New Results for Periodic Solution of High-Order BAM Neural Networks with Continuously Distributed Delays and Impulses

Abstract

By M-matrix theory, inequality techniques, and Lyapunov functional method, certain sufficient conditions are obtained to ensure the existence, uniqueness, and global exponential stability of periodic solution for a new type of high-order BAM neural networks with continuously distributed delays and impulses. These novel conditions extend and improve some previously known results in the literature. Finally, an illustrative example and its numerical simulation are given to show the feasibility and correctness of the derived criteria.

1. Introduction

As is well known, during the hardware implementation of neural networks, time delays are inevitable due to finite switching speeds of the amplifiers and communication time, which may bring about complex influence on the system such as oscillation and instability [1, 2]. On the other hand, impulsive effects wildly exist in many realistic networks [3, 4], which may be caused by witching phenomenon, sudden changes, or other unexpected noise. Therefore, it is more appropriate to consider delay and impulsive effects when modeling neural networks, and many researches on various kinds of neural networks with delays, impulses, or both of them have been available [5–12]. (See Figures 1(a), 1(b), and 1(c)).

(a) State response of the impulse-free system (51)

(b) State response of the impulsive system (51)

(c) Phase plots of system (51) without and with impulses, respectively

Figure 1 

It is confirmed that the system (51) has a unique 𝜋-periodic solution. Here (𝑥1,𝑦1) and (𝑥2,𝑦2) denote two pairs of solution of system (51) with different initial conditions 𝜙1(𝑠)=(−𝑒0.01𝑠+0.4,𝑒0.01𝑠−0.4)𝑇 and 𝜙2(𝑠)=(0.3,−0.3)𝑇 for 𝑠∈[−1,0] and 𝛾=0.5,𝑡𝑘−𝑡𝑘−1=𝜋,𝑘=1,2,….

Bidirectional associative memory (BAM) neural networks, as an extension of the unidirectional autoassociator of Hofield neural network [13], was firstly introduced by Kosko [14]. Due to its wide application in pattern recognition, associative memory, image, and signal processing, BAM neural networks with delays and impulses have been extensively studied in the past few decades [15–22]. In addition, it is worth noting that high-order neural networks structures have advantages of stronger storage capacity, faster convergence rate, and higher fault tolerance, and these merits have been successfully used in pattern recognition [23]. Thus, it is important to investigate BAM neural networks with high-order terms, which is called high-order BAM neural networks.

In this paper, we will consider a new type of high-order BAM neural networks with continuously distributed delays and impulses, which can be described by the following integrodifferential equations: 𝑑𝑥_𝑖(𝑡)𝑑𝑡=−𝑎_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑚_𝑗=1𝑐_𝑖𝑗(𝑡)^∞₀𝐾_𝑖𝑗(𝑠)𝑓_𝑗𝑦_𝑗()+𝑡−𝑠𝑑𝑠_{𝑚𝑚𝑗=1𝑞=1}𝑠_𝑖𝑗𝑞(𝑡)^∞₀𝑅_𝑖𝑗𝑞(𝑠)𝑓_𝑗𝑦_𝑗(𝑡−𝑠)𝑓_𝑞𝑦_𝑞(𝑡−𝑠)𝑑𝑠+𝐼_𝑖(𝑡),𝑡≠𝑡_𝑘,𝑥_𝑖𝑡⁺_𝑘=𝑥_𝑖𝑡⁻_𝑘+Δ𝑥_𝑖𝑡_𝑘=𝐻_𝑖𝑘𝑥_𝑖𝑡⁻_𝑘,𝑖=1,2,…,𝑛,𝑘∈ℤ≜{1,2,…},𝑑𝑦_𝑗(𝑡)𝑑𝑡=−𝑏_𝑗(𝑡)𝑦_𝑗(𝑡)+^𝑛_𝑖=1𝑑_𝑗𝑖(𝑡)^∞₀𝐾_𝑗𝑖(𝑠)𝑔_𝑖𝑥_𝑖()+𝑡−𝑠𝑑𝑠_{𝑛𝑛𝑖=1𝑝=1}𝑒_𝑗𝑖𝑝(𝑡)^∞₀𝑅_𝑗𝑖𝑝(𝑠)𝑔_𝑖𝑥_𝑖(𝑡−𝑠)𝑔_𝑝×𝑥_𝑝(𝑡−𝑠)𝑑𝑠+𝐽_𝑗(𝑡),𝑡≠𝑡_𝑘,𝑦_𝑗𝑡⁺_𝑘=𝑦_𝑗𝑡⁻_𝑘+Δ𝑦_𝑗𝑡_𝑘=𝐸_𝑗𝑘𝑦_𝑗𝑡⁻_𝑘,𝑗=1,2,…,𝑚,𝑘∈ℤ≜{1,2,…}, (1) where Δ𝑥_𝑖(𝑡_𝑘) and Δ𝑦_𝑗(𝑡_𝑘) are the impulses at moments 𝑡_𝑘 and 𝑡₁<𝑡₂<⋯ is a strictly increasing sequence such that lim_𝑘→∞𝑡_𝑘=∞. And 𝑥_𝑖(𝑡) and 𝑦_𝑗(𝑡) are the activations of the 𝑖th neuron and the 𝑗th neuron, respectively; 𝑎_𝑖(𝑡)>0 and 𝑏_𝑗(𝑡)>0 denote the passive decay rates; 𝑐_𝑖𝑗(𝑡),𝑑_𝑗𝑖(𝑡),𝑠_𝑖𝑗𝑞(𝑡),𝑒_𝑗𝑖𝑝(𝑡) are the first- and second-order connection weights of the neural networks, respectively; 𝐼_𝑖(𝑡) and 𝐽_𝑗(𝑡) are the external inputs.

Clearly, system (1) is a more general form of BAM neural networks, which has been widely applied in areas of science and engineering [24], such as neurobiology, image classification, and image recognition. In recent years, studies of such kind of neural networks with delays and impulses have received considerable interest, and some results have been reported in [25–30]. In particular, authors in [25–28] have discussed the stability of equilibrium point for a kind of impulsive high-order BAM neural networks with discrete delays by different methods, such as linear matrix inequality (LMI), Razumikhin technique. Subsequently, Huo et al. [29] and Yang [30] studied the existence of periodic solution and its exponential stability for an impulsive high-order BAM neural network with discrete delays by using the theory of coincidence degree and Lyapunov functional method. However, to the best of our knowledge, there are few results on the existence, uniqueness, and global exponential stability of periodic solution for system (1) with continuously distributed delays.

The main propose of this paper is to study the periodicity of system (1) with distributed delays and general impulsive effects. It should be noticed that some new criteria on the existence and uniqueness of periodic solution for system (1) are established by combining the general ‖⋅‖_𝑟 (see Notations) and analytical techniques, which is different from the conventional continuation theorem of coincidence degree theory used in [29, 30]. In addition, it is worth mentioning that the impulsive part in this paper is not necessarily bounded and linear, which makes its applications more extensive.

The rest of this paper is organized as follows. In Section 2, some assumptions, definitions, and important lemmas are given. In Section 3, the main results and some remarks are presented. In Section 4, an example and its numerical simulation are provided. Finally, some conclusions are summarized in Section 5.

