Periodic Solutions of a Cohen-Grossberg-Type BAM Neural Networks with Distributed Delays and Impulses

Abstract

A class of Cohen-Grossberg-type BAM neural networks with distributed delays and impulses are investigated in this paper. Sufficient conditions to guarantee the uniqueness and global exponential stability of the periodic solutions of such networks are established by using suitable Lyapunov function, the properties of 𝑀-matrix, and some suitable mathematical transformation. The results in this paper improve the earlier publications.

1. Introduction

The research of neural networks with delays involves not only the dynamic analysis of equilibrium point but also that of periodic oscillatory solution. The dynamic behavior of periodic oscillatory solution is very important in learning theory due to the fact that learning usually requires repetition [1, 2].

Cohen and Grossberg proposed the Cohen-Grossberg neural networks (CGNNs) in 1983 [3]. Kosko proposed bi directional associative memory neural networks (BAMNNs) in 1988 [4]. Some important results for periodic solutions of delayed CGNNs have been obtained in [5–10]. Xiang and Cao proposed a class of Cohen-Grossberg BAM neural networks (CGBAMNNs) with distributed delays in 2007 [11]; in addition, many evolutionary processes are characterized by abrupt changes at certain time; these changes are called to be impulsive phenomena, which are included in many neural networks such as Hopfield neural networks, BAM neural networks, CGNNs, and CGBAMNNs and can affect dynamical behaviors of the systems just as time delays. The results for periodic solutions of CGBAMNNs with or without impulses are obtained in [11–15].

The objective of this paper is to study the existence and global exponential stability of periodic solutions of CGBAMNNs with distributed delays by using suitable Lyapunov function, the properties of 𝑀-matrix, and some suitable mathematical transformation. Comparing with the results in [13, 14], improved results are successively obtained, the conditions for the existence and globally exponential stability of the periodic solution of such system without impulses have nothing to do with inputs of the neurons and amplification functions; and we also point that CGBAMNNs model is a special case of CGNNs model, many results of CGBAMNNs can be directly obtained from the results of CGNNs.

The rest of this paper is organized as follows. Preliminaries are given in Section 2. Sufficient conditions which guarantee the uniqueness and global exponential stability of periodic solutions for CGBAMNNs with distributed delays and impulses are given in Section 3. Two examples are given in Section 4 to demonstrate the main results.

2. Preliminaries

Consider the following periodic CGNNs model with distributed delays and impulses:𝑥̇_𝑖(𝑡)=−𝑎_𝑖𝑥_𝑖(𝑡)𝑏_𝑖𝑡,𝑥_𝑖(𝑡)−^𝑛_𝑗=1−𝑝_𝑖𝑗(𝑡)𝑓_𝑗𝜌_𝑗𝑥_𝑗(𝑡)_𝑛_𝑗=1Δ𝑥_𝑖𝑢_𝑖𝑗(𝑡)^0+∞𝑘_𝑖𝑗(𝑠)𝑓_𝑗𝜌_𝑗x_𝑗(𝑡−𝑠)𝑑𝑠−𝐼_𝑖(𝑡),𝑡>0,𝑡≠𝑡_𝑘,𝑡_𝑘=−𝛾_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑡=𝑡_𝑘,𝑘∈𝑍⁺,(2.1) where 1≤𝑖≤𝑛, 𝑡>0, and 𝑍⁺={1,2,…}. 𝑥_𝑖(𝑡) denotes the state variable of the 𝑖th neuron, 𝑓_𝑗(⋅) denotes the signal function of the 𝑗th neuron at time 𝑡; 𝐼_𝑖 denotes input of the 𝑖th neuron at time 𝑡; 𝑎_𝑖(⋅) represents amplification function; 𝑏_𝑖(𝑡,⋅) is appropriately behaved function; 𝑝_𝑖𝑗(𝑡) and 𝑢_𝑖𝑗(𝑡) are connection weights of the neural networks at time 𝑡; respectively, 𝜌_𝑗 is positive constant, which corresponds to the neuronal gain associated with the neuronal activations and 𝑘_𝑖𝑗 corresponds to the delay kernel function; 𝑝_𝑖𝑗(𝑡) and 𝑢_𝑖𝑗(𝑡) are continuously periodic functions on [0,+∞) with common period 𝑇>0.

Δ𝑥_𝑖(𝑡_𝑘)=𝑥_𝑖(𝑡^+𝑘)−𝑥_𝑖(𝑡^−𝑘); 𝑡_𝑘 is called impulsive moment and satisfies 0<𝑡₁<𝑡₂<⋯,lim_𝑘→+∞𝑡_𝑘=+∞; 𝑥_𝑖(𝑡⁻) and 𝑥_𝑖(𝑡⁺) denote the left-hand and right-hand limits at 𝑡_𝑘; respectively, we always assume 𝑥_𝑖(𝑡^−𝑘)=𝑥_𝑖(𝑡_𝑘) and 𝑥^𝑖(𝑡^−𝑘)=𝑥^𝑖(𝑡_𝑘),𝑘∈𝑍⁺.

For system (2.1), we assume the following.(H₁)The amplification function 𝑎_𝑖(⋅) is continuous, and there exist constants 𝑎_𝑖,𝑎_𝑖 such that 0<𝑎_𝑖≤𝑎_𝑖(𝑥_𝑖(𝑡))≤𝑎_𝑖 for 1≤𝑖≤𝑛.(H₂)The behaved function 𝑏_𝑖(𝑡,⋅) is 𝑇-periodic about the first argument; there exists continuous 𝑇-periodic function 𝛼_𝑖(𝑡) such that𝑏_𝑖(𝑡,𝑥)−𝑏_𝑖(𝑡,𝑦)𝑥−𝑦≥𝛼^𝑖(𝑡)>0,(2.2)for all 𝑥≠𝑦,1≤𝑗≤𝑛.(H₃)For activation function 𝑓_𝑗(⋅), there exists positive constant 𝐿_𝑗 such that𝐿_𝑗=sup_𝑥≠y||||𝑓_𝑗(𝑥)−𝑓_𝑗(𝑥)𝑥−𝑦||||,(2.3)for all 𝑥≠𝑦, 1≤𝑗≤𝑛.(H₄) The kernel function 𝑘_𝑖𝑗(𝑠) is nonnegative continuous function on [0,+∞) and satisfies^0+∞𝐾𝑠𝑒^𝜆𝑠𝑘_𝑖𝑗(𝑠)𝑑𝑠<+∞,_𝑖𝑗(𝜆)=^0+∞𝑒^𝜆𝑠𝑘_𝑖𝑗(𝑠)𝑑𝑠(2.4)is differentiable function for 𝜆∈[0,𝑟_𝑖𝑗),0<𝑟_𝑖𝑗<+∞, 𝐾_𝑖𝑗(0)=1 and lim_𝜆→𝑟^−𝑖𝑗𝐾_𝑖𝑗(𝜆)=+∞.(H₅)There exists positive integer 𝑘₀ such that 𝑡_𝑘+𝑘0=𝑡_𝑘+𝑇 and 𝛾_{𝑖(𝑘+𝑘0)}=𝛾_𝑖𝑘 hold.

