Existence and Stability Analysis of Fractional Order BAM Neural Networks with a Time Delay

http://dx.doi.org/10.4236/am.2015.612181

Existence and Stability Analysis of Fractional

Order BAM Neural Networks with

a Time Delay

Yuping Cao

1

, Chuanzhi Bai

2

1_{Department of Basic Courses, Lianyungang Technical College, Lianyungang, China} 2_{Department of Mathematics, Huaiyin Normal University, Huaian, China}

Received 25 September 2015; accepted 22 November 2015; published 25 November 2015 Copyright © 2015 by authors and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Abstract

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 net-works with delays in the leakage terms is investigated in this paper. Some new sufficient condi-tions are established to guarantee the existence and uniqueness of the nontrivial solution. More-over, 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.

Keywords

BAM Neural Networks, Caputo Fractional-Order, Existence, Fixed Point Theorems, Uniform Stability

1. Introduction

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 [1]-[6]. We know that the fractional-order derivative is nonlocal and has weakly singular kernels. It provides an excellent instrument for the description of memory and hereditary properties of dynamical processes where such effects are neglected or difficult to describe to the integer order models.

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 dif-ferentiation. 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 [7]-[14] by many authors. In [11], the authors proposed a fractional-order Hopfield neural network and investigated its stability by using energy function. In [12], the authors investigated stability, multistability, bifurcations, and chaos for fractional-order Hopfield neural networks. In [13], Chen et al. obtained a sufficient condition for uniform stability of a class of fractional-order delayed neural networks. In

[14], we investigated the finite-time stability for Caputo fractional-order BAM neural networks with distributed delay and established a delay-dependent stability criterion by using the theory of fractional calculus and genera-lized Gronwall-Bellman inequality approach. In [15], Song and Cao considered the existence, uniqueness of the nontrivial solution and also uniform stability for a class of neural networks with a fractional-order derivative, by using the contraction mapping principle, Krasnoselskii fixed point theorem and the inequality technique.

The integer-order bidirectional associative memory (BAM) neural networks models, first proposed and stu-died by Kosko [16]. This neural network has been widely studied due to its promising potential for applications in pattern recognition and automatic control. In recent years, integer-order BAM neural networks have been ex-tensively studied [17]-[21]. Recently, some authors considered the uniform stability of delayed neural networks; for example, see [22]-[24] and references therein. However, to the best of our knowledge, there are few results on the uniform stability analysis of fractional-order BAM neural networks.

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

( )

(

)

(

( )

)

( )

(

)

(

( )

)

1

1

, 0, 1, ,

, 0, 1, , ,

m C

i i i ij j j i

j n C

j j j ji i i j

i

D x t c x t a f y t I t i n

D y t d y t b g x t J t j m

α

β

σ

σ

=

=

 _{= − − + + ≥ =}

 

 _{= − − + + ≥ =}



∑

∑





(1)

where 0<α β, <1, C

Dα and C

Dβ denote the Caputo fractional derivative of order ,α β, respectively;

( )

i

x t

(

i=1,,n

)

and y_j

( )

t

(

j=1,,m

)

are the activations of the ith neuron in the neural field F_x and the jth neuron in in the neural field F_y at time t, respectively; f_j

(

y t_j

( )

)

denotes the activation function of the jth neuron from the neural field F_y at time t and g x t_i

(

_i

( )

)

denotes the activation function of the ith neu-ron from the neural field F_x at time t; I_i and J_j are constants, which denote the external inputs on the ith neuron from F_x and the jth neuron from F_y, respectively; the positive constants c_i and d_j denote the rates with which the ith neuron from the neural field F_x and the jth neuron from the neural field F_y will reset their potential to the resting state in isolation when disconnected from the networks and external inputs, respec-tively; the constants a_ij and b_ji represent the connection strengths; the nonnegative constant σ denotes the leakage delay.

This paper is organized as follows. In Section 2, some definitions of fractional-order calculus and some ne-cessary 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. Fi-nally, an example is presented to manifest the effectiveness of our theoretical results in Section 4.

2. Preliminaries

For the convenience of the reader, we first briefly recall some definitions of fractional calculus, for more details, see [1][2][5], for example.