2. Preliminaries

Notations. Throughout this paper, ℜ and ℜ^𝑛 denote the set of real numbers and 𝑛-dimensional vector space, respectively. The symbol (⋅)^𝑇 denotes the transpose of a vector or a matrix. Take ‖𝑥‖_𝑟=(∑^𝑛_𝑖=1|𝑥_𝑖|^𝑟)^1/𝑟 with integer 𝑟∈[1,∞). Clearly, ‖⋅‖₁,‖⋅‖₂ are special cases of ‖⋅‖_𝑟 with 𝑟=1,2, respectively, which are used to investigate the dynamics of various kinds of neural networks in [6–8, 10, 12, 15–17, 19, 21, 22, 25–31]. Denote 𝐶^∗≜{𝜙∶(−∞,0]→ℜ^𝑛+𝑚∣𝜙(𝑠) is continuous for all but at most countable points 𝑠∈(−∞,0] and at these points 𝑠∈(−∞,0], 𝜙(𝑠⁺) and 𝜙(𝑠⁻) exist, 𝜙(𝑠)=𝜙(𝑠⁻)⋅} and define the norm ‖𝜙‖_Δ by 𝜙_Δ=^𝑛_𝑖=1sup_{(]𝑠∈−∞,0}𝜙_𝑥𝑖(𝑠)^𝑟+^𝑚_𝑗=1sup_{(]𝑠∈−∞,0}𝜙_𝑦𝑗(𝑠)^𝑟1/𝑟, (2) where 𝜙=(𝜙^𝑇_𝑥,𝜙^𝑇_𝑦)^𝑇=(𝜙_𝑥1,𝜙_𝑥2,…,𝜙_𝑥𝑛,𝜙_𝑦1,𝜙_𝑦2,…,𝜙_𝑦𝑚)^𝑇, and then 𝐶^∗ is a Banach space with topology of the uniform convergence. In addition, system (1) is supplemented with initial values 𝑥_𝑖(𝑠)=𝜙_𝑥𝑖(𝑠)(]𝑦,𝑠∈−∞,0,𝑖=1,2,…,𝑛,_𝑗(𝑠)=𝜙_𝑦𝑗(𝑠),𝑠∈(−∞,0],𝑗=1,2,…,𝑚. (3)
As usual, we have the following assumptions for system (1).

(S₁) Functions 𝑎_𝑖(𝑡),𝑎_𝑗(𝑡),𝑐_𝑖𝑗(𝑡),𝑑_𝑗𝑖(𝑡),𝑠_𝑖𝑗𝑞(𝑡),𝑒_𝑗𝑖𝑝(𝑡),𝐼_𝑖(𝑡),and𝐽_𝑗(𝑡) are 𝜔-periodic and bounded on [0,+∞) such that 𝑎_𝑖(𝑡)≥𝑎⁻_𝑖>0,𝑏_𝑗(𝑡)≥𝑏⁻_𝑗>0,|𝑐_𝑖𝑗(𝑡)|≤𝑐⁺_𝑖𝑗,|𝑑_𝑗𝑖(𝑡)|≤𝑑⁺_𝑗𝑖,|𝑠_𝑖𝑗𝑞(𝑡)|≤𝑠⁺_𝑖𝑗𝑞,|𝑒_𝑗𝑖𝑝(𝑡)|≤𝑒⁺_𝑗𝑖𝑝,|𝐼_𝑖𝐼(𝑡)|≤_𝑖,|𝐽_𝑗(𝑡)|≤𝐽_𝑗 for 𝑖,𝑝=1,2,…,𝑛,𝑗,𝑞=1,2,…,𝑚.

(S₂) The activation functions 𝑔_𝑖(⋅),𝑔_𝑝(⋅),𝑓_𝑗(⋅), and 𝑓_𝑞(⋅) are bounded and Lipschitz continuous on ℜ; that is, there exist positive numbers 𝑀^𝑓_𝑗,𝑀^𝑓_𝑞,𝑀^𝑔_𝑖,𝑀^𝑔_𝑝 and 𝐿^𝑔_𝑖,𝐿^𝑓_𝑗 such that 𝑓_𝑗(𝑥)≤𝑀^𝑓_𝑗,𝑓_𝑞(𝑥)≤𝑀^𝑓_𝑞,𝑓_𝑗(𝑥)−𝑓_𝑗𝑦≤𝐿^𝑓_𝑗,𝑔𝑥−𝑦_𝑖(𝑥)≤𝑀^𝑔_𝑖,𝑔_𝑝(𝑥)≤𝑀^𝑔_𝑝,𝑔_𝑖(𝑥)−𝑔_𝑖𝑦≤𝐿^𝑔_𝑖𝑥−𝑦, (4) for 𝑖,𝑝=1,2,…,𝑛,𝑗,𝑞=1,2,…,𝑚 and 𝑥,𝑦∈ℜ.

(S₃) The delay kernel functions 𝐾_𝑖𝑗(⋅),𝐾_𝑗𝑖(⋅),𝑅_𝑖𝑗𝑞(⋅), and 𝑅_𝑖𝑗𝑝(⋅) are piecewise continuous functions from [0,∞) to [0,∞) and satisfy 𝐾_𝑖𝑗(𝑠)≤𝒦(𝑠),𝐾_𝑗𝑖(𝑠)≤𝒦(𝑠),𝑅_𝑖𝑗𝑞(⋅)≤ℛ(𝑠), and 𝑅_𝑖𝑗𝑝(𝑠)≤ℛ(𝑠) for 𝑠∈[0,∞), 𝑖,𝑝=1,2,…,𝑛,𝑗,𝑞=1,2,…,𝑚, where 𝒦(𝑠),ℛ(𝑠) satisfy ^∞₀𝒦(𝑠)𝑑𝑠=1,^∞₀𝑒^𝜀𝑠𝒦(𝑠)𝑑𝑠<+∞,_∞0ℛ(𝑠)𝑑𝑠=1,^∞₀𝑒^𝜀𝑠ℛ(𝑠)𝑑𝑠<+∞, (5)

in which 𝜀 denotes some positive constant number. For more information on these delay kernels, one can refer to [5, 9, 15, 18, 22].

(S₄) Ω is an 𝑀-matrix, where 𝑟≥1, ()Ω=𝑟𝐴−𝑟−1𝐺−𝐿^𝑓Π−𝐿^𝑔()𝐹,𝑎Γ𝑟𝐵−𝑟−1𝐴=diag⁻₁,𝑎⁻₂,…,𝑎⁻_𝑛,𝐵=diag𝑏⁻₁,𝑏⁻₂,…,𝑏⁻_𝑚,𝐺=diag𝐺₁,𝐺₂,…,𝐺_𝑛,𝐹=diag𝐹₁,𝐹₂,…,𝐹_𝑚,𝐿^𝑔=diag𝐿^𝑔₁,𝐿^𝑔₂,…,𝐿^𝑔_𝑛,𝐿^𝑓=diag𝐿^𝑓₁,𝐿^𝑓₂,…,𝐿^𝑓_𝑚,𝐺_𝑖=^𝑚_𝑗=1𝑐⁺_𝑖𝑗+^𝑚_𝑞=1𝑠⁺_𝑖𝑗𝑞+𝑠⁺_𝑖𝑞𝑗𝑀^𝑓_𝑞𝐿^𝑓_𝑗,𝐹_𝑗=^𝑛_𝑖=1𝑑⁺_𝑗𝑖+^𝑛_𝑝=1𝑒⁺_𝑗𝑖𝑝+𝑒⁺_𝑗𝑝𝑖𝑀^𝑔_𝑝𝐿^𝑔_𝑖,Π=Π_{𝑖𝑗𝑛×𝑚}=𝑐⁺_𝑖𝑗+^𝑚_𝑞=1𝑠⁺_𝑖𝑗𝑞+𝑠⁺_𝑖𝑞𝑗𝑀^𝑓_𝑞,Γ=Γ_{𝑗𝑖𝑚×𝑛}=𝑑⁺_𝑗𝑖+^𝑛_𝑝=1𝑒⁺_𝑗𝑖𝑝+𝑒⁺_𝑗𝑝𝑖𝑀^𝑔_𝑝. (6)

Definition 1. A function 𝑧(𝑡,𝜙)=(𝑥₁(𝑡,𝜙),𝑥₂(𝑡,𝜙),…,  𝑥_𝑛(𝑡,𝜙),𝑦₁(𝑡,𝜙),𝑦₂(𝑡,𝜙),…,𝑦_𝑚(𝑡,𝜙))^𝑇∈ℜ^𝑛+𝑚 is said to be the solution of system (1) with initial condition 𝜙∈𝐶^∗ if the following two conditions are satisfied.(1)𝑧(𝑡,𝜙) is piecewise continuous with first kind discontinuity at the points 𝑡_𝑘,𝑘∈ℤ. Moreover, 𝑧(𝑡,𝜙) is left continuous at each of the discontinuity points. (2)𝑧(𝑡,𝜙) satisfies system (1) for 𝑡≥0 and 𝑧(𝑠)=𝜙(𝑠) for 𝑠∈(−∞,0].

