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In this paper, 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 result 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.

The following was the famous fixed point theorem introduced by Banach in 1922.

The Banach contraction principle ([

for any

In 1999 Lou proved the following fixed point theorem.

Lou’s fixed point theorem ([

for any

for any

Moreover, in 2002 de Pascale and de Pascale proved the following fixed point theorem.

De Pascale-de Pascale’s fixed point theorem ( [

for any

for any

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 [

In this section, we show a fixed point theorem. It deduces to Lou’s fixed point theorem [

Definition 1. Let I be an arbitrary finite or infinite interval, let J be an interval with

for any

for any

for any

Theorem 1. Let I be an arbitrary finite or infinite interval, let

for any

(H_{1}) for any

(H_{2}) there exist

1)

2)

3)

Then A has a unique fixed point in F.

Proof. By (H_{1}) we obtain

for any _{2}) there exists

Since

Since

and hence

for any

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 [

Remark 1. By Theorem 1 we can obtain Lou’s fixed point theorem [

for any

for any

for any

for any _{1}) holds. Take

Then (1) and (2) of (H_{2}) hold. Moreover, if

if

that is, (3) of (H_{2}) 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 [

for any

for any

for any _{1}) holds. Take

Then (1) and (2) of (H_{2}) hold. Moreover, if

if

that is, (3) of (H_{2}) holds. Therefore, by Theorem 1 A has a unique fixed point in F.

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

for any

Theorem 2. Let

ous mappings from

(H_{f}) there exist

for any

Let

where

have a unique solution in

Proof. Put

Then F is closed. Since

Define a mapping A by

for any

where

_{1}) holds. Take

_{2}) hold. Moreover, since

(3) of (H_{2}) holds. Therefore, by Theorem 1 A has a unique fixed point in F.

By using Theorem 2, we discuss the fractional chaos neuron model [

Example 1. We consider the following fractional differential equation with delay

where

called the fractional chaos neuron model [

Since

f satisfies (H_{f}) for

The authors would like to thank the referee for valuable comments.

ToshiharuKawasaki,MasashiToyoda, (2015) Fixed Point Theorem and Fractional Differential Equations with Multiple Delays Related with Chaos Neuron Models. Applied Mathematics,06,2192-2198. doi: 10.4236/am.2015.613192