Definition 1. The Riemann-Liouville fractional integral of order

α

>0 of a function u: 0,

(

∞ →

)

R is given by

( )

_{( ) (}

)

1

( )

0 1

d

t

I u tα t s α u s s

α

−

= −

Γ

∫

Definition 2. The Caputo fractional derivative of order γ>0 of a function u: 0,

(

∞ →

)

R can be written as

( )

₍

₎

( )

( )

(

)

1 0

1

d , 1 .

n t C

n

u s

D u t s n n

n t s

γ

γ γ

γ + −

= − < <

Γ −

∫

₋

Let X =

{

x x=

(

x x₁, ₂,,x_n

)

T,x_i∈C

(

[ ]

0,T ,

)

}

, Y =

{

y y=

(

y y₁, ₂,,y_m

)

T,y_j∈C

(

[ ]

0,T ,

)

}

. For p, q > 1, we know that X is a Banach space with the norm

(

( )

)

1

0 1

sup

p p n

i t T i p

x = _{≤ ≤}

∑

₌ x t , and Y is a Banach space

with the norm

(

( )

)

1

0 1

sup

q q m

j t T j q

y = ≤ ≤

∑

= y t . It is easy to see that X×Y is a Banach space with the norm

(

,

)

p q

x y = x + y .

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

( )

( )

,

( )

( )

,

[

, 0 ,

]

1, , , 1, , ,

i i j j

x θ =ϕ θ y θ =ψ θ θ∈ −σ i=  n j=  m (2) where

ϕ ψ

i, j∈C

(

[

−

σ

, 0 ,

]



)

.

To prove our results, the following lemmas are needed.

Lemma 1. ([25]). Let

α

>0, then the fractional differential equation

( )

( )

,

C_{D x t}α _{=h t}

has solutions

( )

2 1

0 1 2 1

( ) n ,

n x t =I h tα +c +c t+c t ++c t₋ −

where c_i∈, n=

[ ]

α +1.

Lemma 2. ([26]). Let D be a closed convex and nonempty subset of a Banach space X. Let φ₁, φ₂ be the operators such that

1) φ₁x+φ₂y∈D wherever x y, ∈D; 2) φ₁ is compact and continuous; 3) φ₂ is a contraction mapping.

Then, there exists x∈D such that φ₁x+φ₂x=x.

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

(H1) The neurons activation functions f_i and g_j are Lipschitz continuous, that is, there exist positive con- stants L li, j (i=1,,n, j=1,,m) such that

( )

( )

,

( )

( )

, , .

i i i j j j

f u − f v ≤L u−v g u −g v ≤l u−v ∀u v∈

(H2) For i=1,,n, j=1,,m, there exist M M₁, ₂>0 such that for u v, ∈, f u_i

( )

≤M₁ and

( )

2

j

g v ≤M .

3. Main Results

For convenience, let

0 0 0 0

1 1 _{1 1}

max , max , , ,

m n

i j i ij j ji

i n j m _{j i}

c c d d a a b b

≤ ≤ ≤ ≤ _{= =}

= = =

∑

=

∑

(3)

0 0 0 0

1 1 1 1

max _i , max _j , max _i, max ,_j

i n j m i n j m

I I J J L L l l

≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤

= = = = (4)

( )1 ( )1 ( 1) ( 1)

1 1

, .

q q p p

m _{q q} n _{p p}

i ij j ji

j i

a b

η ξ

− −

− −

= =

   

=  = 

 



∑



∑

(5)

(

) (

)

(

)

(

) (

)

(

)

1 1

1 1

0 0 0 0

1 1

max , 1.

1 1 1 1

q p

p m q n

q p

j i

j i

c n l T d m L T

B T T

β α

α β

σ ξ σ η

α β = β α =

 _{   } 

 

= _ − + _{ } − + _{  }<

Γ + Γ +   Γ + Γ +   

 



∑

∑

(6)

Proof. Define

(

F G,

)

:X× →Y X×Y as

(

)( )( )

(

( )( )

( )( ) ( )( )

( )( )

)

T

1 1

, , , , , n , , , , , m , ,

F G x y t = F x y t  F x y t G x y t  G x y t

where

( )( )

( )

(

_{( )}

)

1

(

)

(

( )

)

0

1

, 0 d ,

m t

i i i i ij j j i

j

t s

F x y t c x s a f y s I s

α

φ

σ

α

−

=

−  

= + _− − + + _

Γ _

∑

_

∫

( )( )

( )

(

)

( )

(

)

(

( )

)

1

0

1

, 0 d .

n t

j j j j ji i i j

i

t s

G x y t d y s b g x s J s

β

ψ

σ

β

−

=

−  

= + _− − + + _

Γ 

∑



∫

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

(

F G B,

)

δ ⊂Bδ, where Bδ =

{

( )

x y, ∈ ×X Y:

( )

x y, ≤δ

}

. Set

(

1

) (

,

)

, 1

A C

B

φ ψ

δ ≥ + +

− (7)

where

(

) (

)

1 1

0 0

max , ,

1 1

p q

c n d m

A

α β

σ σ

α β

 

 

=  

Γ + Γ +

 

  (8)

and

(

)

(

)

(

)

(

)

1 1

1 1

0 0 0 0

0 0

1 1

.