Remark 2.1. A typical example of kernel function is given by 𝑘_𝑖𝑗(𝑠)=(𝑠^𝑟/𝑟!)𝑟^{𝑟+1𝑖𝑗}𝑒^{−𝑟𝑖𝑗𝑠} for 𝑠∈[0,+∞), where 𝑟_𝑖𝑗∈(0,+∞),𝑟∈{0,1,…,𝑛}. These kernel functions are called as the gamma memory filter [16] and satisfy condition (H₄).
For any continuous function 𝑆(𝑡) on [0,𝑇], 𝑆 and 𝑆 denote min_{𝑡∈[0,𝑇]}{|𝑆(𝑡)|} and max_{𝑡∈[0,𝑇]}{|𝑆(𝑡)|}, respectively.
For any 𝑥(𝑡)=(𝑥_𝑖(𝑡),𝑥₂(𝑡),…,𝑥_𝑘(𝑡))^𝑇∈𝑅^𝑘, 𝑡>0, define ‖𝑥(𝑡)‖=∑^𝑘𝑖=1|𝑥_𝑖(𝑡)|, and for any 𝜑(𝑠)=(𝜑₁(𝑠),𝜑₂(𝑠),…,𝜑_𝑘(𝑠))^𝑇∈𝑅^𝑘, 𝑠∈(−∞,0], define ‖𝜑(𝑠)‖=sup_{𝑠∈(−∞,0]}∑^𝑘𝑖=1|𝜑_𝑖(𝑠)|.
Denote PC(−∞,0],𝑅^𝑘={𝜓m𝜓os𝑠∶t[a−fi∞ni,te0n]u⟶mbe𝑅r^𝑘of∣𝜓po(i𝑠n)tsis𝑠b∈ou(n−d∞ed,0a]n,dacnodntaitnuthoeussefpoorianltlsb𝑠u,₊tat,𝜓(𝑠⁻)existand𝜓(𝑠⁻)=𝜓(𝑠)}.(2.5) Then PC((−∞,0],𝑅^𝑘) is a Banach space with respect to ‖⋅‖.
The initial conditions of system (2.1) are given by 𝑥_𝑖(𝑠)=𝜑_𝑖(𝑠),−∞≤𝑠≤0,1≤𝑖≤𝑛,(2.6) where 𝜑(𝑠)=(𝜑₁(𝑠),𝜑₂(𝑠),…,𝜑_𝑛(𝑠))∈PC([−∞,0],𝑅^𝑛).
Let 𝑥(𝑡,𝜑)=(𝑥₁(𝑡,𝜑),𝑥₂(𝑡,𝜑),…,𝑥_𝑛(𝑡,𝜑)^𝑇) denote any solution of the system (2.1) with initial value 𝜑∈PC((−∞,0],𝑅^𝑛).

Definition 2.2. A solution 𝑥(𝑡,𝜑) of system (2.1) is said to be globally exponentially stable, if there exist two constants 𝜆>0,𝑀>0 such that ‖𝑥(𝑡,𝜓)−𝑥(𝑡,𝜑)‖≤𝑀‖𝜓−𝜑‖𝑒^−𝜆𝑡,𝑡>0,(2.7) for any solutions 𝑥(𝑡,𝜓) of system (2.1).

Definition 2.3. A real matrix 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛is said to be a nonsingular 𝑀-matrix if 𝑎_𝑖𝑗≤0(𝑖,𝑗=1,2,…,𝑛,𝑖≠𝑗), and all successive principle minors of 𝐴 are positive.

Lemma 2.4 (see [17]). A matrix with nonpositive off-diagonal elements 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛 is a nonsingular 𝑀-matrix if and if only there exists a vector 𝑝=(𝑝_𝑖)_1×𝑛>0 such that 𝑝^𝑇𝐴>0 or 𝐴𝑝 holds.

Lemma 2.5. Under assumptions (H₁)–(H₅), system (2.1) has a 𝑇-periodic solution which is globally exponentially stable, if the following conditions hold.(H₆)ℳ=𝐴−𝐶 is a nonsingular 𝑀-matrix, where𝐴=diag𝛼₁,𝛼₂,…,𝛼_𝑛,𝐶=𝑐^𝑖𝑗_𝑛×𝑛,𝑐_𝑖𝑗=𝑝_𝑖𝑗+𝑢_𝑖𝑗_{𝜌𝑗𝐿𝑗}.(2.8)(H₇)𝑎_𝑖((1−𝛾_𝑖𝑘)𝑠)≥|1−𝛾_𝑖𝑘|𝑎_𝑖(𝑠), for all 𝑠∈R,𝑖=1,2,…,𝑛.

