Fixed Point Theorem and Fractional Differential Equations with Multiple Delays Related with Chaos Neuron Models ()
Received 9 July 2015; accepted 27 November 2015; published 30 November 2015


1. Introduction
The following was the famous fixed point theorem introduced by Banach in 1922.
The Banach contraction principle ([1] ). Let
be a complete metric space, let F be a nonempty closed subset of X and let A be a mapping from F into itself. Suppose that there exist
such that

for any
. Then A has a unique fixed point in F.
In 1999 Lou proved the following fixed point theorem.
Lou’s fixed point theorem ([2] ). Let
, let
be a Banach space, let
be the Ba- nach space consisting of all continuous mappings from I into E with norm

for any
, let F be a nonempty closed subset of
and let A be a mapping from F into itself. Suppose that there exist
and
such that

for any
and for any
. Then A has a unique fixed point in F.
Moreover, in 2002 de Pascale and de Pascale proved the following fixed point theorem.
De Pascale-de Pascale’s fixed point theorem ( [3] ). Let
, let
be a Banach space, let
be the Banach space consisting of all bounded continuous mappings from I into E with norm
![]()
for any
, let F be a nonempty closed subset of
and let A be a mapping from F into itself. Suppose that there exist
,
and
such that
![]()
for any
and for any
. Then A has a unique fixed point in F.
In this paper, using the Banach contraction principle, we show a fixed point theorem which deduces to both of Lou’s fixed point theorem and de Pascale and de Pascale’s fixed point theorem. Moreover, our results can be applied to show the existence and uniqueness of solutions for fractional differential equations with multiple delays. Using the theorem, we discuss the fractional chaos neuron model [4] .
2. Fixed Point Theorem
In this section, we show a fixed point theorem. It deduces to Lou’s fixed point theorem [2] and de Pascale and de Pascale’s fixed point theorem [3] .
Definition 1. Let I be an arbitrary finite or infinite interval, let J be an interval with
, let
be a Banach space, let
be the Banach space consisting all bounded continuous mappings from I into E with norm
![]()
for any
, let
be the Banach space consisting all bounded continuous mappings from J into E with norm
![]()
for any
, let F be a nonempty closed subset of
, and let
be a mapping from
into E. Define a mapping
by
![]()
for any
. We say F satisfies (*) for
if (*)
holds for any
.
Theorem 1. Let I be an arbitrary finite or infinite interval, let
be intervals with
, let
be a Banach space, let
be the Banach space consisting all bounded continuous mappings from I into E with norm
![]()
for any
, and let F be a nonempty closed subset of
. Suppose that there exists a mapping
from
into E such that F satisfies (*) for
. Let A be a mapping from F into itself. Suppose that there exist
, a mapping G from
into
integrable with respect to the second variable for any the first variable, mappings
from I into
with
,
, and mappings
for any
such that
(H1) for any
and for any ![]()
![]()
(H2) there exist
,
,
with
and
such that
1)
;
2)
for any
and for any
;
3)
for any
.
Then A has a unique fixed point in F.
Proof. By (H1) we obtain
![]()
for any
and for any
. By (H2) there exists
, that is,
for any
. Define a new norm
in
by
![]()
Since
![]()
is equivalent of
. Define a metric d in F by
![]()
Since
for any
, we obtain
![]()
and hence
is a complete metric space. We obtain
![]()
for any
and for any
. Since
for any
, we obtain
![]()
that is, A is a contraction mapping. By the Banach contraction principle A has a unique fixed point in F.
The following remarks show that our fixed point theorem derives Lou’s fixed point theorem [2] and de Pascale and de Pascale’s fixed point theorem [3] . The proofs are owed to [5] .
Remark 1. By Theorem 1 we can obtain Lou’s fixed point theorem [2] . Actually let
, let
be a Banach space, let
be the Banach space consisting of all continuous mappings from I into E with norm
![]()
for any
, and let F be a nonempty closed subset of
. F satisfies (*) for the null mapping. Note that, since I is a finite interval,
is equivalent to
. Let A be a mapping from F into itself. Suppose that there exist
and
such that
![]()
for any
and for any
. Note that A is continuous. Therefore by the l’Hopital theorem we obtain
![]()
for any
. Put
![]()
,
,
and
. Then we obtain
![]()
for any
and for any
, that is, (H1) holds. Take
satisfying
. Put
,
,
, and
![]()
Then (1) and (2) of (H2) hold. Moreover, if
, then
![]()
if
, then
![]()
that is, (3) of (H2) holds. Therefore, by Theorem 1 A has a unique fixed point in F.
Remark 2. By Theorem 1 we can obtain de Pascale and de Pascale’s fixed point theorem [3] . Actually let
, let
be a Banach space, let
be the Banach space consisting of all bounded continuous mappings from I into E with norm
![]()
for any
, and let F be a nonempty closed subset of
. F satisfies (*) for the null mapping. Let A be a mapping from F into itself. Suppose that there exist
,
and
such that
![]()
for any
and for any
. Put
![]()
,
,
and
. Then we obtain
![]()
for any
and for any
, that is, (H1) holds. Take
and
satisfying
. Put
,
,
,
and
![]()
Then (1) and (2) of (H2) hold. Moreover, if
, then
![]()
if
, then
![]()
that is, (3) of (H2) holds. Therefore, by Theorem 1 A has a unique fixed point in F.
3. Fractional Differential Equations with Multiple Delays
In this section, by using Theorem 1, we show the existence and uniqueness of solutions for fractional differential equations with multiple delays. Throughout this paper, the fractional derivative means the Caputo-Riesz derivative
defined by
![]()
for any
and for any function u, where
is the gamma function and m is a natural number with
; for instance, see [6] .
Theorem 2. Let
be a Banach space, let
be the space consisting of all continu-
ous mappings from
into E and let
satisfying
(Hf) there exist
such that
![]()
for any
and for any
.
Let
be the Banach space consisting of all continuous mappings from
into E, let
be the space consisting of all continuous mappings from
into
and let
be the space consisting of all continuous mappings from
into E. Then the following fractional differential equation with multiple delays
![]()
where
,
is the
-order Caputo-Riesz derivative,
and
,
have a unique solution in
.
Proof. Put
,
,
and
![]()
Then F is closed. Since
and
for any
, we obtain
for any
. Therefore, F satisfies (*) for
. By direct computations,
is a solution of the equation above if and only if it is a solution of the following integral equation:
![]()
Define a mapping A by
![]()
for any
. Since
, we obtain
. We show that A has a unique fixed point. Indeed, we obtain
![]()
where
,
and
. Put
,
![]()
and
. Then (H1) holds. Take
with
and take c with
. Put
,
,
and
. Then (1) and (2) of (H2) hold. Moreover, since
![]()
(3) of (H2) holds. Therefore, by Theorem 1 A has a unique fixed point in F.
By using Theorem 2, we discuss the fractional chaos neuron model [4] .
Example 1. We consider the following fractional differential equation with delay
![]()
where
,
,
and
. In this equation,
is an internal state of the neuron at time t,
is a dissipative parameter and
is delay time. Moreover, we use a sinusoidal function with a periodic parameter
as an activation to be related to the output of the neuron. This equation is
called the fractional chaos neuron model [4] . Put
,
,
and
.
Since
![]()
f satisfies (Hf) for
and
. Therefore, by Theorem 2 the equation above has a unique solution in
. For analysis of neural networks using fixed point theorems, see [7] [8] .
Acknowledgements
The authors would like to thank the referee for valuable comments.