1 1 1 1

q p

p q

p q

i j

i j

I n T J m T f T g T

C a b

α β α β

α β α = β =

 

 

=_{Γ +} +_{Γ +} +_{Γ +}   +_{Γ +}  



∑

 

∑

 (9)

By Minkowski inequality, we have

( )

( )

(

_{( )}

)

(

)

(

( )

)

( )

(

_{( )}

)

(

)

(

)

( )

( )

1 1

0

0 1 1

1 1

1

0 0

1 1

1

0

0 1 1

, sup 0 d

0 sup d

0 d sup

p p

n _t m

i i i ij j j i

p

t T i j

p p p

n n _t

p _i

i i

t T

i i

p

n m

t ij

j t T i j

t s

F x y c x s a f y s I s

c t s

x s s

a t s

f s

α

α

α

φ σ

α

φ σ

α

α

−

≤ ≤ = =

−

≤ ≤

= =

−

≤ ≤ = =

 _{−  } 

 

=  + _Γ − − + +  

 

 

 

 _{ − } 

  _{   }

≤_{ } + −

 

 Γ 

  _{   }

  ₋ 

 

+

 Γ 

 



∑

∫

∑

∑

∑ ∫

∑ ∑

∫

(

)

( )

(

( )

)

( )

(

)

( )

1

1 1

0

0 1 1

1 1

0

0 1

0 d sup

d .

sup

p

p p

n m

t ij

j j j

t T i j

p p n _t

i t T i

a t s

f y s f s

t s

I s

α

α α

α

−

≤ ≤ = =

−

≤ ≤ =



 

 

 _

 _{ − } 

 _{ } 

+ _ − _

 Γ 

 

 

 

 _{ − } 

 _{ } 

+

 

 _ Γ _ 

 

∑ ∑

∫

∑ ∫

(10)

(

)

( )

(

)

( )

(

(

)

(

)

)

( )

(

(

)

(

)

)

( )

(

(

)

( )

)

( )

1 1 0 0 1 1 1 1 1 0 0 0

0 1 0 1

1

0 1

0 0

0 1 0 1

d sup d d sup sup d sup sup p p n _t i i t T i

p p

n p n _t p

i i

t T i t T i

p

n p n

i

t T i t T i

c t s

x s s

c c

t s s s t s x s s

c c

t

α

σ α α

σ

α σ

σ α

φ σ σ

α α

σ θ φ θ θ

α α − ≤ ≤ = − − ≤ ≤ = ≤ ≤ = − − ≤ ≤ = ≤ ≤ =  _{ − }   _{ − }   

 _ Γ _ 

 

   

≤ _ − − _ + _ − − _

Γ   Γ  

 

= _ − − _ +

Γ   Γ

∑ ∫

∑

∫

∑

∫

∑

∫

∑

(

(

)

( )

)

( )

( )

( )

( )

(

)

(

)

1 1 0 1 1

1 1 1 1

0 0

0

0 1 0 1 0

1 0

d

d d

sup sup sup sup

( ) . 1 p p t i p p n n t t

p _p p _p

i _t i

t T t T t T

i i

p

p p

t x

c c

n x t n

c n T x σ α δ α α σ

σ θ σ

α α

σ θ θ θ

φ θ γ γ γ γ

α α

σ φ σ

α − − − − − − − ≤ ≤ = ≤ ≤ ≤ ≤ = ≤ ≤  _{− −}          ≤ _{ } ⋅ + _{ } ⋅

Γ   Γ  

= + −

Γ +

∫

∑

∫

∑

∫

(11)

Similar to (11) and the proof of Theorem 1 in [15], we have by (H1), (4) and (5) that

(

)

( )

(

)

1 1 ₁ 0 0 0 1 d , sup 1 p p p n t i t T i

t s I n T

I s α _α α α − ≤ ≤ =  _{ − }   _{ }  _≤  

 _ Γ _  Γ +



∑ ∫

 (12)

(

)

( )

( )

(

)

1 1 1 0 0 0

0 1 1 1

0 d ,

sup

1

p p

p n _t m

ij _p

j i

t T i j i

a t s _{f T}

f s a

α _α α α − ≤ ≤ = = =  _{ − }     _{ }  _≤  

 _{ Γ }  _{Γ +  }

 

 

 

∑ ∑

∫

∑

(13)

and

(

)

( )

