be nondecreasing on
. Suppose also that for each
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ10_HTML.gif)
and there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ11_HTML.gif)
Then, the zero solution of (1.4) is stable.
Theorem 1.4 (Becker and Burton [3]).
Suppose is odd, strictly increasing, and satisfies a Lipschitz condition on an interval
and that
is nondecreasing on
. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ12_HTML.gif)
where is the unique solution of
, and if a continuous function
exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ13_HTML.gif)
on , then the zero solution of (1.5) is stable at
. Furthermore, if
is continuously differentiable on
with
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ14_HTML.gif)
then the zero solution of (1.4) is asymptotically stable.
In the present paper, we adopt the contraction mapping principle to study the boundedness and stability of (1.1) and (1.2). That means we investigate how the stability property will be when (1.3) and (1.4) are added to the perturbed term . We obtain their differences about the stability of the zero solution, and we also improve and generalize the special case
. Finally, we give an example to illustrate our theory.
2. Main Results
From existence theory, we can conclude that for each continuous initial function there is a continuous solution
on an interval
for some
and
on
. Let
denote the set of all continuous functions
and
. Stability definitions can be found in [8].
Theorem 2.1.
Suppose that the following conditions are satisfied:
(i), and there exists a constant
so that if
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ15_HTML.gif)
(ii)there exists a constant and a continuous function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ16_HTML.gif)
(iii) 
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ17_HTML.gif)
Then, the zero solution of (1.1) is asymptotically stable if and only if
(iv) 
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ18_HTML.gif)
Proof.
First, suppose that (iv) holds. We set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ19_HTML.gif)
Let , then
is a Banach space.
Multiply both sides of (1.1) by , and then integrate from 0 to
to obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ20_HTML.gif)
By performing an integration by parts, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ21_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ22_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ23_HTML.gif)
Then, is a complete metric space with metric
for
. For all
, define the mapping
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ24_HTML.gif)
By (i) and ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ25_HTML.gif)
Thus, when ,
.
We now show that as
. Since
and
as
, for each
, there exists a
such that
implies
. Thus, for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ26_HTML.gif)
Hence, as
. And
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ27_HTML.gif)
By (ii) and (iv), there exists such that
implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ28_HTML.gif)
Apply (ii) to obtain . Thus,
as
. Similarly, we can show that the rest term in (2.10) approaches zero as
. This yields
as
, and hence
.
Also, by (ii), is a contraction mapping with contraction constant
. By the contraction mapping principle,
has a unique fixed point
in
which is a solution of (1.1) with
on
and
as
.
In order to prove stability at , let
be given. Then, choose
so that
. Replacing
with
in
, we see there is a
such that
implies that the unique continuous solution
agreeing with
on
satisfies
for all
. This shows that the zero solution of (1.1) is asymptotically stable if (iv) holds.
Conversely, suppose (iv) fails. Then, by (iii), there exists a sequence ,
as
such that
for some
. We may choose a positive constant
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ29_HTML.gif)
for all . To simplify the expression, we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ30_HTML.gif)
for all . By (ii), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ31_HTML.gif)
This yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ32_HTML.gif)
The sequence is bounded, so there exists a convergent subsequence. For brevity of notation, we may assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ33_HTML.gif)
for some and choose a positive integer
so large that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ34_HTML.gif)
for all , where
satisfies
.
By (iii), in (2.5) is well defined. We now consider the solution
of (1.1) with
and
for
. We may choose
so that
for
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ35_HTML.gif)
It follows from (2.10) with that for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ36_HTML.gif)
On the other hand, if the solution of (1.1) as
, since
as
and (ii) holds, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ37_HTML.gif)
which contradicts (2.22). Hence, condition (iv) is necessary for the asymptotically stability of the zero solution of (1.1). The proof is complete.
When , a constant,
, we can get the following.
Corollary 2.2.