Proof. Let 𝑥(𝑡,𝜓₁) and 𝑥(𝑡,𝜓₂) be two solutions of system (2.1) with initial value 𝜓₁=(𝜑₁,𝜑₂,…,𝜑_𝑛) and 𝜓₂=(𝜁₁,𝜁₂,…,𝜁_𝑛)∈PC((−∞,0],𝑅^𝑛), respectively.
Let𝐹_𝑖(𝜃)=𝜇_𝑖𝛼^𝑖−𝜃𝑎_𝑖−^𝑛_𝑗=1𝜇_𝑗𝑝_𝑗𝑖+𝑢_𝑗𝑖𝐾_𝑗𝑖(𝜃)𝜌_𝑖𝐿_𝑖,𝑖=1,2,…,𝑛.(2.9) Since ℳ is a nonsingular 𝑀-matrix according to condition (H₆), ℳ^𝑇 is also a nonsingular 𝑀-matrix; we know from Lemma 2.4 that there exists a vector 𝑝=(𝜇₁,𝜇₂,…,𝜇_𝑛)^𝑇 such that ℳ^𝑇𝑝>0; that is, 𝜇_𝑖𝛼_𝑖−^𝑛_𝑗=1𝜇_𝑗𝑝_𝑗𝑖+𝑢_𝑗𝑖𝜌_𝑖𝐿_𝑖>0,(2.10) for 1≤𝑖≤𝑛, which indicates that 𝐹_𝑖(0)>0. Since 𝐹_𝑖(𝜃) are continuous and differential on [0,𝑟_𝑗𝑖) and lim_𝜃→𝑟^−𝑗𝑖𝐹_𝑖(𝜃)=−∞ according to condition (H₄), 𝐹^𝑖(𝜃)<0 for 𝜃∈[0,𝑢_𝑗𝑖). There exist constants 𝜃_𝑖 such that 𝐹_𝑖(𝜃_𝑖)=0 for 𝑖=1,2,…,𝑛. So we can choose 0<𝜆≤min𝜃₁,𝜃₂,…,𝜃_𝑛,(2.11) such that 𝐹_𝑖(𝜆)≥0.(2.12) Define 𝑋_𝑖(𝑡)=||𝑥_𝑖𝑡,𝜓₂−𝑥_𝑖𝑡,𝜓₁||.(2.13)
Now we define a Lyapunov function 𝑉(𝑡) by 𝑉(𝑡)=^𝑛_𝑖=1𝜇_𝑖𝑉^𝑖(𝑡)+^𝑛_𝑗=1𝑢_𝑖𝑗𝐿_𝑗𝜌_𝑗^0+∞𝑘_𝑖𝑗(𝑠)^{𝑡𝑡−𝑠}𝑋_𝑗(𝜇)𝑒^{𝜆(𝑠+𝜇)}𝑑𝜇𝑑𝑠,(2.14) in which 𝑉_𝑖(𝑡)=𝑒^𝜆𝑡sign𝑥_𝑖𝑡,𝜓₂−𝑥_𝑖𝑡,𝜓₁^{𝑥𝑖(𝑡,𝜓2)𝑥}_{𝑖𝑡,𝜓1}1𝑎_𝑖(𝑠)𝑑𝑠.(2.15) for 𝑖=1,2,…,𝑛.
When 𝑡≠𝑡_𝑘,𝑘∈𝑍⁺, calculating the upper right derivative of 𝑉(𝑡) along solution of (2.1), similar to proof of Theorem  3.1 in [10], corresponding to case in which 𝑟→1,𝑣_𝑖𝑗𝑙(𝑡)=0 in [10], we obtain from (2.12)–(2.15) that𝐷⁺𝑉(𝑡)∣_(2.1)≤−𝑒^𝜆𝑡^𝑛_𝑖=1𝜇_𝑖𝛼^𝑖−𝜆𝑎_𝑖𝑋^𝑖(𝑡)−^𝑛_𝑛𝑖=1_𝑗=1=−𝑒_𝜆𝑡𝜇_𝑖𝑝_𝑖𝑗+𝑢_𝑖𝑗𝐾_𝑖𝑗(𝜆)𝜌_𝑗𝐿_𝑗𝑋_𝑗(𝑡)^𝑛_𝑖=1𝜇_𝑖𝛼^𝑖−𝜆𝑎_𝑖𝑋^𝑖(𝑡)−^𝑛_𝑛𝑖=1_𝑗=1=−𝑒_𝑛𝜆𝑡𝜇_𝑖𝑝_𝑗𝑖+𝑢_𝑗𝑖𝐾_𝑗𝑖(𝜆)𝜌_𝑖𝐿_𝑖𝑋_𝑖(𝑡)_𝑖=1𝐹_𝑖(𝜆)𝑋_𝑖(𝑡)≤0.(2.16) When 𝑡=𝑡_𝑘,𝑘∈𝑍⁺, we have 𝑉_𝑖𝑡^+𝑘=𝑒^𝜆𝑡+sign𝑥_𝑖𝑡^+𝑘,𝜓₂−𝑥_𝑖𝑡^+𝑘,𝜓₁^{𝑥𝑖(𝑡+𝑘,𝜓2)𝑥}_{𝑖(𝑡+𝑘,𝜓1)}1=𝑒_𝜆𝑡𝑎_𝑖(𝑠)𝑑𝑠sign1−𝛾_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝜓₂−𝑥_𝑖𝑡_𝑘,𝜓₁^{(1−𝛾𝑖𝑘)𝑥𝑖(𝑡𝑘,𝜓2)(1−𝛾}_{𝑖𝑘)𝑥𝑖(𝑡𝑘,𝜓1)}1=𝑒_𝜆𝑡𝑎_𝑖(𝑠)𝑑𝑠,sign𝑥_𝑖𝑡_𝑘,𝜓₂−𝑥_𝑖𝑡_𝑘,𝜓₁^{𝑥𝑖(𝑡𝑘,𝜓2)𝑥}_{𝑖(𝑡𝑘,𝜓1)}||1−𝛾_𝑖𝑘||𝑎_𝑖1−𝛾_𝑖𝑘𝑠𝑑𝑠,(2.17) which, together with (H₇), leads to 𝑉_𝑖𝑡_𝑘−𝑉_𝑖𝑡^+𝑘=𝑒^𝜆𝑡sign𝑥_𝑖𝑡_𝑘,𝜓₂−𝑥_𝑖𝑡_𝑘,𝜓₁^{𝑥𝑖(𝑡𝑘,𝜓2)𝑥}_{𝑖(𝑡𝑘,𝜓1)}𝑎_𝑖1(𝑠)−||1−𝛾^𝑖𝑘||≥𝑒_𝜆𝑡𝑎_𝑖1−𝛾_𝑖𝑘𝑠𝑑𝑠,sign𝑥_𝑖𝑡_𝑘,𝜓₂−𝑥_𝑖𝑡_𝑘,𝜓₁^{𝑥𝑖(𝑡𝑘,𝜓2)𝑥}_{𝑖(𝑡𝑘,𝜓1)}𝑎_𝑖1−𝛾_𝑖𝑘𝑠−||1−𝛾_𝑖𝑘||𝑎_𝑖(𝑠)𝑎_𝑖(𝑠)𝑎_𝑖1−𝛾_𝑖𝑘𝑠𝑑𝑠≥0,(2.18) that is, 𝑉_𝑖𝑡^+𝑘≤𝑉_𝑖𝑡_𝑘.(2.19) It follows that 𝑉𝑡^+𝑘=^𝑛_𝑖=1𝜇_𝑖𝑉^𝑖≤𝑡⁺+^𝑛_𝑗=1𝑢_𝑖𝑗𝐿_𝑗𝜌_𝑗^0+∞𝑘_𝑖𝑗(𝑠)^𝑡+_𝑡+−𝑠𝑋_𝑗(𝜇)𝑒^{𝜆(𝑠+𝜇)}𝑑𝜇𝑑𝑠_𝑛_𝑖=1𝜇_𝑖𝑉^𝑖(𝑡)+^𝑛_𝑗=1𝑢_𝑖𝑗𝐿_𝑗𝜌_𝑗^0+∞𝑘_𝑖𝑗(𝑠)^{𝑡𝑡−𝑠}𝑋_𝑗(𝜇)𝑒^{𝜆(𝑠+𝜇)}𝑑𝜇𝑑𝑠=𝑉𝑡_𝑘.(2.20) Then we have 𝑉(𝑡)≤𝑉(0).(2.21) On the other hand, from (2.14), we have 𝑉(𝑡)≥𝑚₀𝑒^𝑛𝜆𝑡_𝑖=1||𝑥_𝑖𝑡,𝜓₂−𝑥_𝑖𝑡,𝜓₁||,𝑉(0)≤𝑀₀sup_{𝑛𝑙∈(−∞,0]}_𝑖=1||𝜑_𝑖(𝑙)−𝜁_𝑖(𝑙)||,(2.22) in which 𝑚₀=min_{1≤𝑖≤𝑛}𝜇𝑎^𝑖_𝑖,𝑀⁰=max𝑀₁,𝑀₂,𝑀₁=max_{1≤𝑖≤𝑛}𝑀₂𝜇𝑎^𝑖_𝑖,=^𝑛_𝑗=1𝜇_𝑗max_{1≤𝑖≤𝑛}𝑢_𝑗𝑖𝜌_𝑖𝐿_𝑖^0+∞𝑠𝑒^𝜆𝑠max_{1≤𝑖≤𝑛}𝑘_𝑗𝑖(𝑠)𝑑𝑠.(2.23) Hence, from (2.21) and (2.22), we know that the following inequality holds for 𝑡>0: ‖‖𝑥𝑡,𝜓₂−𝑥𝑡,𝜓₁‖‖≤𝑀‖‖𝜓₂−𝜓₁‖‖𝑒^−𝜆𝑡,(2.24) in which 𝑀=𝑀₀/𝑚₀.
We can always choose a positive integer 𝑁 such that 𝑒^{−𝜆0𝑁𝑇}𝑀≤1/2 and define a Poincaré mapping 𝑃∶𝐶→𝐶 by 𝑃(𝜉)=𝑥_𝑇(𝜉); we have ‖‖𝑃^𝑁𝜓₂−𝑃^𝑁𝜓₁‖‖≤12‖‖𝜓²−𝜓₁‖‖,(2.25) which implies that 𝑃^𝑁 is a contraction mapping. Similar to [10], using contraction mapping principle, we know that system (2.1) has a 𝑇-periodic solution which is globally exponentially stable. This completes the proof.

