Applied Mathematics
Vol.06 No.13(2015), Article ID:61592,7 pages
10.4236/am.2015.613192
Fixed Point Theorem and Fractional Differential Equations with Multiple Delays Related with Chaos Neuron Models
Toshiharu Kawasaki, Masashi Toyoda
Faculty of Engineering, Tamagawa University, Tokyo, Japan
Copyright © 2015 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
Received 9 July 2015; accepted 27 November 2015; published 30 November 2015
ABSTRACT
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.
Keywords:
Fixed Point Theorem, Ordinary Differential Equation, Delay Differential Equation, Fractional Differential Equation, Fractional Chaos Neuron Model
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.
Cite this paper
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
References
- 1. Banach, S. (1922) Sur les opérations dans les ensembles abstraits et leur application aux équations integrals. Fundamenta Mathematicae, 3, 133-181.
- 2. Lou, B. (1999) Fixed Points for Operators in a Space of Continuous Functions and Applications. Proceedings of the American Mathematical Society, 127, 1159-2264.
http://dx.doi.org/10.1090/S0002-9939-99-05211-9 - 3. de Pascale, E. and de Pascale, L. (2002) Fixed Points for Some Non-Obviously Contractive Operators. Proceedings of the American Mathematical Society, 130, 3249-3254.
http://dx.doi.org/10.1090/S0002-9939-02-06704-7 - 4. Matsuzaki, T. and Nakagawa, M. (2003) A Chaos Neuron Model with Fractional Differential Equation. Journal of the Physical Society of Japan, 72, 2678-2684.
http://dx.doi.org/10.1143/JPSJ.72.2678 - 5. Suzuki, T. (2006) Lou’s Fixed Point Theorem in a Space of Continuous Mappings. Journal of the Mathematical Society of Japan, 58, 769-774.
http://dx.doi.org/10.2969/jmsj/1156342037 - 6. Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006) Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam.
- 7. Lai, X. and Zhang, Y. (2012) Fixed Point and Asymptotic Analysis of Cellular Neural Networks. Journal of Applied Mathematics, 2012, Article ID: 689845.
- 8. Zhang, Y. and Luo, Q. (2013) Global Exponential Stability of Impulsive Cellular Neural Networks with Time-Varying Delays via Fixed Point Theory. Advances in Difference Equations, 2013, 2013:23.