(

( )

)

( )

(

)

( )

( )

(

)

( )

( ) ( )

( )

1 1 1 1 0 0

0 1 1 0 1 1

1 1 1

0

0 1 1 1

0 d d

sup sup

sup

p p

p p

n _tm n _tm

ij ij

j j j j j

t T i j t T i j

q q q q

n _t m m _q

ij j

j

t T i j j

a t s a t s

f y s f s L y s s

a L t s

y s α α α α α α − − ≤ ≤ = = ≤ ≤ = = − − − ≤ ≤ = = =  _{ − }   _{ − }   _{ − }  _≤  _{ } 

 _{ Γ }   _{ Γ } 

             _{ − }     _{ }  ≤ _{  ⋅}

 Γ  _

 _{ }   

∑ ∑

∫

∑ ∑

∫

∑

∫

∑

∑

₍

₎

1 1 1 0 1 d . 1 p p q p n p i q i L T s y α η α =  _{ }   _{ }  _{ }  _{ ≤}  

   _{Γ + } 

             

∑

(14)

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

( )

₍

₎

₍

₎

(

)

(

(

)

)

(

)

1 1 0 0 0 1 1 1 0 0 1 , 1 1 . 1 1 p p p i p p i p p n p i

p p q

i

I n T f T

F x y a

c n L T

T x y

α α α α α φ α α

σ φ σ η

α α

=

=

 

≤ + + _{ }

Γ + Γ _{+ } 

 

+ + − + _{ }

Γ + Γ _{+ } 

∑

∑

(15)

Similarly, we obtain

( )

₍

₎

₍

₎

(

)

(

)

(

)

1 1 0 0 0 1 1 1 0 0 1 , 1 1 ( ) . 1 1 q q q j q q j q q m q j

q p p

j

J m T g T

G x y b

d m l T

T y x

β β

β

β β

ψ

β β

σ ψ σ ξ

β β

=

=

 

≤ + + _{ }

Γ + Γ _{+  }

 

+ + − + _{ }

Γ + Γ _{+  }

∑

∑

(16)

(

)(

)

(

)

(

)

(

) (

)

₍

₎

₍

₎

(

)

(

)

(

)

(

) (

)

1 1 0 0 1 1 0 0 0 0 1 1

, , , , 1 ,

1 1

,

1 1

1 , .

p q p q q p p q i j i j

I n T J m T

F G x y F x y G x y A

f T g T

a b B x y

A C B

α β

α β

φ ψ

α β

α β

φ ψ δ δ

= =

= + ≤ + + +

Γ + Γ +

 

 

+ _{ } + _{ } +

Γ +   Γ +  

≤ + + + ≤

∑

∑

Secondly, we prove that

(

F G,

)

:X× →Y X×Y is a contraction mapping. Let

(

x y,

) ( )

, u v, ∈ ×X Y, similar to the above process, we get

(

)( ) (

)( )

( )

( )

( )

( )

(

)

( )

(

)

(

)

(

(

( )

)

(

( )

)

)

(

)

( )

(

)

(

)

(

(

( )

)

(

( )

)

)

1 1 0

0 1 1

1

0

0 1 1

, , , , , , , , d sup d sup p q p p n m t

i i i ij j j j j

t T i j

q

m _t n

j j j ji i i i i

t T j i

F G x y F G u v F x y F u v G x y G u v

t s

c x s u s a f y s f v s s

t s

d y s v s b g x s g u s s

α β σ σ α σ σ β − ≤ ≤ = = − ≤ ≤ = = − = − + −  _{−  }    = _{ }− _ − − − _+ − _{ }

Γ _{ }

 

 

 ₋ 

 

 _{ } 

+  _{Γ }− _ − − − _+ − _ 

   

∑

∫

∑

∑

∫

∑

(

)

(

)

(

)

( )

(

)

( )

( )

( )

(

)

(

)

(

)

( )

1 1 1 0 0 1 1 1 0

0 1 1

1 1 0 0 1 0 d sup d sup d sup sup q p p n _t

i i i

t T i

p p n m

t ij j

j j

t T i j

q q m _t

j j j

t T j

t T j

t s c x s u s

s

t s a L

y s v s s

t s d y s v s

s α α β σ σ α α σ σ β − ≤ ≤ = − ≤ ≤ = = − ≤ ≤ = ≤ ≤   _{ − − − − }   _{ }  ≤ _{ }

 Γ 

       _{ − }   _{ }  + _ − _

 Γ 

       _{ − − − − }   _{ }  + _{ }

 Γ 

      +

∑ ∫

∑ ∑

∫

∑ ∫

(

)