Remark 2.6. The result above also holds for (2.1) without impulses, and the existence and globally exponential stability of the periodic solution for (2.1) have nothing to do with amplification functions and inputs of the neuron. The results in [5] have more restrictions than Lemma 2.5 in this paper because conditions for the ones in [5] are relevant to amplification functions.

3. Periodic Solutions of CGBAMNNs with Distributed Delays and Impulses

Consider the following periodic CGBAMNNs model with distributed delays:𝑥̇_𝑖(𝑡)=−𝑎_𝑖𝑥_𝑖(𝑡)𝑏_𝑖𝑡,𝑥_𝑖(𝑡)−^𝑚_𝑗=1−𝑝_𝑖𝑗(𝑡)𝑓_𝑗𝜌_𝑗𝑦_𝑗(𝑡)_𝑚_𝑗=1Δ𝑥_𝑖𝑢_𝑖𝑗(𝑡)^0+∞𝑘_𝑖𝑗(𝑠)𝑓_𝑗𝜌_𝑗𝑦_𝑗(𝑡−𝑠)𝑑𝑠−𝐼_𝑖(𝑡),𝑡>0,𝑡≠𝑡_𝑘,𝑦̇_𝑗𝑡_𝑘=−𝛾_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑡=𝑡_𝑘,𝑘∈𝑍⁺,(𝑡)=−𝑐_𝑗𝑦_𝑗(𝑡)𝑑_𝑗𝑡,𝑦_𝑗(𝑡)−^𝑛_𝑖=1−𝑞_𝑗𝑖(𝑡)𝑔_𝑖𝜌̃_𝑖𝑥_𝑖(𝑡)_𝑛_𝑖=1Δ𝑦_𝑗𝑣_𝑗𝑖(𝑡)^0+∞𝑘̃_𝑗𝑖(𝑠)𝑔_𝑖𝜌̃_𝑖𝑥_𝑖(𝑡−𝑠)𝑑𝑠−𝐽_𝑗(𝑡),𝑡>0,𝑡≠𝑡_𝑘,𝑡_𝑘=−𝛿_𝑗𝑘𝑦_𝑗𝑡_𝑘,𝑡=𝑡_𝑘,𝑘∈𝑍⁺(3.1) for 1≤𝑖≤𝑛,1≤𝑗≤𝑚, and 𝑍⁺={1,2,…}; 𝑥_𝑖(𝑡) and 𝑦_𝑗(𝑡) denote the state variable of the 𝑖th neuron from the neural field 𝐹_𝑋 and the 𝑗th neuron from the neural field 𝐹_𝑌 at time 𝑡;  𝑓_𝑗(⋅) and 𝑔_𝑖(⋅) denote the signal functions of the 𝑗th neuron from the neural field 𝐹_𝑌 and the 𝑖th neuron from the neural field 𝐹_𝑋 at time t; respectively, 𝐼_𝑖 and 𝐽_𝑗 denote inputs of the 𝑖th neuron from the neural field 𝐹_𝑋 and the 𝑗th neuron from the neural field 𝐹_𝑌 at time 𝑡; respectively, 𝑎_𝑖(⋅) and 𝑐_𝑗(⋅) represent amplification functions; 𝑏_𝑖(𝑡,⋅) and 𝑑_𝑗(𝑡,⋅) are appropriately behaved functions; 𝑝_𝑖𝑗(𝑡),𝑞_𝑗𝑖(𝑡),𝑢_𝑖𝑗(𝑡), and 𝑣_𝑗𝑖(𝑡) are the connection weights; 𝜌_𝑗,𝜌̃_𝑖 are positive constants, which correspond to the neuronal gains associated with the neuronal activations; 𝑘_𝑖𝑗 and 𝑘̃_𝑗𝑖 correspond to the delay kernel functions; 𝑢_𝑖𝑗(𝑡),𝑣_𝑗𝑖(𝑡),𝑝_𝑖𝑗(𝑡),𝑞_𝑗𝑖(𝑡), 𝐼_𝑖(𝑡), and 𝐽_𝑗(𝑡) are all continuously periodic functions on [0,+∞) with common period 𝑇>0.

