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] .

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.

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

(H_f) 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 (H_f) 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] .

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