Definition 2. The periodic solution 𝑧^∗(𝑡,𝜙^∗) of system (1) is said to be globally exponentially stable, if there exist constants ℳ≥1 and 𝛼>0 such that any other solution 𝑧(𝑡,𝜙) of system (1) satisfies 𝑧𝑡,𝜙−𝑧^∗𝑡,𝜙^∗_𝑟≤ℳ𝑒^−𝛼𝑡𝜙−𝜙^∗_Δ,𝑡≥0. (7)

Lemma 3 (see [32]). Let 𝐴∈ℤ^𝑛×𝑛, where ℤ^𝑛×𝑛 is a set of 𝑛×𝑛 matrices with nonpositive off-diagonal elements. 𝐴 is an 𝑀-matrix if and only if there exists a positive vector 𝜆=(𝜆₁,𝜆₂,…,𝜆_𝑛)^𝑇 such that ∑^𝑛_𝑗=1𝑎_𝑗𝑖𝜆_𝑗>0 or ∑^𝑛_𝑗=1𝑎_𝑖𝑗𝜆_𝑗>0.

Lemma 4. Assume that assumptions (S₃) and (S₄) hold; then there exist positive constants 𝜀 and 𝜆₁,𝜆₂,…,𝜆_𝑛+𝑚 such that 𝑃_𝑖(𝜀)=−𝜀+𝑟𝑎⁻_𝑖−(𝑟−1)𝐺_𝑖𝜆_𝑖−^𝑚_𝑗=1𝜆_𝑛+𝑗𝑑⁺_𝑗𝑖𝐿^𝑔_𝑖^∞₀𝒦(𝑠)𝑒^𝜀𝑠−𝑑𝑠_{𝑚𝑛𝑗=1𝑝=1}𝜆_𝑛+𝑗𝑒⁺_𝑗𝑖𝑝+𝑒⁺_𝑗𝑝𝑖𝑀^𝑔_𝑝𝐿^𝑔_𝑖×^∞₀ℛ(𝑠)𝑒^𝜀𝑠𝑄𝑑𝑠≥0,_𝑗(𝜀)=−𝜀+𝑟𝑏⁻_𝑗−(𝑟−1)𝐹_𝑗𝜆_𝑛+𝑗−^𝑛_𝑖=1𝜆_𝑖𝑐⁺_𝑖𝑗𝐿^𝑓_𝑗^∞₀𝒦(𝑠)𝑒^𝜀𝑠−𝑑𝑠_{𝑛𝑚𝑖=1𝑞=1}𝜆_𝑖𝑠⁺_𝑖𝑗𝑞+𝑠⁺_𝑖𝑞𝑗𝑀^𝑓_𝑞𝐿^𝑓_𝑗×^∞₀ℛ(𝑠)𝑒^𝜀𝑠𝑑𝑠≥0, (8) for 𝑖,𝑝=1,2,…,𝑛,𝑗,𝑞=1,2,…,𝑚.

Proof. Construct the aided functions as follows: 𝑃_𝑖𝜗_𝑖=−𝜗_𝑖+𝑟𝑎⁻_𝑖−(𝑟−1)𝐺_𝑖𝜆_𝑖−^𝑚_𝑗=1𝜆_𝑛+𝑗𝑑⁺_𝑗𝑖𝐿^𝑔_𝑖^∞₀𝒦(𝑠)𝑒^𝜗𝑖𝑠−𝑑𝑠_{𝑚𝑛𝑗=1𝑝=1}𝜆_𝑛+𝑗𝑒⁺_𝑗𝑖𝑝+𝑒⁺_𝑗𝑝𝑖𝑀^𝑔_𝑝𝐿^𝑔_𝑖^∞₀ℛ(𝑠)𝑒^𝜗𝑖𝑠𝑄𝑑𝑠,_𝑗𝜇_𝑗=−𝜇_𝑗+𝑟𝑏⁻_𝑗−(𝑟−1)𝐹_𝑗𝜆_𝑛+𝑗−^𝑛_𝑖=1𝜆_𝑖𝑐⁺_𝑖𝑗𝐿^𝑓_𝑗^∞₀𝒦(𝑠)𝑒^𝜇𝑗𝑠−𝑑𝑠_{𝑛𝑚𝑖=1𝑞=1}𝜆_𝑖𝑠⁺_𝑖𝑗𝑞+𝑠⁺_𝑖𝑞𝑗𝑀^𝑓_𝑞𝐿^𝑓_𝑗^∞₀ℛ(𝑠)𝑒^𝜇𝑗𝑠𝑑𝑠 (9) for 𝑖,𝑝=1,2,…,𝑛,𝑗,𝑞=1,2,…,𝑚. Using Lemma 3 and assumptions((S₃)-(S₄)), it is easy to deduce that (8) hold by similar proof in [7, 15–18]. For concise, it is omitted here.

Lemma 5. Let the integer 𝑟≥1; then the inequality holds as follows: (𝑛+𝑚)^{(1−𝑟)/𝑟𝑛+𝑚}_𝑖=1𝑧_𝑖≤^𝑛+𝑚_𝑖=1𝑧_𝑖^𝑟1/𝑟 (10) for all 𝑧=(𝑧₁,𝑧₂…,𝑧_𝑛+𝑚)^𝑇∈ℜ^𝑛+𝑚.

Proof. Obviously, the inequality (10) with 𝑟=1 is trivial. When 𝑟>1, consider the aided function 𝑔(𝑥)=𝑥^𝑟,𝑥≥0. It is claimed that 𝑔(𝑥) is convex since 𝑔^′′(𝑥)=𝑟(𝑟−1)𝑥^𝑟−1>0, 𝑟>1 for 𝑥>0. Let 𝑥=(1/(𝑛+𝑚))∑^𝑛+𝑚_𝑖=1|𝑧_𝑖|; by Jensen's inequality, we have 1(𝑛+𝑚)^{𝑟𝑛+𝑚}_𝑖=1𝑧_𝑖^𝑟≤1𝑛+𝑚^𝑛+𝑚_𝑖=1𝑧_𝑖^𝑟, (11) which implies that the inequality (10) holds. This completes the proof.

3. Main Results

Firstly, let 𝑧(𝑡,𝜙)=(𝑥₁(𝑡,𝜙),𝑥₂(𝑡,𝜙),…,𝑥_𝑛(𝑡,𝜙),𝑦₁(𝑡,𝜙),𝑦₂(𝑡,𝜙),…,𝑦_𝑚(𝑡,𝜙))^𝑇 and 𝑧(𝑡,𝜑)=(𝑥₁(𝑡,𝜑),𝑥₂(𝑡,𝜑),…,𝑥_𝑛(𝑡,𝜑),𝑦₁(𝑡,𝜑),𝑦₂(𝑡,𝜑),…,𝑦_𝑚(𝑡,𝜑))^𝑇 be any two solutions of system (1) through 𝜙,𝜓∈𝐶^∗, respectively; then we have the following useful lemma.

Lemma 6. Under assumptions ((S₁)–(S₄)), if the following two conditions hold: (S₅)𝐻_𝑖𝑘(⋅),𝐸_𝑗𝑘(⋅) are Lipschitz continuous on ℜ; that is, there exist positive constants 𝐻_𝑖𝑘,𝐸_𝑗𝑘 such that |𝐻_𝑖𝑘(𝑥)−𝐻_𝑖𝑘(𝑦)|≤𝐻_𝑖𝑘|𝑥−𝑦|,|𝐸_𝑗𝑘(𝑥)−𝐸_𝑗𝑘(𝑦)|≤𝐸_𝑗𝑘|𝑥−𝑦| for 𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚,𝑘∈ℤ and 𝑥,𝑦∈ℜ;(S₆)there exists 𝜃 such that ln𝜂^𝑟_𝑘/(𝑡_𝑘−𝑡_𝑘−1)≤𝜃<𝜀, where 𝑡₀=0, 𝜂_𝑘=max{1,max_{1≤𝑖≤𝑛}𝐻_𝑖𝑘,max_{1≤𝑗≤𝑚}𝐸_𝑗𝑘},𝑘∈ℤ, and the scalar 𝜀 is estimated by (8).Then, the following inequality holds: 𝑧𝑡,𝜙−𝑧𝑡,𝜑_𝑟≤ℳ𝑒^−𝛼𝑡𝜙−𝜑_Δ,𝑡≥0, (12) where the constants ℳ≥1 and 𝛼>0 are to be determined later.