Δ𝑥_𝑖(𝑡_𝑘)=𝑥_𝑖(𝑡^+𝑘)−𝑥_𝑖(𝑡^−𝑘), Δ𝑦_𝑗(𝑡_𝑘)=𝑦_𝑗(𝑡^+𝑘)−𝑦_𝑗(𝑡^−𝑘); 𝑡_𝑘 is called impulsive moment and satisfies 0<𝑡₁<𝑡₂<⋯,lim_𝑘→+∞𝑡_𝑘=+∞; 𝑥_𝑖(𝑡⁻),𝑦_𝑗(𝑡⁻) and 𝑥_𝑖(𝑡⁺),𝑦_𝑗(𝑡⁺) denote the left-hand and right-hand limits at 𝑡_𝑘; respectively, we always assume 𝑥_i(𝑡^−𝑘)=𝑥_𝑖(𝑡_𝑘),𝑦_𝑗(𝑡^−𝑘)=𝑦_𝑗(𝑡_𝑘),𝑥^𝑖(𝑡^−𝑘)=𝑥^𝑖(𝑡_𝑘), and 𝑦^𝑗(𝑡^−𝑘)=𝑦^𝑗(𝑡_𝑘),𝑘∈𝑍⁺.

For system (3.1), we assume the following.(H₈) Amplification functions 𝑎_𝑖(⋅) and 𝑐_𝑗(⋅) are continuous and there exist constants 𝑎_𝑖,𝑎_𝑖 and 𝑐_𝑗,𝑐_𝑗 such that 0<𝑎_𝑖≤𝑎_𝑖(𝑥_𝑖(𝑡))≤𝑎_𝑖,0<𝑐_𝑗≤𝑐_𝑗(𝑦_𝑗(𝑡))≤𝑐_𝑗,1≤𝑖≤𝑛,1≤𝑗≤𝑚.(H₉)𝑏_𝑖(𝑡,𝑢),𝑑_𝑗(𝑡,𝑢) are 𝑇-periodic about the first argument, there exist continuous, 𝑇-periodic functions 𝛼_𝑖(𝑡) and 𝛽_𝑗(𝑡) such that 𝑏_𝑖(𝑡,𝑥)−𝑏_𝑖(𝑡,𝑦)𝑥−𝑦≥𝛼^𝑖(𝑡)>0,𝑑^𝑗(𝑡,𝑥)−𝑑_𝑗(𝑡,𝑦)𝑥−𝑦≥𝛽^𝑗(𝑡)>0(3.2) for all 𝑥≠𝑦,1≤𝑖≤𝑛,1≤𝑗≤𝑚.(H₁₀)For activation functions 𝑓_𝑗(⋅) and 𝑔_𝑖(⋅), there exist constant 𝐿_𝑗 and 𝐿_𝑖 such that 𝐿_𝑗=sup_𝑥≠𝑦||||𝑓_𝑗(𝑥)−𝑓_𝑗(𝑦)𝑥−𝑦||||,𝐿^𝑖=sup_𝑥≠𝑦||||𝑔_𝑖(𝑥)−𝑔_𝑖(𝑦)𝑥−𝑦||||,∀𝑥≠𝑦∈𝑅,1≤𝑖≤𝑛,1≤𝑗≤𝑚.(3.3)(H₁₁)The kernel functions 𝑘_𝑖𝑗(𝑠) and 𝑘̃_𝑗𝑖(𝑠) are nonnegative continuous functions on [0,+∞) and satisfy ^0+∞𝐾𝑠𝑒^𝜆𝑠𝑘_𝑖𝑗(𝑠)𝑑𝑠<+∞,^0+∞𝑠𝑒^𝜆𝑠𝑘̃_𝑗𝑖(𝑠)𝑑𝑠<+∞,_𝑖𝑗(𝜆)=^0+∞𝑒^𝜆𝑠𝑘_𝑖𝑗(𝑠)𝑑𝑠,𝐾_𝑗𝑖(𝜆)=^0+∞𝑒^𝜆𝑠𝑘̃_𝑗𝑖(𝑠)𝑑𝑠,(3.4) are differentiable functions for 𝜆∈[0,𝑟_𝑖𝑗) and 𝜆∈[0,𝑟̃_𝑗𝑖); respectively, 0<𝑟_𝑖𝑗<+∞,0<𝑟̃_𝑗𝑖<+∞,  𝐾_𝑖𝑗(0)=1,𝐾_𝑗𝑖(0)=1, lim_𝜆→𝑟^−𝑖𝑗𝐾_𝑖𝑗(𝜆)=+∞ and lim_𝜆→𝑟̃^−𝑗𝑖𝐾_𝑗𝑖(𝜆)=+∞.(H₁₂)There exists positive integer 𝑘₀ such that 𝑡_𝑘+𝑘0=𝑡_𝑘+𝑇 and 𝛾_{𝑖(𝑘+𝑘0)}=𝛾_𝑖𝑘,𝛿_{𝑗(𝑘+𝑘0)}=𝛿_𝑗𝑘 hold.

We assume that system (3.1) has the following initial conditions:𝑥_𝑖(𝑠)=𝜑_𝑖(𝑠),𝑦_𝑗(𝑠)=𝜙_𝑗(𝑠),𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚,−∞≤𝑠≤0,(3.5) where 𝜓=(𝜑,𝜙)∈PC((−∞,0],𝑅^𝑛+𝑚), 𝜑(𝑠)=(𝜑₁(𝑠),𝜑₂(𝑠),…,𝜑_𝑛(𝑠)),𝜙(𝑠)=(𝜙₁(𝑠),𝜙₂(𝑠),…,𝜙_𝑚(𝑠)).

Let 𝑍(𝑡,𝜓)=(𝑥(𝑡,𝜓),𝑦(𝑡,𝜓)) denote any solution of the system (3.1) with initial value 𝜓=(𝜑,𝜙)∈PC, 𝑥(𝑡,𝜓)=(𝑥₁(𝑡,𝜓),𝑥₂(𝑡,𝜓),…,𝑥(𝑡_𝑛,𝜓)), 𝑦(𝑡,𝜓)=(𝑦₁(𝑡,𝜓),𝑦₂(𝑡,𝜓),…,𝑦_𝑚(𝑡,𝜓)).