( )

( )

( )

(

) (

)

(

)

(

) (

)

(

)

( ) ( )

( ) ( )

1 1 0 1 1 1 1 1 1

0 0 0 0

1 1

d

max , , ,

1 1 1 1

, , ,

q q m _t n

ji i

i i

i

q p

p m q n

q p

j i

j i

t s b l

x s u s s

c n l T d m L T

T T x y u v

B x y u v

β

β α

α β

β

σ ξ σ η

α β β α

−

= =

= =

 _{ − } 

 _{ − } 

 _{ Γ } 

              ≤ _ − + _{ } − + _{  } −

Γ + Γ +   Γ + Γ _{+ }  

 



= −

∑ ∑

∫

∑

∑

By (6), we conclude that

(

F G,

)

is a contraction mapping. It follows from the contraction mapping principle that system (1) has a unique solution. The proof is completed.

Theorem 4. Assume that (H2) holds. If there exist real numbers p q, >1 such that

(

) (

)

(

) (

)

1 1

0 0

max , 1,

1 1

p q

c n d m

T σ α T σ β

α β

 

 _{− −} _<

_{Γ + Γ +} 

 

  (17)

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

[ ]

0,T .

Proof. Let

(

) (

)

₍

₎

₍

₎

₍

₎

₍

₎

(

)

(

)

1 1

1 1

0 0 2 1

1 1

1 1

0 0

1 ,

1 1 1 1

.

1 max ( ) , ( )

1 1

q p

p q m n

q p

j i

j i

p q

I n T J m T M T M T

A

c n d m

T T

α β β α

α β

φ ψ ξ η

α β β α

δ σ σ α β = =     + + + + _{ } + _{ }

Γ + Γ + Γ +   Γ _{+ } 

≥

 

 

− _{Γ +} − _{Γ +} − 

 

 

∑

∑

Define two operators

( )

L L, and

(

N N,

)

on B_δ as follows

( )

( )( )

(

( )( )

( )( ) ( )( )

( )( )

)

T

1 1

, , , , , n , , , , , m , ,

L L x y t = L x y t  L x y t L x y t  L x y t

(

)

( )( )

(

( )( )

( )( )

( )( )

( )( )

)

T

1 1

, , , , , n , , , , , m , ,

N N x y t = N x y t  N x y t N x y t  N x y t

where

( )( )

,

( )

0 ₀t

(

_{( )}

)

1

(

)

d ,

i i i i i

t s

L x y t c x s I s

α

φ

σ

α

− − = + _− − + _ Γ

∫

( )( )

₀

(

_{( )}

)

1

(

( )

)

1

, d , 1, , ,

m t

i ij j j

j

t s

N x y t a f y s s i n

α

α

− = − = = Γ

∑

∫



( )( )

,

( )

0 ₀t

(

_{( )}

)

1

(

)

d ,

j j j j j

t s

L x y t d y s J s

β

ψ

σ

β

− − _{ } = + _− − + _ Γ

∫

( )( )

₀

(

_{( )}

)

1

(

( )

)

1

, d , 1, , .

n t

j ji i i

i

t s

N x y t b g x s s j m

β

β

− = − = = Γ

∑

∫



Firstly, we will prove

( )

L L,

( )

x y, +

(

N N,

)

( )

x y, ∈Bδ. In fact, using Minkowski inequality and (18) gives

that

( )

( )

(

)

( )

( )

(

_{( )}

)

(

)

(

( )

)

( )

(

_{( )}

)

(

)

(

( )

)

( )

( )

1 1 0

0 1 1

1 1

0

0 1 1

1 1

1 1

, , , ,

sup 0 d

sup 0 d

0 0

p p

n _t m

i i i ij j j i

t T i j

q q

m _t n

i j j ji i i j

t T j i

q p

n _p m _q

i i

i j

L L x y N N x y

t s

c x s a f y s I s

t s

d y s b g x s J s

α β φ σ α ψ σ β φ ψ − ≤ ≤ = = − ≤ ≤ = = = = +  _{−  }    = _ + _− − + + _{ }

Γ _{ }

 

 

 _{−  } 

 

+  + _− − + + _ 

Γ  

        ≤_{ } +_{ } +    

∑

∫

∑

∑

∫

∑

∑

∑

(

_{( )}

)

(

)

(

)

( )

(

)

(

)

( )

(

( )

)

(

)

