Finite-Time Stability of Fractional-Order BAM Neural Networks with Distributed Delay

and Applied Analysis 3 mapping the interval [−τ, 0] into Rn with the norm defined as follows: for every φ ∈ C1([−τ, 0],R), 󵄩 󵄩 󵄩 󵄩 φ 󵄩 󵄩 󵄩 󵄩1 = max {󵄩󵄩󵄩 󵄩 φ 󵄩 󵄩 󵄩 󵄩 , 󵄩 󵄩 󵄩 󵄩 󵄩 φ 󸀠 󵄩 󵄩 󵄩 󵄩 󵄩 } = max{ sup θ∈[−τ,0] 󵄨 󵄨 󵄨 󵄨 φ (θ) 󵄨 󵄨 󵄨 󵄨 , sup θ∈[−τ,0] 󵄨 󵄨 󵄨 󵄨 󵄨 φ 󸀠 (θ) 󵄨 󵄨 󵄨 󵄨 󵄨 } . (10) The initial conditions associated with (6) are given by x i (θ) = φ i (θ) , x 󸀠 i (θ) = φ 󸀠 i (θ) , y j (θ) = ψ j (θ) , y 󸀠 j (θ) = ψ 󸀠 j (θ) , θ ∈ [−τ, 0] , (11) where φ i , ψ j ∈ C 1 ([−τ, 0],R). In order to obtain main result, we make the following assumptions. (H1) For i, j = 1, . . . , n, the functions r ij (⋅) and p ji (⋅) are continuous on [0, τ]. (H2) The neurons activation functions f i and g j (i, j = 1, . . . , n) are bounded. (H3) The neurons activation functions f i and g j are Lipschitz continuous; that is, there exist positive constants h i , l j (i, j = 1, . . . , n) such that 󵄨 󵄨 󵄨 󵄨 f i (u) − f i (V)󵄨󵄨󵄨 󵄨 ≤ h i |u − V| , 󵄨 󵄨 󵄨 󵄨 󵄨 g j (u) − g j (V) 󵄨 󵄨 󵄨 󵄨 󵄨 ≤ l j |u − V| , ∀u, V ∈ R. (12) Since the Caputo’s fractional derivative of a constant is equal to zero, the equilibriumpoint of system (6) is a constant vector (x∗, y∗) = (x∗ 1 , x ∗ 2 , . . . , x ∗ n , y ∗ 1 , y ∗ 2 , . . . , y ∗ n ) T ∈ R2n which satisfies the system


Introduction
Fractional calculus (integral and differential operations of noninteger order) was firstly introduced 300 years ago.Due to lack of application background and the complexity, it did not attract much attention for a long time.In recent decades fractional calculus is applied to physics, applied mathematics, and engineering [1][2][3][4][5][6].Since the fractionalorder derivative is nonlocal and has weakly singular kernels, it provides an excellent instrument for the description of memory and hereditary properties of dynamical processes.Nowadays, study on the complex dynamical behaviors of fractional-order systems has become a very hot research topic.
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, fractional-order neural networks have been presented and designed to distinguish the classical integerorder models [7][8][9][10].Currently, some excellent results about fractional-order neural networks have been investigated, such as Kaslik and Sivasundaram [11,12], Zhang et al. [13], Delavari et al. [14], and Li et al. [15,16].On the other hand, time delay is one of the inevitable problems on the stability of dynamical systems in the real word [17][18][19][20].But till now, there are few results on the problems for fractional-order delayed neural networks; Chen et al. [21] studied the uniform stability for a class of fractional-order neural networks with constant delay by the analytical approach; Wu et al. [22] investigated the finite-time stability of fractional-order neural networks with delay by the generalized Gronwall inequality and estimates of Mittag-Leffler functions; Alofi et al. [23] discussed the finite-time stability of Caputo fractional-order neural networks with distributed delay.
The integer-order bidirectional associative memory (BAM) model known as an extension of the unidirectional autoassociator of Hopfield [24] was first introduced by Kosko [25].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 extensively studied [26][27][28][29].However, to the best of our knowledge, there is no effort being made in the literature to study the finite-time stability of fractional-order BAM neural networks so far.
Motivated by the above-mentioned works, we were devoted to establishing the finite-time stability of Caputo fractional-order BAM neural networks with distributed

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  > 0 of a function  : (0, ∞) →  is given by provided that the right side is pointwise defined on (0, ∞), where Γ(⋅) is the Gamma function.(2) Definition 3. The Mittag-Leffler function in two parameters is defined as where  > 0,  > 0, and  ∈ C, where C denotes the complex plane.In particular, for  = 1, one has The Laplace transform of Mittag-Leffler function is where  and  are, respectively, the variables in the time domain and Laplace domain and L{⋅} stands for the Laplace transform.
In this paper, we are interested in the finite-time stability of fractional-order BAM neural networks with distributed delay by the following state equations: or in the matrix-vector notation where 1 < ,  < 2. The model ( 6) is made up of two neural fields   and   , where   () and   () are the activations of the th neuron in   and the th neuron in   , respectively; is the state vector of the network at time ; the functions are the activation functions of the neurons at time ;  = diag(  ) is a diagonal matrix;   > 0 represents the rate with which the th unit will reset its potential to the resting state in isolation when disconnected from the network and external The initial conditions associated with ( 6) are given by where In order to obtain main result, we make the following assumptions.
(H3) The neurons activation functions   and   are Lipschitz continuous; that is, there exist positive constants ℎ  ,   (,  = 1, . . ., ) such that Since the Caputo's fractional derivative of a constant is equal to zero, the equilibrium point of system (6) By using the Schauder fixed point theorem and assumptions (H1)-(H3), it is easy to prove that the equilibrium points of system (6) exist.We can shift the equilibrium point of system (6) to the origin.Denoting then system (6) can be written as with the initial conditions where Similarly, by using the matrix-vector notation, system (15) can be expressed as with the initial condition where Define the functions as follows: where ,  = 1, . . ., .From assumption (H3), we can obtain |ℎ  ()| ≤ ℎ  , |  ()| ≤   .By (21), we have Thus, system (18) can be further written as the following form: where () = diag{ℎ  ()}, () = diag{  ()}. implies where  is a positive real number and  > 0,  < ,  0 denotes the initial time of observation of the system, and  denotes time interval  = [ 0 ,  0 + ).
A technical result about norm upper-bounding function of the matrix function  , is given in [30] where − ( > 0) is the largest eigenvalue of the diagonal stability matrix .

Main Result
We first give a key lemma in the proof of our main result as follows.
as follows.