Proof. To be convenient, let 𝑋_𝑖(𝑡)=𝑥_𝑖𝑡,𝜙−𝑥_𝑖𝑌𝑡,𝜑,𝑖=1,2,…,𝑛,_𝑗(𝑡)=𝑦_𝑗𝑡,𝜙−𝑦_𝑗𝑡,𝜑,𝑗=1,2,…,𝑚. (13) It follows from ((S₁)–(S₃)) that 𝐷⁺𝑋_𝑖(𝑡)≤−𝑎⁻_𝑖𝑋_𝑖(𝑡)+^𝑚_𝑗=1𝑐⁺_𝑖𝑗𝐿^𝑓_𝑗^∞₀𝒦(𝑠)𝑌_𝑗()+𝑡−𝑠𝑑𝑠_{𝑚𝑚𝑗=1𝑞=1}𝑠⁺_𝑖𝑗𝑞×^∞₀ℛ(𝑠)𝑓_𝑗𝑥_𝑗𝑡−𝑠,𝜙𝑓_𝑞𝑥_𝑞𝑡−𝑠,𝜙−𝑓_𝑗𝑥_𝑗𝑡−𝑠,𝜑𝑓_𝑞𝑥_𝑞𝑡−𝑠,𝜙+𝑓_𝑗𝑥_𝑗𝑡−𝑠,𝜑𝑓_𝑞𝑥_𝑞𝑡−𝑠,𝜙−𝑓_𝑗𝑥_𝑗𝑡−𝑠,𝜑𝑓_𝑞𝑥_𝑞𝑡−𝑠,𝜑𝑑𝑠≤−𝑎⁻_𝑖𝑋_𝑖(𝑡)+^𝑚_𝑗=1𝑐⁺_𝑖𝑗𝐿^𝑓_𝑗^∞₀𝒦(𝑠)𝑌_𝑗()+𝑡−𝑠𝑑𝑠_{𝑚𝑚𝑗=1𝑞=1}𝑠⁺_𝑖𝑗𝑞+𝑠⁺_𝑖𝑞𝑗𝑀^𝑓_𝑞𝐿^𝑓_𝑗^∞₀ℛ(𝑠)𝑌_𝑗(𝑡−𝑠)𝑑𝑠, (14) for 𝑡≥0,𝑡≠𝑡_𝑘,𝑘∈ℤ, 𝑖=1,2,…,𝑛. Similarly, we have 𝐷⁺𝑌_𝑗(𝑡)≤−𝑏⁻_𝑗𝑌_𝑗(𝑡)+^𝑚_𝑗=1𝑑⁺_𝑗𝑖𝐿^𝑔_𝑖^∞₀𝒦(𝑠)𝑋_𝑖()+𝑡−𝑠𝑑𝑠_{𝑛𝑛𝑖=1𝑝=1}𝑒⁺_𝑗𝑖𝑝+𝑒⁺_𝑗𝑝𝑖𝑀^𝑔_𝑝𝐿^𝑔_𝑖^∞₀ℛ(𝑠)𝑋_𝑖(𝑡−𝑠)𝑑𝑠, (15) for 𝑡≥0,𝑡≠𝑡_𝑘,𝑘∈ℤ, 𝑗=1,2,…,𝑚. Also, 𝑋_𝑖𝑡_𝑘+0=𝐻_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝜙−𝐻_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝜑≤𝐻_𝑖𝑘𝑋_𝑖𝑡_𝑘𝑌,𝑘∈ℤ,𝑖=1,2,…,𝑛,_𝑗𝑡_𝑘+0=𝐸_𝑗𝑘𝑦_𝑗𝑡_𝑘,𝜙−𝐸_𝑗𝑘𝑦_𝑗𝑡_𝑘,𝜑≤𝐸_𝑗𝑘𝑌_𝑗𝑡_𝑘,𝑘∈ℤ,𝑗=1,2,…,𝑚. (16)
Now define 𝑈_𝑖(𝑡)=𝑒^𝜀𝑡𝑋_𝑖(𝑡)^𝑟𝑉,𝑖=1,2,…,𝑛,_𝑗(𝑡)=𝑒^𝜀𝑡𝑌_𝑗(𝑡)^𝑟,𝑗=1,2,…,𝑚. (17) By using Young inequality 𝑎^𝑝𝑏^𝑞≤𝑝𝑎+𝑞𝑏, where 𝑎,𝑏,𝑝,𝑞>0 and 1/𝑝+1/𝑞=1, it follows from (14)-(15) that 𝐷⁺𝑈_𝑖(𝑡)≤𝜀−𝑟𝑎⁻_𝑖𝑈_𝑖(𝑡)+𝑟𝑒^𝑚𝜀𝑡_𝑗=1𝑐⁺_𝑖𝑗𝐿^𝑓_𝑗×^∞₀𝒦(𝑠)𝑋_𝑖(𝑡)^{𝑟(𝑟−1)/𝑟}𝑌_𝑗(𝑡−𝑠)^𝑟1/𝑟𝑑𝑠+𝑟𝑒^{𝑚𝜀𝑡𝑚}_𝑗=1𝑞=1𝑠⁺_𝑖𝑗𝑞+𝑠⁺_𝑖𝑞𝑗𝑀^𝑓_𝑞𝐿^𝑓_𝑗×^∞₀ℛ(𝑠)𝑋_𝑖(𝑡)^{𝑟(𝑟−1)/𝑟}𝑌_𝑗(𝑡−𝑠)^𝑟1/𝑟≤𝑑𝑠𝜀−𝑟𝑎⁻_𝑖+(𝑟−1)𝐺_𝑖𝑈_𝑖(𝑡)+^𝑚_𝑗=1𝑐⁺_𝑖𝑗𝐿^𝑓_𝑗^∞₀𝒦(𝑠)𝑒^𝜀𝑠𝑉_𝑗()+𝑡−𝑠𝑑𝑠_{𝑚𝑚𝑗=1𝑞=1}𝑠⁺_𝑖𝑗𝑞+𝑠⁺_𝑖𝑞𝑗𝑀^𝑓_𝑞𝐿^𝑓_𝑗^∞₀ℛ(𝑠)𝑒^𝜀𝑠𝑉_𝑗(𝑡−𝑠)𝑑𝑠, (18) for 𝑡≥0,𝑡≠𝑡_𝑘,𝑘∈ℤ, 𝑖=1,2,…,𝑛. Similarly, we have 𝐷⁺𝑉_𝑗(𝑡)≤𝜀−𝑟𝑏⁻_𝑗+(𝑟−1)𝐹_𝑗𝑉_𝑗(𝑡)+^𝑛_𝑖=1𝑑⁺_𝑗𝑖𝐿^𝑔_𝑖^∞₀𝒦(𝑠)𝑒^𝜀𝑠𝑈_𝑖()+𝑡−𝑠𝑑𝑠_{𝑛𝑛𝑖=1𝑝=1}𝑒⁺_𝑗𝑖𝑝+𝑒⁺_𝑗𝑝𝑖𝑀^𝑔_𝑝𝐿^𝑔_𝑖^∞₀ℛ(𝑠)𝑒^𝜀𝑠𝑈_𝑖(𝑡−𝑠)𝑑𝑠, (19) for 𝑡≥0,𝑡≠𝑡_𝑘,𝑘∈ℤ, 𝑗=1,2,…,𝑚. Also 𝑈_𝑖𝑡_𝑘+0≤𝐻^𝑟_𝑖𝑘𝑈_𝑖𝑡_𝑘𝑉,𝑘∈ℤ,𝑖=1,2,…,𝑛,_𝑗𝑡_𝑘+0≤𝐸^𝑟_𝑗𝑘𝑉_𝑗𝑡_𝑘,𝑘∈ℤ,𝑗=1,2,…,𝑚. (20) Consider the candidate Lyapunov-Krasovskii functional as follows: 𝕍(𝑡)=𝕍₁(𝑡)+𝕍₂(𝑡), (21) where 