1 1 0 0 1 1 1 1 1 0 0

0 1 0 1 1

1

0

0 1 1

sup d

sup d sup d

sup p p n _t i i t T i

p

q _p

q

m _t n _t m

ij j

j j j

t T j t T i j

m n t ji t T j i

c t s

x s s

a t s

d t s

y s s f y s s

b t s

α α β β σ α σ β α − ≤ ≤ = − − ≤ ≤ = ≤ ≤ = = − ≤ ≤ = =  _{ − }   _{ − }   

 _ Γ _ 

     _{ − }  _{ − }         + _ − _ + _{ }

 Γ   Γ 

            − + Γ

∑ ∫

∑

∫

∑ ∑

∫

∑ ∑

∫

_{( )}

(

( )

)

(

_{( )}

)

(

)

( )

(

) (

)

₍

₎

₍

₎

₍

_{) (}

)

₍

_{) (}

)

1 1 1 0 0 1 1 1 0 0 1

1 1 1 1

0 0 0 0

d sup d

sup d

1 , max ,

1 1 1 1

q p

q p

n t

i i i

t T i

q q m _t

j t T j

p q p q

t s

g x s s I s

t s

J s

I n T J m T c n d m

A T T

M α β α β α β β α β

φ ψ σ σ δ

α β α β

− ≤ ≤ = − ≤ ≤ =       ₋       +    

 _ _   _ Γ _ 

   

 

 _{ − } 

 _{ } 

+ _{ }

 _ Γ _ 

 

 

 

≤ + + + + _ − − _

Γ + Γ + _Γ + Γ + _

+

∑ ∫

∑ ∫

(

)

(

)

1 1 2 1 1 1 . 1 1 q p m n q p j i j i

Tβ _ξ M Tα _{η δ}

β = α =

  ₊   _≤

   

Thus, we conclude that

( )

L L,

( )

x y, +

(

N N,

)

( )

x y, ∈B_δ. Secondly, for any

(

x u,

) (

, y v,

)

∈Bδ, we have

( )

( )

( ) ( )

( )

( )

(

)

( )

(

)

(

)

(

)

( )

(

)

(

)

(

) (

)

(

) (

) ( ) ( )

1 1 0 0 1 1 1 0 0 1 1 1 0 0 , , , , , , d sup d sup

max , , , ,

1 1 p q p p n t _i i i

t T i

q q m _t

j

j j

t T j

p q

L L x y L x y L u v L x y L x y

c t s

x s u s s

d t s

y s v s s

c n d m

T T x u y v

α β α β σ σ α σ σ β σ σ α β − ≤ ≤ = − ≤ ≤ = = − + −  _{ − }   _{ }  ≤ − − −  

 _ Γ _ 

 

 _{ − } 

 _{ } 

+ _ − − − _

 Γ 

          ≤  − −  −

Γ + Γ +

 

 

∑ ∫

∑ ∫

which implies that

( )

L L, is a contraction mapping by (17).

Thirdly, we prove that

(

N N,

)

is continuous and compact. Since f_j,g_i, j=1,,m, i=1,,n, are con- tinuous, it is obvious that

(

N N,

)

is also continuous. Let

(

x y,

)

∈Bδ, we get by (H2) that

(

)

( )

( )

( )

(

)

( )

(

( )

)

(

)

( )

(

( )

)

(

)

(

)

1 1 1 1 0 0

0 1 1 0 1 1

1 1 2 1 1 1 , , , , d d sup sup , 1 1 p q p q p q

n _tm m _tn

ij ji

j j i i

t T i j t T j i

q p

m n

q p

j i

j i

N N x y N x y N x y

a t s b t s

f y s s g x s s

M T M T

α β β α α β ξ η β α − − ≤ ≤ = = ≤ ≤ = = = = = +  _{ − }   _{ − }   _{ }   _{ }  = _{ } + _{ }

 Γ   Γ 

   

   

   

   

≤ _{ } + _{ }

Γ +   Γ _{+ } 

∑ ∑

∫

∑ ∑

∫

∑

∑

which implies that

(

N N,

)

is uniformly bounded on Bδ. Moreover, we can show that

(

N N,

)

( )( )

x y t, is

equicontinuous. In fact, for

(

x y,

)

∈Bδ, 0< <t2 t1, we obtain

(

)

( )( )

(

)

( )( )

( )( )

( )( )

( )( )

( )( )

(

)

( )

(

( )

)

(

)

( )

(

( )

)

(

)

( )

(

( )

)

(

)

( )

(

( )

)