Theorem 3.1. Under assumptions (H₈)–(H₁₂), there exists a 𝑇-periodic solution which is asymptotically stable, if the following conditions hold.(H₁₃)The following ℳ is a nonsingular 𝑀-matrix, and ℳ=⎛⎜⎜⎝−𝐴𝐶−𝐴𝐶⎞⎟⎟⎠,(3.6) in which 𝐶=𝑒_𝑖𝑗𝐴=diag𝛼₁,𝛼₂,…,𝛼_𝑛,𝐴=diag𝛽₁,𝛽₂,…,𝛽_𝑚,_𝑚×𝑛,𝐶=𝑒̃_𝑖𝑗_𝑛×𝑚,𝑒̃_𝑖𝑗=𝑞_𝑗𝑖+𝑣_𝑗𝑖𝜌̃_𝑖𝐿_𝑖,𝑒_𝑖𝑗=𝑝_𝑖𝑗+𝑢_𝑖𝑗𝜌_𝑗𝐿_𝑗.(3.7)(H₁₄)𝑎_𝑖((1−𝛾_𝑖𝑘)𝑠)≥|1−𝛾_𝑖𝑘|𝑎_𝑖(𝑠),𝑐_𝑗((1−𝛿_𝑗𝑘)𝑠)≥|1−𝛿_𝑗𝑘|𝑐_𝑗(𝑠),∀𝑠∈𝑅,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚.

Proof. Let 𝑥𝑝_𝑛+𝑗(𝑡)=𝑦_𝑗(𝑡),𝑎_𝑛+𝑗𝑡,𝑥_𝑛+𝑗(𝑡)=𝑐_𝑗𝑡,𝑦_𝑗(𝑡),𝑏_𝑛+𝑗𝑡,𝑥_𝑛+𝑗(𝑡)=𝑑_𝑗𝑡,𝑦_𝑗(𝑡)_{𝑛+𝑗,𝑖},𝑆(𝑡)=𝑞_𝑗𝑖(𝑡),𝑝_{𝑖,𝑛+𝑗}(𝑡)=𝑝_𝑖𝑗(𝑡),𝑢_{𝑛+𝑗,𝑖}(𝑡)=𝑣_𝑗𝑖(𝑡),𝑢_{𝑖,𝑛+𝑗}(𝑡)=𝑢_𝑖𝑗(𝑡),_𝑖𝑥_𝑖(𝑡)𝐼=𝑔_𝑖𝑥_𝑖(𝑡),𝑆_𝑛+𝑗𝑥_𝑛+𝑗(𝑡)=𝑓_𝑗𝑥_𝑗(𝑡),𝜑_𝑛+𝑗(𝑠)=𝜓_𝑗(𝑠),_𝑛+𝑗=𝛼𝐽_𝑗(𝑡),𝑘_{𝑛+𝑗,𝑖}(𝑠)=𝑘̃_𝑗𝑖(𝑠),𝑘_𝑖𝑗(𝑠)=𝑘_{𝑖,𝑛+𝑗}(𝑠),_𝑛+𝑗(𝑡)=𝛽_𝑗(𝑡),𝐿_𝑛+𝑗=𝐿_𝑗,𝜌̃_𝑛+𝑗=𝜌_𝑗.(3.8) It follows that system (3.1) can be rewritten as 𝑥̇_𝑖(𝑡)=−𝑎_𝑖𝑥_𝑖(𝑡)𝑏_𝑖𝑡,𝑥_𝑖(𝑡)−^𝑚_𝑗=1−𝑝_{𝑖,𝑛+𝑗}(𝑡)𝑆_𝑛+𝑗𝜌̃_𝑛+𝑗𝑥_𝑛+𝑗(𝑡)_𝑚_𝑗=1Δ𝑥_𝑖𝑢_{𝑖,𝑛+𝑗}(𝑡)^0+∞𝑘_{𝑖,𝑛+𝑗}(𝑠)𝑆_𝑛+𝑗𝜌̃_𝑛+𝑗𝑥_𝑛+𝑗(𝑡−𝑠)𝑑𝑠−𝐼_𝑖(𝑡),𝑡≠𝑡_𝑘,𝑥̇_𝑛+𝑗𝑡_𝑘=−𝛾_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑡=𝑡_𝑘,𝑘∈𝑍⁺,(𝑡)=−𝑎_𝑛+𝑗𝑥_𝑛+𝑗(𝑡)𝑏_𝑛+𝑗𝑡,𝑥_𝑛+𝑗(𝑡)−^𝑛_𝑖=1−𝑝_{𝑛+𝑗,𝑖}(𝑡)𝑆_𝑖𝜌̃_𝑖𝑥_𝑖(𝑡)_𝑛_𝑖=1Δ𝑥_𝑛+𝑗𝑢_{𝑛+𝑗,𝑖}(𝑡)^0+∞𝑘_{𝑛+𝑗,𝑖}(𝑠)𝑆_𝑖𝜌̃_𝑖𝑥_𝑖(𝑡−𝑠)𝑑𝑠−𝐼_𝑛+𝑗(𝑡),𝑡≠𝑡_𝑘,𝑡_𝑘=−𝛾_{𝑛+𝑗,𝑘}𝑥_𝑛+𝑗𝑡_𝑘,𝑡=𝑡_𝑘,𝑘∈𝑍⁺,(3.9) for 1≤𝑖≤𝑛,1≤𝑗≤𝑚.
Initial conditions are given by 𝑥_𝑖(𝑠)=𝜑_𝑖(𝑠),𝑠∈(−∞,0],𝑖=1,2,…,(𝑛+𝑚).(3.10) Thus system (3.9) is a special case of system (2.1) in mathematical form, under conditions (H₈)–(H₁₄), we obtain from Lemma 2.5 that system (3.9) has a 𝑇-periodic solution which is globally exponentially stable if 𝑎_𝑖((1−𝛾_𝑖𝑘)𝑠)≥|1−𝛾_𝑖𝑘|𝑎_𝑖(𝑠) and the following matrix ℳ′ is a 𝑀-matrix, and ℳ′=𝐴′−𝐶′,(3.11) where 𝐷_𝐴′=diag𝛼₁,𝛼₂,…,𝛼_𝑛+𝑚,=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝0⋯0𝑤^1,𝑛+1⋯𝑤^{1,𝑛+𝑚}⋮0⋯⋮0𝑤⋮_{𝑛,𝑛+1}⋮⋯𝑤^{𝑛,𝑛+𝑚}𝑤𝑤⋮^𝑛+1,1⋯𝑤⋮^{𝑛+1,𝑛}⋮0⋯⋮_{𝑛+𝑚,1}0⋯𝑤^{𝑛+𝑚,𝑛}0⋯0⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,(3.12) in which 𝑤^𝑖𝑗=(𝑝_𝑖𝑗+𝑢_𝑖𝑗)𝜌̃_𝑗𝐿_𝑗.
Then, we know from (3.8) and (3.11) that Theorem 3.1 holds.
If 𝑎_𝑖(𝑥_𝑖(𝑡))=𝑐_𝑗(𝑦_𝑗(𝑡))=1, 𝑏_𝑖(𝑡,𝑥_𝑖(𝑡))=𝑏_𝑖(𝑡)𝑥_𝑖(𝑡) and 𝑑_𝑗(𝑡,𝑦_𝑗(𝑡))=𝑑_𝑗(𝑡)𝑦_𝑗(𝑡), where 𝑏_𝑖(𝑡) and 𝑑_𝑗(𝑡) are positive continuous 𝑇-periodic functions for 𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚. System (3.1) reduces to the following Hopfield-type BAM neural networks model: 𝑥̇_𝑖(𝑡)=−𝑏_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑚_𝑗=1+𝑝_𝑖𝑗(𝑡)𝑓_𝑗𝜌_𝑗𝑦_𝑗(𝑡)_𝑚_𝑗=1Δ𝑥_𝑖𝑢_𝑖𝑗(𝑡)^0+∞𝑘_𝑖𝑗(𝑠)𝑓_𝑗𝜌_𝑗𝑦_𝑗(𝑡−𝑠)𝑑𝑠+𝐼_𝑖(𝑡),𝑡>0,𝑡≠𝑡_𝑘,𝑦̇_𝑗𝑡_𝑘=−𝛾_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑡=𝑡_𝑘,𝑘∈𝑍⁺,(𝑡)=−𝑑_𝑗(𝑡)𝑦_𝑗(𝑡)+^𝑛_𝑖=1+𝑞_𝑗𝑖(𝑡)𝑔_𝑖𝜌̃_𝑖𝑥_𝑖(𝑡)_𝑛_𝑖=1Δ𝑦_𝑗𝑣_𝑗𝑖(𝑡)^0+∞𝑘̃_𝑗𝑖(𝑠)𝑔_𝑖𝜌̃_𝑖𝑥_𝑖(𝑡−𝑠)𝑑𝑠+𝐽_𝑗(𝑡),𝑡>0,𝑡≠𝑡_𝑘,𝑡_𝑘=−𝛿_𝑗𝑘𝑦_𝑗𝑡_𝑘,𝑡=𝑡_𝑘,𝑘∈𝑍⁺.(3.13)