𝕍₁(𝑡)=^𝑛_𝑖=1𝜆_𝑖𝑈_𝑖(𝑡)+^𝑚_𝑗=1𝑐⁺_𝑖𝑗𝐿^𝑓_𝑗×^∞₀𝒦(𝑠)𝑒^𝜀𝑠𝑡_𝑡−𝑠𝑉_𝑗(𝑧)+𝑑𝑧𝑑𝑠_{𝑚𝑚𝑗=1𝑞=1}𝑠⁺_𝑖𝑗𝑞+𝑠⁺_𝑖𝑞𝑗𝑀^𝑓_𝑞𝐿^𝑓_𝑗×^∞₀ℛ(𝑠)𝑒^𝜀𝑠𝑡_𝑡−𝑠𝑉_𝑗(𝑧),𝕍𝑑𝑧𝑑𝑠₂(𝑡)=^𝑛_𝑗=1𝜆_𝑛+𝑗𝑉_𝑗(𝑡)+^𝑛_𝑖=1𝑑⁺_𝑗𝑖𝐿^𝑔_𝑖×^∞₀𝒦(𝑠)𝑒^𝜀𝑠𝑡_𝑡−𝑠𝑈_𝑖(𝑧)+𝑑𝑧𝑑𝑠_{𝑛𝑛𝑖=1𝑝=1}𝑒⁺_𝑗𝑖𝑝+𝑒⁺_𝑗𝑝𝑖𝑀^𝑔_𝑝𝐿^𝑔_𝑖×^∞₀ℛ(𝑠)𝑒^𝜀𝑠𝑡_𝑡−𝑠𝑈_𝑖(𝑧)𝑑𝑧𝑑𝑠. (22) When 𝑡≠𝑡_𝑘,𝑘∈ℤ, calculating the upper right Dini derivative of 𝕍₁(𝑡) along the solutions of system (1), we get 𝐷⁺𝕍₁(𝑡)=^𝑛_𝑖=1𝜆_𝑖𝐷⁺𝑈_𝑖(𝑡)+^𝑚_𝑗=1𝑐⁺_𝑖𝑗𝐿^𝑓_𝑗^∞₀𝒦(𝑠)𝑒^𝜀𝑠𝑉_𝑗(𝑡)−𝑉_𝑗()+𝑡−𝑠𝑑𝑠_{𝑚𝑚𝑗=1𝑞=1}𝑠⁺_𝑖𝑗𝑞+𝑠⁺_𝑖𝑞𝑗𝑀^𝑓_𝑞𝐿^𝑓_𝑗×^∞₀ℛ(𝑠)𝑒^𝜀𝑠𝑉_𝑗(𝑡)−𝑉_𝑗()≤𝑡−𝑠𝑑𝑠_𝑛𝑖=1𝜆_𝑖𝜀−𝑟𝑎⁻_𝑖+(𝑟−1)𝐺_𝑖𝑈_𝑖(𝑡)+^𝑚_𝑗=1𝑐⁺_𝑖𝑗𝐿^𝑓_𝑗^∞₀𝒦(𝑠)𝑒^𝜀𝑠𝑑𝑠𝑉_𝑗(𝑡)+^𝑚𝑚_𝑗=1𝑞=1𝑠⁺_𝑖𝑗𝑞+𝑠⁺_𝑖𝑞𝑗𝑀^𝑓_𝑞𝐿^𝑓_𝑗×^∞₀ℛ(𝑠)𝑒^𝜀𝑠𝑑𝑠𝑉_𝑗(𝑡). (23) Similarly, we have 𝐷⁺𝕍₂(𝑡)≤^𝑛_𝑗=1𝜆_𝑛+𝑗𝜀−𝑟𝑏⁻_𝑗+(𝑟−1)𝐹_𝑗𝑉_𝑗(𝑡)+^𝑛_𝑖=1𝑑⁺_𝑗𝑖𝐿^𝑔_𝑖^∞₀𝒦(𝑠)𝑒^𝜀𝑠𝑑𝑠𝑈_𝑖(𝑡)+^𝑛𝑛_𝑖=1𝑝=1𝑒⁺_𝑗𝑖𝑝+𝑒⁺_𝑗𝑝𝑖𝑀^𝑔_𝑝𝐿^𝑔_𝑖×^∞₀ℛ(𝑠)𝑒^𝜀𝑠𝑑𝑠𝑈_𝑖(𝑡). (24) Therefore, by Lemma 4, we obtain that 𝐷⁺𝕍(𝑡)≤𝐷⁺𝕍₁(𝑡)+𝐷⁺𝕍₂(𝑡)≤−^𝑛_𝑖=1𝑃_𝑖(𝜀)𝑈_𝑖(𝑡)−^𝑚_𝑗=1𝑄_𝑗(𝜀)𝑉_𝑗(𝑡)≤0. (25) When 𝑡=𝑡_𝑘,𝑘∈ℤ, we have 𝕍𝑡_𝑘=+0_𝑛𝑖=1𝜆_𝑖𝑈_𝑖𝑡_𝑘++0_𝑚𝑗=1𝑐⁺_𝑖𝑗𝐿^𝑓_𝑗^∞₀𝒦(𝑠)𝑒^{𝜀𝑠𝑡𝑘}_𝑡⁺⁰_{𝑘+0−𝑠}𝑉_𝑗(𝑧)+𝑑𝑧𝑑𝑠_{𝑚𝑚𝑗=1𝑞=1}𝑠⁺_𝑖𝑗𝑞+𝑠⁺_𝑖𝑞𝑗𝑀^𝑓_𝑞𝐿^𝑓_𝑗×^∞₀ℛ(𝑠)𝑒^{𝜀𝑠𝑡𝑘}_𝑡⁺⁰_{𝑘+0−𝑠}𝑉_𝑗(𝑧)𝑑𝑧𝑑𝑠_𝑛𝑗=1,𝜆_𝑛+𝑗𝑉_𝑗𝑡_𝑘++0_𝑛𝑖=1𝑑⁺_𝑗𝑖𝐿^𝑔_𝑖^∞₀𝒦(𝑠)𝑒^{𝜀𝑠𝑡𝑘}_𝑡⁺⁰_{𝑘+0−𝑠}𝑈_𝑖(𝑧)+𝑑𝑧𝑑𝑠_{𝑛𝑛𝑖=1𝑝=1}𝑒⁺_𝑗𝑖𝑝+𝑒⁺_𝑗𝑝𝑖𝑀^𝑔_𝑝𝐿^𝑔_𝑖×^∞₀ℛ(𝑠)𝑒^{𝜀𝑠𝑡𝑘}_𝑡⁺⁰_{𝑘+0−𝑠}𝑈_𝑖(𝑧),≤𝑑𝑧𝑑𝑠_𝑛𝑖=1𝜆_𝑖𝜂^𝑟_𝑘𝑈_𝑖(𝑡)+^𝑚_𝑗=1𝑐⁺_𝑖𝑗𝐿^𝑓_𝑗^∞₀𝒦𝑒^𝜀𝑠𝑡_𝑡−𝑠𝑉_𝑗(𝑧)+𝑑𝑧𝑑𝑠_{𝑚𝑚𝑗=1𝑞=1}𝑠⁺_𝑖𝑗𝑞+𝑠⁺_𝑖𝑞𝑗𝑀^𝑓_𝑞𝐿^𝑓_𝑗×^∞₀ℛ(𝑠)𝑒^𝜀𝑠𝑡_𝑡−𝑠𝑉_𝑗(𝑧)𝑑𝑧𝑑𝑠_𝑛𝑗=1,𝜆_𝑛+𝑗𝜂^𝑟_𝑘𝑉_𝑗(𝑡)+^𝑛_𝑖=1𝑑⁺_𝑗𝑖𝐿^𝑔_𝑖^∞₀𝒦(𝑠)𝑒^𝜀𝑠𝑡_𝑡−𝑠𝑈_𝑖(𝑧)+𝑑𝑧𝑑𝑠_{𝑛𝑛𝑖=1𝑝=1}𝑒⁺_𝑗𝑖𝑝+𝑒⁺_𝑗𝑝𝑖𝑀^𝑔_𝑝𝐿^𝑔_𝑖×^∞₀ℛ(𝑠)𝑒^𝜀𝑠𝑡_𝑡−𝑠𝑈_𝑖(𝑧)𝑑𝑧𝑑𝑠≤𝜂^𝑟_𝑘𝕍𝑡_𝑘. (26) Now, we claim that 𝕍𝑡_𝑘+0≤𝜂^𝑟_𝑘𝜂^𝑟_𝑘−1⋯𝜂^𝑟₁𝕍𝑡₀,𝑡∈𝑡_𝑘−1,𝑡_𝑘,𝑘∈ℤ. (27) In fact, for 𝑡∈(𝑡₀,𝑡₁], noticing that 𝜂₁≥1 and (25), we have 𝜂^𝑟₁𝕍𝑡₁≤𝜂^𝑟₁𝕍𝑡₀. (28) On the other hand, from (26), we have 𝕍𝑡₁+0≤𝜂^𝑟₁𝕍𝑡₁. (29) Combining (28) and (29), we obtain 𝕍𝑡₁+0≤𝜂^𝑟₁𝕍𝑡₀, (30) which implies that (27) holds for 𝑘=1. Assume that (27) holds for 𝑘=𝑚, that is, 𝕍𝑡_𝑚+0≤𝜂^𝑟_𝑚𝜂^𝑟_𝑚−1⋯𝜂^𝑟₁𝕍𝑡₀. (31) Then, for 𝑡∈(𝑡_𝑚,𝑡_𝑚+1], from (25), we have 𝕍𝑡_𝑚+1≤𝕍𝑡_𝑚+0. (32) On the other hand, from (26), we have 𝕍𝑡_𝑚+1+0≤𝜂^𝑟_𝑚+1𝕍𝑡_𝑚+1. (33) From (31)–(33), we obtain 𝕍𝑡_𝑚+1+0≤𝜂^𝑟_𝑚+1𝜂^𝑟_𝑚⋯𝜂^𝑟₁𝕍𝑡₀. (34) This shows that (27) holds for 𝑘=𝑚+1. Hence, by mathematical induction, (27) holds for all 𝑘∈ℤ. Combining (25) and (27), we obtain 𝕍(𝑡)≤𝕍𝑡_𝑘+0≤𝜂^𝑟_𝑘𝜂^𝑟_𝑘−1,⋯,𝜂^𝑟₁𝕍𝑡₀, (35) for all 𝑡∈(𝑡_𝑘,𝑡_𝑘+1],𝑘∈ℤ. Noticing that 𝜂^𝑟_𝑘≤𝑒^{𝜃(𝑡𝑘−𝑡𝑘−1)},𝑘∈ℤ in (𝑆₆), we have 𝕍(𝑡)≤𝕍𝑡₀𝑒^{𝜃(𝑡𝑘−𝑡𝑘−1)}𝑒^{𝜃(𝑡𝑘−1−𝑡𝑘−2)}⋯𝑒^{𝜃(𝑡1−𝑡0)}≤𝕍(0)𝑒^𝜃𝑡, (36) for all 𝑡∈(𝑡_𝑘,𝑡_𝑘+1],𝑘∈ℤ. On the other hand, it follows from (21) that 𝕍(𝑡)≥𝜆^−𝑛_𝑖=1𝑈_𝑖(𝑡)+^𝑚_𝑗=1𝑉_𝑗(𝑡)=𝜆^−𝑛_𝑖=1𝑥_𝑖𝑡,𝜙−𝑥_𝑖𝑡,𝜑^𝑟+^𝑚_𝑗=1𝑦_𝑗𝑡,𝜙−𝑦_𝑗𝑡,𝜑^𝑟𝑒^𝜀𝑡,𝕍(0)=^𝑛_𝑖=1𝜆_𝑖𝑈_𝑖(0)+^𝑚_𝑗=1𝑐⁺_𝑖𝑗𝐿^𝑓_𝑗^∞₀𝒦(𝑠)𝑒^𝜀𝑠0_−𝑠𝑉_𝑗(𝑧)+𝑑𝑧𝑑𝑠_{𝑚𝑚𝑗=1𝑞=1}𝑠⁺_𝑖𝑗𝑞+𝑠⁺_𝑖𝑞𝑗𝑀^𝑓_𝑞𝐿^𝑓_𝑗×^∞₀ℛ(𝑠)𝑒^𝜀𝑠0_−𝑠𝑉_𝑗(𝑧)+𝑑𝑧𝑑𝑠_𝑛𝑗=1𝜆_𝑛+𝑗𝑉_𝑗(0)+^𝑛_𝑖=1𝑑⁺_𝑗𝑖𝐿^𝑔_𝑖^∞₀𝒦(𝑠)𝑒^𝜀𝑠0_−𝑠𝑈_𝑖(𝑧)+𝑑𝑧𝑑𝑠_{𝑛𝑛𝑖=1𝑝=1}𝑒⁺_𝑗𝑖𝑝+𝑒⁺_𝑗𝑝𝑖𝑀^𝑔_𝑝𝐿^𝑔_𝑖×^∞₀ℛ(𝑠)𝑒^𝜀𝑠0_−𝑠𝑈_𝑖(𝑧)𝑑𝑧𝑑𝑠≤𝜆⁺Λ^𝑚_𝑗=1sup_{(]𝑠∈−∞,0}𝜙_𝑥𝑖(𝑠)−𝜓_𝑥𝑖(𝑠)^𝑟+^𝑛_𝑖=1sup_{(]𝑠∈−∞,0}𝜙_𝑦𝑗(𝑠)−𝜓_𝑦𝑗(𝑠)^𝑟, (37) where 𝜆⁺=max𝜆₁,𝜆₂,…,𝜆_𝑛+𝑚,𝜆⁻=min𝜆₁,𝜆₂,…,𝜆_𝑛+𝑚,Λ=max1+^𝑛_𝑖=1max_{1≤𝑗≤𝑚}𝑑⁺_𝑗𝑖𝐿^𝑔_𝑖^∞₀𝒦(𝑠)𝑒^𝜀𝑠0_−𝑠𝑒^𝜀𝑧+𝑑𝑧𝑑𝑠_{𝑛𝑛𝑖=1𝑝=1}max_{1≤𝑗≤𝑚}𝑒⁺_𝑗𝑖𝑝+𝑒⁺_𝑗𝑝𝑖𝑀^𝑔_𝑝𝐿^𝑔_𝑖×^∞₀ℛ(𝑠)𝑒^𝜀𝑠0_−𝑠𝑒^𝜀𝑧𝑑𝑧𝑑𝑠,1+^𝑚_𝑗=1max_{1≤𝑖≤𝑛}𝑐⁺_𝑖𝑗𝐿^𝑓_𝑗^∞₀𝒦(𝑠)𝑒^𝜀𝑠0_−𝑠𝑒^𝜀𝑧+𝑑𝑧𝑑𝑠_{𝑚𝑚𝑗=1𝑞=1}max_{1≤𝑖≤𝑛}𝑠⁺_𝑖𝑗𝑞+𝑠⁺_𝑖𝑞𝑗𝑀^𝑓_𝑞𝐿^𝑓_𝑗×^∞₀ℛ(𝑠)𝑒^𝜀𝑠0_−𝑠𝑒^𝜀𝑧𝑑𝑧𝑑𝑠. (38) Together with (36)-(37), we have ^𝑛_𝑖=1𝑥_𝑖𝑡,𝜙−𝑥_𝑖𝑡,𝜑^𝑟+^𝑚_𝑗=1𝑦_𝑗𝑡,𝜙−𝑦_𝑗𝑡,𝜑^𝑟≤𝜆⁺𝜆⁻Λ𝑒^{−(𝜀−𝜃)𝑡}𝜙−𝜑^𝑟_Δ, (39) for all 𝑡≥0. Let ℳ=((𝜆⁺/𝜆⁻)Λ)^1/𝑟, 𝛼=(𝜀−𝜃)/𝑟 and then we have 𝑧(𝑡,𝜙)−𝑧𝑡,𝜑_𝑟≤ℳ𝑒^−𝛼𝑡𝜙−𝜑_Δ. (40) This completes the proof.