1 2 1 2 1 2

1 2 1 2

1

1 1

1 2

0 0

1 1 1

1 1

1 2

0 0

1 1 1

, , , , , , , , d d d d p q p p

n _t m _t m

ij ij

j j j j

i j j

m _t n _t n

ji ji

i i i i

j i i

N N x y t N N x y t

N x y t N x y t N x y t N x y t

a t s a t s

f y s s f y s s

b t s b t s

g x s s g x s s

α α β β α α β β − − = = = − − = = = − = − + −  _{ − − }   _{ }  ≤_ − _

 Γ Γ 

       _{− −}   + −

 Γ Γ

 

∑ ∑

∫

∫

∑

∑ ∑

∫

∫

∑

(

)

(

)

( )

(

( )

)

(

)

(

)

( )

(

( )

)

(

)

2 1 2 2 1 2 1 1

1 1 1

2 1 1

1 ₀

1 1

1

1 1 1

2 1 1

2 ₀ 1 1 2 =1 d d d d 1 q q p p

n m _{t t}

ij _t

i j

q q

m n _{t t}

ji _t

j i

m

j

t s t s t s

M a s s

t s t s t s

M b s s

M T

α α α

β β β

β α α β β ξ β − − − = = − − − = =    _   _       _{  − − − − }   _{  }  ≤_ + _  

 _ Γ Γ _

       _{  − − − − }   _{  }  +_{ } + _{ }

 _ Γ Γ _

      ≤ Γ +

∑ ∑

∫

∫

∑ ∑

∫

∫

∑

1

(

)

₍

1

₎

1

(

)

1 2 2 1 1 2 2 1

=1

2 2 0,

1 q p n q p j i i M T

t t t t t t t t

α

β β β _η α α α

α

  _{− + − +}   _{− + − →}

  _{Γ + } 



 

∑

by Lemma 2 we have that system (1) has at least one solution.

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

[ ]

0,T .

Proof. Assume that

(

x t

( ) ( )

,y t

)

T and

(

u t

( ) ( )

,v t

)

T are any two solutions of system (1) with the initial con- ditions

(

φ ψ,

)

and

( )

φ ψ

, , respectively. Then

( ) ( )

(

)

T

(

)( )( )

(

( ) ( )

)

T

(

)( )( )

, , , , , , , ,

x t y t = F G x y t u t v t = F G u v t

that is,

( )

( )

(

_{( )}

)

(

)

(

( )

)

( )

( )

(

_{( )}

)

(

)

(

( )

)

1 0 1 1 0 1

0 d ,

0 d ,

m t

i i i i ij j j i

j n t

j j j j ji i i j

i

t s

x t c x s a f y s I s

t s

y t d y s b g x s J s

α β φ σ α ψ σ β − = − =  _{−  } = + − − + +  _{ } Γ     −  _{= +} _{− − + +} 

 _Γ _ _



∑

∫

∑

∫

( )

( )

(

_{( )}

)

(

)

(

( )

)

( )

( )

(

_{( )}

)

(

)

(

( )

)

1 0 1 1 0 1

0 d ,

0 d ,

m t

i i i i ij j j i

j n t

j j j j ji i i j

i

t s

u t c u s a f v s I s

t s

v t d v s b g u s J s

α β φ σ α ψ σ β − = − =  _{−  } = + − − + +  _{ } Γ     −  _{= +} _{− − + +} 

 _Γ _ _



∑

∫

∑

∫

which implies

( ) ( )

(

)

( )

(

)

(

)

(

)

( )

(

( )

)

(

( )

)

(

)

( )

(

)

(

)

1 1 0 0 1 1 1 0

0 1 1

1 0 0 1 , , d sup d sup d sup p q p p n _t i i i p

t T i

p p n _t m

ij

j j j j

t T i j

m t j

j j

q

t T j

x y u v x u y v

c t s

x s u s s

a t s

f y s f v s s

d t s

y s v s s

α

α

β

φ φ σ σ

α

α

ψ ψ σ σ

β − ≤ ≤ = − ≤ ≤ = = − ≤ ≤ = − = − + −  _{ − }   _{ }  ≤ − + − − −  

 _ Γ _ 

 

 _{ − } 

 _{ } 

+ _ − _

 Γ 

       ₋    + − + − − −  Γ  

∑ ∫

∑ ∑

∫

∑ ∫

(

)

( )

(

( )

)

(

( )

)

(

)

( )

₍

_{) (}

)

₍

₎

(

)

(

)

(

)

1 1 1 0

0 1 1

1 1 0 0 1 1 1 0 0 1 d sup , , 1 1 ( ) 1 1 , , q q q q m _t n

ji

i i i i

t T j i

q p m q j p j p q n p i q i

b t s

g x s g u s s

c n l T

T x u

d m L T

T y v

β β α α β β

φ ψ φ ψ σ ξ

α β

σ η

β α

φ ψ φ

− ≤ ≤ = = = =             _{ − }   _{ }  + _ − _

 Γ 

       _{ }    ≤ − + − + _{ } −

Γ + Γ _{+  } 

 

 _{ } 

+_ − + _{ } _ −

Γ + Γ _{+ } 

 

≤ −

∑ ∑

∫

∑

∑

( )

ψ +B x u

( ) ( )

, − y v, . Hence, we have

( ) ( )

1

(

)

( )

, , , , .