Corollary 3.2. Under assumptions (H₉)–(H₁₂), there exists a 𝑇-periodic solution which is globally asymptotically stable, if the following conditions hold. (H^13) The following ℳ is a nonsingular 𝑀-matrix, and ℳ=⎛⎜⎜⎝−A𝐶−𝐴C⎞⎟⎟⎠,(3.14) in which 𝐶=𝑒_𝑖𝑗𝐴=diag𝑏₁,𝑏₂,…,𝑏_𝑛,𝐴=diag𝑑₁,𝑑₂,…,𝑑_𝑚,_𝑚×𝑛,𝐶=𝑒̃_𝑖𝑗_𝑛×𝑚,𝑒_𝑖𝑗=𝑝_𝑖𝑗+𝑢_𝑖𝑗𝜌_𝑗𝐿_𝑗,𝑒̃_𝑖𝑗=𝑞_𝑗𝑖+𝑣_𝑗𝑖𝜌̃_𝑖𝐿_𝑖.(3.15)(H^14)0≤𝛾_𝑖𝑘≤2,0≤𝛿_j𝑘≤2 for 𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚,𝑘∈𝑍⁺.

Proof. As 𝑏_𝑖(𝑡,𝑥_𝑖(𝑡))=𝑏_𝑖(𝑡)𝑥_𝑖(𝑡) and 𝑑_𝑗(𝑡,𝑦_𝑗(𝑡))=𝑑_𝑗(𝑡)𝑦_𝑗(𝑡), we obtain 𝛼_𝑖(𝑡)=𝑏_𝑖(𝑡) and 𝛽_𝑗(𝑡)=𝑑_𝑗(𝑡) in (H₂), (H^13) implies (H₁₃) holds. Since 𝑎_𝑖(𝑥_𝑖(𝑡))=𝑐_𝑗(𝑦_𝑗(𝑡))≡1, then condition (H₁₄) reduces to (H^14). Corollary 3.2 Holds from Theorem 3.1.

Remark 3.3. The conditions for the existence and globally exponential stability of the periodic solution of (3.1) without impulses have nothing to do with inputs of the neuron and amplification functions. The results in [13, 14] have more restrictions than Theorem 3.1 in this paper because conditions for the ones in [13, 14] are relevant to amplification functions and inputs of neurons our results should be better. In addition, Corollary 3.2 is similar to Theorem  2.1 in [15]; our results generalize the results in [15].

Remark 3.4. In view of proof of Theorem 3.1, since CGBAMNNs model is a special case of CGNNs model in form as BAM neural networks model is a special case of Hopfield neural networks model, many results of CGBAMNNs can be directly obtained from the ones of CGNNs, needing no repetitive discussions. Since system (3.1) reduces to autonomous system, Theorem 3.1 still holds, which means that system (3.1) has a equilibrium which is globally asymptotically stable; we know that many results in [18] can be directly obtained from the results in [19].

4. Two Simple Examples

Example 4.1. Consider the following CGNNs model with distributed delays: 𝑥𝑥̇₂̇₁(𝑡)=−2𝑥₁(𝑡)−0.2tanh𝑥₁(𝑡)−sin𝑡^0+∞𝑒^−𝑠tanh𝑥₂(𝑡−𝑠)𝑑𝑠,(𝑡)=−2+cos𝑥₂(𝑡)𝑥₂(𝑡)−0.3^0+∞𝑒^−𝑠tanh𝑥₁(𝑡−𝑠)𝑑𝑠−5.(4.1) Obviously, system (4.1) satisfies (H₁)–(H₅).
Note that ℳ=⎛⎜⎜⎝−01.3−11.2⎞⎟⎟⎠,(4.2) it is a nonsingular 𝑀-matrix and system (4.1) also satisfies condition (H₆). According to Lemma 2.5, system (4.1) has a 2𝜋-periodic solution which is globally exponentially stable. Figure 1 shows the dynamic behaviors of system (4.1) with initial condition (0.1,0.2).
However, It is easy to check that system (4.1) does not satisfy Theorem 4.3 or  4.4 in [5], so theorems in [5] cannot are used to ascertain the existence and stability of periodic solutions of system (4.1).

(a)

(b)

Figure 1 

Time response of state variables 𝑥1,𝑥2 and phase plot in space (𝑡,𝑥1,𝑥2) for system (4.1).