In the following, we will study the existence, uniqueness, and global exponential stability of periodic solution of system (1) by exploiting Lemmas 5 and 6.

Theorem 7. Assume that assumptions (S₁)–(S₆) hold, then system (1) has a unique 𝜔-periodic solution, which is globally exponentially stable.

Proof. Firstly, we prove the existence of periodic solution of system (1). To this end, let 𝑧(𝑡,𝜑)=(𝑥₁(𝑡,𝜑),𝑥₂(𝑡,𝜑),…,𝑥_𝑛(𝑡,𝜑),𝑦₁(𝑡,𝜑), 𝑦₂(𝑡,𝜑),…,𝑦_𝑚(𝑡,𝜑))^𝑇 be an arbitrary solution of system (1) through (0,𝜑), where 𝜑∈𝐶^∗. Define 𝑧(𝑡,𝜙)=𝑧(𝑡+𝜔,𝜑), where 𝜙=𝑧(𝑠+𝜔,𝜑),𝑠≤0. We can know that 𝜙∈𝐶^∗ and 𝑧(𝑡,𝜙)=(𝑥₁(𝑡,𝜙),𝑥₂(𝑡,𝜙),…,𝑥_𝑛(𝑡,𝜙),𝑦₁(𝑡,𝜙),𝑦₂(𝑡,𝜙),…,𝑦_𝑚(𝑡,𝜙))^𝑇 is also a solution of system (1) through (0,𝜙). By virtue of Lemma 6, we have ^𝑛_𝑖=1𝑥_𝑖𝑡,𝜙−𝑥_𝑖𝑡,𝜑^𝑟+^𝑚_𝑗=1𝑦_𝑖𝑡,𝜙−𝑦_𝑖𝑡,𝜑^𝑟1/𝑟≤ℳ𝑒^−𝛼𝑡𝜙−𝜑_Δ, (41) for 𝑡≥0. So, we have ^𝑛_𝑖=1𝑥_𝑖𝑡+𝜔,𝜑−𝑥_𝑖𝑡,𝜑^𝑟+^𝑚_𝑗=1𝑦_𝑗𝑡+𝜔,𝜑−𝑦_𝑗𝑡,𝜑^𝑟1/𝑟≤ℳ𝑒^−𝛼𝑡𝜙−𝜑_Δ, (42) for 𝑡≥0. It follows from Lemma 5 that ^𝑛_𝑖=1𝑥_𝑖𝑡+𝜔,𝜑−𝑥_𝑖𝑡,𝜑+^𝑚_𝑗=1𝑦_𝑗𝑡+𝜔,𝜑−𝑦_𝑗≤()𝑡,𝜑𝑛+𝑚^1−1/𝑟ℳ𝑒^−𝛼𝑡𝜙−𝜑_Δ. (43) Noticing that for 𝑖=1,2,…,𝑛, 𝑥_𝑖𝑡+𝑘𝜔,𝜑=𝑥_𝑖𝑡,𝜑+^𝑘_𝑠=1𝑥_𝑖𝑡+𝑠𝜔,𝜑−𝑥_𝑖𝑡+(𝑠−1)𝜔,𝜑. (44) It follows from (43)-(44) that lim_𝑘→∞𝑥_𝑖𝑡+𝑘𝜔,𝜑=𝑥_𝑖𝑡,𝜑+lim^𝑘_{𝑘→∞𝑠=1}𝑥_𝑖𝑡+𝑠𝜔,𝜑−𝑥_𝑖𝑡+(𝑠−1)𝜔,𝜑≤𝑥_𝑖𝑡,𝜑+(𝑛+𝑚)^(1−1/𝑟)ℳ𝜙−𝜑_Δlim^𝑘_{𝑘→∞𝑠=1}𝑒^{−𝛼(𝑡+(𝑠−1)𝜔)}≤𝑥_𝑖𝑡,𝜑+(𝑛+𝑚)^(1−1/𝑟)ℳ𝑒^−𝛼𝑡𝜙−𝜑_Δ^∞_𝑠=1𝑒^{−𝛼(𝑠−1)𝜔}<∞, (45) which implies that lim_𝑘→∞𝑥_𝑖(𝑡+𝑘𝜔,𝜑) exists. Similar to (44) and (45), we obtain that lim_𝑘→∞𝑦_𝑗(𝑡+𝑘𝜔,𝜑) exists. Let 𝑧^∗𝑡,𝜑^∗=𝑥^∗₁𝑡,𝜑^∗,𝑥^∗₂𝑡,𝜑^∗,…,𝑥^∗_𝑛𝑡,𝜑^∗,𝑦^∗₁𝑡,𝜑^∗,𝑦^∗₂𝑡,𝜑^∗,…,𝑦^∗_𝑚𝑡,𝜑^∗𝑇, (46) where lim_𝑘→∞𝑥_𝑖(𝑡+𝑘𝜔,𝜑)=𝑥^∗_𝑖(𝑡,𝜑^∗),lim_𝑘→∞𝑦_𝑗(𝑡+𝑘𝜔,𝜑)=𝑦^∗_𝑗(𝑡,𝜑^∗) for 𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚. Then 𝑧^∗(𝑡,𝜑^∗) is an 𝜔-periodic solution for system (1).
Secondly, we prove the uniqueness of periodic solution of system (1). Assume that 𝑧(𝑡,𝜁)=(𝑥₁(𝑡,𝜁),𝑥₂(𝑡,𝜁),…,𝑥_𝑛(𝑡,𝜁),𝑦₁(𝑡,𝜁),𝑦₂(𝑡,𝜁),…,𝑦_𝑚(𝑡,𝜁))^𝑇 is another 𝜔-periodic solution of system (1) through (0,𝜁), where 𝜁∈𝐶^∗. By a minor modification of the proof of (43), we have ^𝑛_𝑖=1𝑥^∗_𝑖𝑡,𝜑^∗−𝑥_𝑖(𝑡,𝜁)+^𝑚_𝑗=1𝑦^∗_𝑗𝑡,𝜑^∗−𝑦_𝑗(=𝑡,𝜁)_𝑛𝑖=1𝑥^∗_𝑖𝑡+𝑘𝜔,𝜑^∗−𝑥_𝑖(+𝑡,𝜁)_𝑚𝑗=1𝑦^∗_𝑗𝑡+𝑘𝜔,𝜑^∗−𝑦_𝑗(≤()𝑡,𝜁)𝑛+𝑚^1−1/𝑟ℳ𝑒^{−𝛼(𝑡+𝑘𝜔)}𝜑^∗−𝜁_Δ,𝑡≥0. (47) Taking 𝑘→∞, we have 𝑥^∗_𝑖𝑡,𝜑^∗=𝑥_𝑖()𝑦𝑡,𝜁,𝑡≥0,𝑖=1,2,…,𝑛,_∗𝑗𝑡,𝜑^∗=𝑦_𝑗(𝑡,𝜁),𝑡≥0,𝑗=1,2,…,𝑚, (48) which implies that system (1) has a unique 𝜔-periodic solution.
Finally, since 𝑧^∗(𝑡,𝜑^∗) is a unique 𝜔-periodic solution of system (1), let 𝑧(𝑡,𝜓) be any other solution of system (1) through (0,𝜓). From Lemma 6, we obtained that 𝑧𝑡,𝜓−𝑧^∗𝑡,𝜑^∗_𝑟≤ℳ𝑒^−𝛼𝑡𝜓−𝜑^∗_Δ,𝑡≥0, (49) where 𝜓,𝜑^∗∈𝐶^∗ and 𝛼,ℳ are the same as defined in Lemma 6. It follows from Definition 2 that the 𝜔-periodic solution 𝑧^∗(𝑡,𝜑^∗) is globally exponentially stable. Up to now, we conclude that system (1) has a unique 𝜔-periodic solution 𝑧^∗(𝑡,𝜑^∗), which is globally exponentially stable. This completes the proof.