1

x y u v

B φ ψ φ ψ

− ≤ −

− (19)

( ) ( )

x y, − u v, <

ε

,

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

[ ]

0,T .

4. An Illustrative Example

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

( )

(

)

(

( )

)

(

( )

)

( )

(

)

(

( )

)

(

( )

)

( )

(

)

(

( )

)

(

( )

)

( )

(

)

(

( )

)

(

( )

)

1 1 1 1 2 2

2 2 1 1 2 2

1 1 1 1 2 2

2 2 1 1 2 2

0.3 0.2 0.1 0.4,

0.2 0.3 0.2 0.3,

0.4 0.4 0.2 0.5,

0.5 0.1 0.3 0.2

C C C C

D x t x t f y t f y t

D x t x t f y t f y t

D y t y t g x t g x t

D y t y t g x t g x t

α α β β

σ

σ

σ

σ

 = − − − + +



= − − + + −



 _{= − − + + −}



 _{= − − + − +}



(20)

with the initial condition

( )

( )

( )

( )

( )

( )

( )

( )

[

]

1 1 , 2 2 , 1 1 , 2 2 , , 0 ,

x t =φ t x t =φ t y t =ψ t y t =ψ t t∈ −σ

where

φ ψ

i, i∈C

(

[

−

σ

, 0 ,

]



)

, i=1, 2, α =0.8,β =0.9,

σ

=0.3, n= =m 2,

( )

(

)

1

1 1

2

j j j j

f x = x + − x − ,

( )

tanh

( )

i i i

g x = x

(

i j, =1, 2

)

. Let T=1, p= =q 2, from (3)-(5), it is easy to check that

0 0.3, 0 0.5, 10 0.3, 20 0.5, 10 0.6, 10 0.4,

c = d = a = a = b = b =

2 2 2 2

0 0 1, 1 11 12 0.2236, 2 21 22 0.3606,

L = =l η = a +a = η = a +a =

2 2 2 2

1 b11 b12 0.4472, 2 b21 b22 0.3162.

ξ = + = ξ = + =

Thus,

(

)

(

)

(

)

(

)

( )

( )

( )

( )

1 1

1 1

0 0 0 0

1 1

0.8 2 2 0.9 2 2

max ( ) , ( )

1 1 1 1

0.3 2 0.7 0.4472 0.3162 0.5 2 0.7 0.2236 0.3606

max ,

1.8 1.9 1.9 1.9

0.9745 1,

q p

p m q n

q p

j i

j i

c n l T d m L T

B T T

β α

α β

σ ξ σ η

α β = β α =

     

 

= _ − + _{ } − + _{  }

Γ + Γ +   Γ + Γ +   

 



 _{× × + × × +} 

 

=  _Γ + _{Γ Γ} + _Γ 

 

 

= <

∑

∑

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

[ ]

0,1 .

In the following, we show the simulation result for model (20). We consider four cases: Case 1 with the initial values

( ) ( ) ( )

( )

(

)

T

(

)

T

1 t , 2 t , 1 t , 2 t 0.3,1.5, 0.9, 0.8

φ φ ψ ψ ≡ − − for t∈ −

[

0.3, 0

]

, Case 2 with the initial values

( ) ( ) ( )

( )

(

)

T

(

)

T

1 t , 2 t , 1 t , 2 t 1.2, 0.9,1.4, 1.3

φ φ ψ ψ ≡ − − for t∈ −

[

0.3, 0

]

, Case 3 with the initial values

( ) ( ) ( )

( )

(

)

T

(

)

T

1 t , 2 t , 1 t , 2 t 0.9,1.6, 0.7,1.5

φ φ ψ ψ ≡ for t∈ −

[

0.3, 0

]

,

Case 4 with the initial values

( ) ( ) ( )

( )

(

)

T

(

)

T

1 t , 2 t , 1 t , 2 t 0.5, 1.3, 1.9, 2.2

(a) (b)

[image:11.595.71.542.78.461.2]

(c) (d)

Figure 1.Transient states of the fractional-order BAM neural networks (20) with α =0.8, β =0.9, and σ =0.3.

Acknowledgements

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

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