Example 4.2. Consider the following CGBAMNNs model with distributed delays and impulses: 𝑥̇₁(𝑡)=−2+sin𝑥Δ₁(𝑥𝑡)₁2𝑥₁(𝑡)−sin𝑡^0+∞𝑒^−𝑠||𝑦₁(𝑡−𝑠)||𝑑𝑠−1,𝑡>0,𝑡≠𝑡_𝑘,𝑦̇₁𝑡_𝑘=−𝛾_1𝑘𝑥_𝑖𝑡_𝑘,𝑡=𝑡_𝑘,𝑘∈𝑍⁺,(𝑡)=−3+cos𝑦₁(𝑡)Δ𝑦₁(3+cos𝑡)𝑦₁(𝑡)−sin𝑡^0+∞𝑒^−𝑠||𝑥₁(𝑡−𝑠)||𝑑𝑠−1,𝑡>0,𝑡≠𝑡_𝑘,𝑡_𝑘=−𝛿_1𝑘𝑦₁𝑡_𝑘,𝑡=𝑡_𝑘,𝑘∈𝑍⁺,(4.3) where 𝑡_𝑘=𝜋𝑘,𝑘∈𝑍⁺.
Obviously, system (4.3) satisfies (H₈)–(H₁₂).

Case 1. 𝛾_1𝑘=0,𝛿_1𝑘=0. Note that ℳ=⎛⎜⎜⎝−21−21⎞⎟⎟⎠,(4.4) it is a nonsingular 𝑀-matrix and system (4.3) also satisfies condition (H₁₃). According to Theorem 3.1, system (4.3) without impulses has a 2𝜋-periodic solution which is globally exponentially stable. Figure 2 shows the dynamic behaviors of system (4.3) with initial condition (0.1,0.2).
However, it is easy to check that system (4.3) without impulses does not satisfy Theorem  1 in [13] and theorems in [14]; so theorems in [13, 14] cannot be used to ascertain the existence and stability of periodic solutions of system (4.3).

(a)

(b)

Figure 2 

Time response of state variables 𝑥1,𝑦1 and phase plot in space (𝑡,𝑥1,𝑦1) for system (4.3) without impulsive effects.

Case 2. 𝛾_1𝑘=0.7,𝛿_1𝑘=(1−0.5sin(𝑡_𝑘+1)). Note that 𝑎₁(𝑠)=2+sin𝑠,𝑐₁(𝑠)=3+cos𝑠, and 𝑎₁/𝑎₁=0.5>|1−𝛾_1𝑘|=0.3 and |1−𝛿_1𝑘|<0.5<𝑐₁/𝑐₁=2/3, which means condition (H₁₄) also holds for system (4.3). Hence, system (4.3) with impulses still has that there exists a 2𝜋-periodic solution which is globally asymptotically stable. Figure 3 shows the dynamic behaviors of system (4.3) with initial condition (0.1,0.2).
This example illustrates the feasibility and effectiveness of the main results obtained in this paper, and it also shows that the conditions for the existence and globally exponential stability of the periodic solutions of CGBAMNNs without impulses have nothing to do with inputs of the neurons and amplification functions. If impulsive perturbations exist, the periodic solutions still exist and they are globally exponentially stable when we give some restrictions on impulsive perturbations.

(a)

(b)

Figure 3 

Time response of state variables 𝑥1,𝑦1 and phase plot in space (𝑡,𝑥1,𝑦1) for system (4.3) with impulsive effects.

5. Conclusions

A class of CGBAMNNs with distributed delays and impulses are investigated by using suitable Lyapunov functional, the properties of 𝑀-matrix, and some suitable mathematical transformation in this paper. Sufficient conditions to guarantee the uniqueness and global exponential stability of the periodic solutions of such networks are established without assuming the boundedness of the activation functions. Lemma 2.5 improves the results in [5], and Theorem 3.1 improves the results in [13, 14] and generalize the results in [15]. In addition, we point that CGBAMNNs model is a special case of CGNNs model; many results of CGBAMNNs can be directly obtained from the ones of CGNNs, needing no repetitive discussions. Our results are new, and two examples have been provided to demonstrate the effectiveness of our results.

Appendix

The source program (MATLAB 7.0) of Figure 1 is given as follows [14].clearT=70;N=7000;h=T/N;m=40/h;for i=1:mU(:,i)=[0.1; 0.2];endfor i=(m+1):(N+m)r(i)=i*h-40;x(i)=r(i);I=2;J=2+cos(U(2,i-1));A=[-I,0; 0,-J];B=[0,sin(x(i))*I; 0.3*J,0];U(:,i)=h*A*[(U(1,i-1)-0.2*tanh(U(1,i-1))); U(2,i-1)]+U(:,i-1);P(:,1)=[0; 0];for k=1:mP(:,1)=P(:,1)+h*exp(-(40-(k-1)*h))*[(tanh(U(1,i-m+k-1)));(tanh(U(2,i-m+k-1)))];endU(:,i)=U(:,i)+B*h*[(P(1,1));(P(2,1))]+h*[0; 5*J];endy=U(1,:);z=U(2,:);hold onplot(r,y,’:’)hold onplot(r,z)hold onplot3(r,y,z)

The source program (MATLAB 7.0) of Figures 2 and 3 is given as follows [14].clearT=70;N=7000;h=T/N;m=40/h;for i=1:mU(:,i)=[0.1;0.2];endfor i=(m+1):(N+m)r(i)=i*h-40;x(i)=r(i);I=2+sin(U(1,i-1));J=3+cos(U(2,i-1));A=[-I,0;0,-J];B=[0,I*sin(x(i)); J*sin(x(i)),0];U(:,i)=h*A*[2*U(1,i-1);(3+cos(x(i)))*U(2,i-1)]+U(:,i-1);P(:,1)=[0;0];for k=1:mP(:,1)=P(:,1)+h*exp(-(40-(k-1)*h))*[(abs(U(1,i-m+k-1)));(abs(U(2,i-m+k-1)))];endU(:,i)=U(:,i)+B*h*[(P(1,1));(P(2,1))]+[I; J]*h;if mod(i-m,314)==0U(:,i)=[0.3,0; 0,1/2*(sin(x(i)+1))]*U(:,i);endendy=U(1,:);z=U(2,:)hold onplot(r,y,’:’)hold onplot(r,z)hold onplot3(r,y,z)

Acknowledgment

The authors would like to thank the editor and the reviewers for their valuable suggestions and comments which greatly improved the original paper. Projects supported by the National Natural Science Foundation of China (no. 11071254).