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Based on the theory of fractional calculus, the contraction mapping principle, Krasnoselskii fixed point theorem and the inequality technique, a class of Caputo fractional-order BAM neural networks with delays in the leakage terms is investigated in this paper. Some new sufficient conditions are established to guarantee the existence and uniqueness of the nontrivial solution. Moreover, uniform stability of such networks is proposed in fixed time intervals. Finally, an illustrative example is also given to demonstrate the effectiveness of the obtained results.

Fractional order calculus was firstly introduced 300 years ago, but it did not attract much attention for a long time since it lack of application background and the complexity. In recent decades, the study of fractional-order calculus has re-attracted tremendous attention of much researchers because it can be applied to physics, applied mathematics and engineering [

We know that the next state of a system not only depends upon its current state but also upon its history information. Since a model described by fractional-order equations possesses memory, it is precise to describe the states of neurons. Moreover, the superiority of the Caputo’s fractional derivative is that the initial conditions for fractional differential equations with Caputo derivatives take on the similar form as those for integer-order differentiation. Therefore, it is necessary and interesting to study fractional-order neural networks both in theory and in applications.

Recently, some important and interesting results on fractional-order neural networks have been obtained and various issues have been investigated [

The integer-order bidirectional associative memory (BAM) neural networks models, first proposed and studied by Kosko [

Motivated by the above-mentioned works, this paper considers the uniform stability of a class of fractional-order BAM neural networks with delays in the leakage terms described by

where

This paper is organized as follows. In Section 2, some definitions of fractional-order calculus and some necessary lemmas are given. In Section 3, some new sufficient conditions to ensure the existence, uniqueness of the nontrivial solution and also uniform stability of the fractional-order BAM neural networks 1 is obtained. Finally, an example is presented to manifest the effectiveness of our theoretical results in Section 4.

For the convenience of the reader, we first briefly recall some definitions of fractional calculus, for more details, see [

Definition 1. The Riemann-Liouville fractional integral of order

provided the right side is pointwise defined on

Definition 2. The Caputo fractional derivative of order

Let

The initial conditions associated with system (1) are of the form

where

To prove our results, the following lemmas are needed.

Lemma 1. ([

has solutions

where

Lemma 2. ([

1)

2)

3)

Then, there exists

In order to obtain main result, we make the following assumptions.

(H1) The neurons activation functions

(H2) For

For convenience, let

Theorem 3. Under assumption (H1), the system (1) has a unique solution on

Proof. Define

where

By Lemma 1, we know that the fixed point of (F, G) is a solution of system (1) with initial conditions (2). In the following, we will using the contraction mapping principle to prove that the operator (F, G) has a unique fixed point.

Firstly, we prove

where

and

By Minkowski inequality, we have

By direct computation, we obtain by (3) that

Similar to (11) and the proof of Theorem 1 in [

and

Substitute (11)-(14) into (10), we get

Similarly, we obtain

Thus, from (15), (16) and (7), we have

Secondly, we prove that

By (6), we conclude that

Theorem 4. Assume that (H2) holds. If there exist real numbers

then the system (1) has at least one solution on

Proof. Let

Define two operators

where

Firstly, we will prove

Thus, we conclude that

Secondly, for any

which implies that

Thirdly, we prove that

which implies that

as

Theorem 5. Assume that (H1) and condition (6) hold. Then the solution of system (1) is uniformly stable on

Proof. Assume that

ditions

that is,

which implies

Hence, we have

For any

which implies that the solution of system (1) is uniformly stable on

In this section, we give an example to illustrate the effectiveness of our main results.

Consider the following two-state Caputo fractional BAM type neural networks model with leakage delay

with the initial condition

where

Thus,

that is, condition (6) holds. By utilizing Theorems 3.1 and 3.3, we can obtain that the system (20) has a unique solution which is uniformly stable on

In the following, we show the simulation result for model (20). We consider four cases:

Case 1 with the initial values

Case 2 with the initial values

Case 3 with the initial values

Case 4 with the initial values

The time responses of state variables are shown in

This work is supported by Natural Science Foundation of Jiangsu Province (BK2011407) and Natural Science Foundation of China (11571136 and 11271364).

YupingCao,ChuanzhiBai, (2015) Existence and Stability Analysis of Fractional Order BAM Neural Networks with a Time Delay. Applied Mathematics,06,2057-2068. doi: 10.4236/am.2015.612181