Suppose that the following conditions are satisfied:
(i), and there exists a constant
so that if
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ38_HTML.gif)
-
(ii)
there exists a constant
such that for all
, one has
(2.25)
-
(iii)
(2.26)
Then, the zero solution of (1.1) is asymptotically stable if and only if
(iv) 
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ41_HTML.gif)
Remark 2.3.
We can also obtain the result that is bounded by
on
. Our results generalize Theorems 1.1 and 1.2.
Theorem 2.4.
Suppose that a continuous function exists such that
and that the inverse function
of
exists. Suppose also that the following conditions are satisfied:
(i)there exists a constant such that
,
-
(ii)
there exists a constant
such that
satisfy a Lipschitz condition with constant
on an interval
,
(iii) and
are odd, increasing on
.
is nondecreasing on
,
(iv)for each , one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ42_HTML.gif)
Then, the zero solution of (1.2) is stable.
Proof.
By (iv), there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ43_HTML.gif)
Let be the space of all continuous functions
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ44_HTML.gif)
where is a constant. Then,
is a Banach space, which can be verified with Cauchy's criterion for uniform convergence.
The equation (1.2) can be transformed as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ45_HTML.gif)
By the variation of parameters formula, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ46_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ47_HTML.gif)
then is a complete metric space with metric
for
. For all
, define the mapping
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ48_HTML.gif)
By (i), (iii), and (2.29), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ49_HTML.gif)
Thus, there exists such that
and
. Hence,
.
We now show that is a contraction mapping in
. For all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ50_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ51_HTML.gif)
we have . That means
. Hence,
is a contraction mapping in
with constant
. By the contraction mapping principle,
has a unique fixed point
in
, which is a solution of (1.2) with
on
and
.
In order to prove stability at , let
be given. Then, choose
so that
. Replacing
with
in
, we see there is a
such that
implies that the unique continuous solution
agreeing with
on
satisfies
for all
. This shows that the zero solution of (1.2) is stable. That completes the proof.
When , a constant, we have the following.
Corollary 2.5.
Suppose that the following conditions are satisfied:
(i)there exists a constant such that
,
(ii)there exists a constant such that
satisfy a Lipschitz condition with constant
on an interval
,
(iii) and
are odd, increasing on
.
is nondecreasing on
,
(iv)for each , one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ52_HTML.gif)
Then, the zero solution of the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ53_HTML.gif)
is stable.
Corollary 2.6.
Suppose that the following conditions are satisfied:
(i)there exists a constant such that
,
(ii)there exists a constant such that
,
,
satisfy a Lipschitz condition with constant
on an interval
,
(iii) and
are odd, increasing on
.
is nondecreasing on
,
(iv)for each , one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ54_HTML.gif)
Then, the zero solution of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ55_HTML.gif)
is stable.
Remark 2.7.
The zero solution of (1.2) is not as asymptotically stable as that of (1.1). The key is that is not complete under the weighted metric when added the condition to
that
as
.
Remark 2.8.
Theorem 2.4 makes use of the techniques of Theorems 1.3 and 1.4.
3. An Example
We use an example to illustrate our theory. Consider the following differential equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ56_HTML.gif)
where ,
,
,
, and
,
. This equation comes from [4].
Choosing , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F195916/MediaObjects/13663_2010_Article_1224_Equ57_HTML.gif)
Let , when
is sufficiently small,
. Then, the condition (ii) of Theorem 2.1 is satisfied.
Let , then the condition (i) of Theorem 2.1 is satisfied.
And , then the condition (iii) and (iv) of Theorem 2.1 are satisfied.
According to Theorem 2.1, the zero solution of (3.1) is asymptotically stable.
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Ding, L., Li, X. & Li, Z. Fixed Points and Stability in Nonlinear Equations with Variable Delays. Fixed Point Theory Appl 2010, 195916 (2010). https://doi.org/10.1155/2010/195916
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DOI: https://doi.org/10.1155/2010/195916