Remark 8. In assumption (S₅), we only assume that the impulsive operators 𝐻_𝑖𝑘(⋅) and 𝐸_𝑗𝑘(⋅) are Lipschitz continuous, which remove the usual assumptions that the boundedness and linearity of the impulsive operators are required in [18, 19, 21, 29–31]. Thus, our results have wider adaptive range. Particularly, if we take the linear operators Δ𝑥_𝑖(𝑡_𝑘)=−𝛾_𝑖𝑘𝑥_𝑖(𝑡⁻_𝑘) and Δ𝑦_𝑗(𝑡_𝑘)=−𝛼_𝑗𝑘𝑦_𝑗(𝑡⁻_𝑘) as considered in [18, 19, 21, 29–31], that is,(S₇)𝐻_𝑖𝑘(𝑥_𝑖(𝑡⁻_𝑘))=(1−𝛾_𝑖𝑘)𝑥_𝑖(𝑡⁻_𝑘),𝛾_𝑖𝑘𝐸∈(0,2),𝑖=1,2,…,𝑛,𝑘=1,2,…,_𝑗𝑘(𝑦_𝑗(𝑡⁻_𝑘))=(1−𝛼_𝑗𝑘)𝑦_𝑗(𝑡⁻_𝑘),𝛼_𝑗𝑘∈(0,2),𝑗=1,2,…𝑚,𝑘=1,2,…,then we have 𝐻_𝑖𝑘=|1−𝛾_𝑖𝑘|<1,𝐸_𝑗𝑘=|1−𝛼_𝑗𝑘|<1. So we can choose 𝜂_𝑘=1 and 𝜃=0 to satisfy assumption (S₆). In this case, we have the following interesting corollary.

Corollary 9. Assume that assumptions ((S₁)–(S₄)) and (S₇) hold; then system (1) has a unique 𝜔-periodic solution, which is globally exponentially stable.

Remark 10. Note that when 𝑡_𝑘−𝑡_𝑘−1=∞,𝑘=1,2,… in assumption (S₆), which implies that there are no impulsive effects on system (1). Correspondingly, we call system (1) an impulse-free. In this case, we have the following corollary.

Corollary 11. Assume that assumptions ((S₁)–(S₄)) hold; then the impulse-free system (1) has a unique 𝜔-periodic solution, which is globally exponentially stable.

Remark 12. Clearly, based on the general ‖⋅‖_𝑟 and Lemma 5, a general criterion ensuring the existence of periodic solution and its global exponential stability of system (1) with and without impulses has been established. Compared with results in [6, 7, 15–17], it is easy to see that our results are extended and improved because their results can be viewed as the special case of 𝑟=1 in assumption (S₄). In addition, since the nonnetwork parameter 𝑟 is introduced in the condition (S₄), it can allow much broader applications for designing the circuit of a convergent impulsive network.

Remark 13. In assumption (S₃), if the kernel is a delta function of the form: 𝐾_𝑖𝑗(𝑠)=𝑅_𝑖𝑗𝑞(𝑠)(),𝐾=𝛿𝑠−𝜏_𝑗𝑖(𝑠)=𝑅_𝑗𝑖𝑝(𝑠)=𝛿(𝑠−𝜎), (50) where 𝜏≥0 and 𝜎≥0, then system (1) with continuously distributed delays reduces to the model with discrete delays in [29]. According to Lemma 3, we know that the condition (H₅) of Theorem 3.1 in [29] implies that Ω with 𝑟=1 is an 𝑀-matrix but not vice versa. Thus, our results are new and complementary to their results.

4. An Example

In this section, an example and its numerical simulation are given to illustrate the correctness of the obtained theoretical results.

An Example. Consider the following high-order BAM neural networks with infinite distributed delays and impulses: 𝑑𝑥₁(𝑡)()𝑥𝑑𝑡=−8+sin2𝑡₁(𝑡)+(2+cos2𝑡)^∞₀𝑒^−𝑠𝑓₁𝑦₁(𝑡−𝑠)𝑑𝑠+2sin2𝑡^∞₀𝑠𝑒^−𝑠𝑓₁𝑦₁(𝑡−𝑠)𝑓₁𝑦₁(𝑡−𝑠)𝑑𝑠+cos2𝑡,𝑡≠𝑡_𝑘,𝑥₁𝑡⁺_𝑘=𝑥₁𝑡⁻_𝑘+Δ𝑥₁𝑡_𝑘=𝐻_1𝑘𝑥₁𝑡⁻_𝑘,𝑘=1,2,…,𝑑𝑦₁(𝑡)()𝑦𝑑𝑡=−10−3cos2𝑡₁(𝑡)+(1+sin2𝑡)^∞₀𝑒^−𝑠𝑔₁𝑥₁(𝑡−𝑠)𝑑𝑠+2cos2𝑡^∞₀𝑠𝑒^−𝑠𝑔₁𝑥₁(𝑡−𝑠)𝑔₁𝑥₁(𝑡−𝑠)𝑑𝑠+sin2𝑡,𝑡≠𝑡_𝑘,𝑦₁𝑡⁺_𝑘=𝑦₁𝑡⁻_𝑘+Δ𝑦₁𝑡_𝑘=𝐸_1𝑘𝑦₁𝑡⁻_𝑘,𝑘=1,2,…, (51) where 𝑓₁(𝑢)=𝑔₁(𝑢)=tanh(𝑢). By simple calculation, we obtain that 𝐿^𝑔₁=𝐿^𝑓₁=𝑀^𝑓₁=𝑀^𝑔₁=1 and Σ=7−7−66+𝑟. (52) If the integer 𝑟≥1, then Σ is an 𝑀-matrix. Thus, assumptions ((S₁)–(S₄)) are satisfied for system (51). For the impulsive part, the following two cases are considered.

Case 1. When 𝑡_𝑘−𝑡_𝑘−1=∞,𝑘=1,2,…, by Corollary 11, we conclude that the impulse-free system (51) has a unique 𝜋-periodic solution, which is globally exponentially stable.

Case 2. When the impulsive parts are taken as the nonlinear operators such that Δ𝑥₁(𝑡_𝑘)=𝛾tan(𝑥₁(𝑡⁻_𝑘)),𝛾∈(0,2) and Δ𝑦₁(𝑡_𝑘)=𝛾tan(𝑦₁(𝑡⁻_𝑘)),𝛾∈(0,2), that is, 𝐻_1𝑘𝑥_𝑖𝑡⁻_𝑘=𝑥₁𝑡⁻_𝑘−𝛾tan𝑥₁𝑡⁻_𝑘,()𝐸𝛾∈0,2,𝑘=1,2,…,_1𝑘𝑦_𝑗𝑡⁻_𝑘=𝑦₁𝑡⁻_𝑘−𝛾tan𝑦₁𝑡⁻_𝑘,𝛾∈(0,2),𝑘=1,2,…, (53) which satisfy the assumption (S₅) with Lipschitz constants 𝐻_1𝑘=𝐸_1𝑘≤|1−𝛾|<1,𝑘=1,2,…. So we can choose 𝜂_𝑘=1 and 𝜃=0 to satisfy the assumption (S₆). According to Theorem 7, we conclude that the impulsive system (51) has a unique 𝜋-periodic solution, which is globally exponentially stable. However, in this case, results in [18, 19, 21, 29–31] are ineffective because the function tan(⋅) is nonlinear. Moreover, the condition (H₅) of Theorem 3.1 in [29] cannot be applied to system (51), since 𝑏⁻₁−𝑐⁺₁₁𝐿^𝑓₁−2𝑠⁺₁₁₁𝑀^𝑓₁𝐿^𝑓₁=0>0̸.

5. Conclusions

In this paper, we have studied the existence, uniqueness, and global exponential stability of periodic solution for a kind of high-order BAM neural networks with continuously distributed delays and general impulses. It should be noted that some extended and improved criteria have been derived by exploiting the general ‖⋅‖_𝑟, Lemma 5, and the Lyapunov functional method. In addition, these criteria are in terms of 𝑀-matrix, which can be easily checked by many equivalent conditions listed in [32]. Finally, an example and its numerical simulation are given to show the feasibility and correctness of the obtained results.

Acknowledgments

The authors appreciate the anonymous reviewer for her/his insightful comments and valuable suggestions, which have led to a great improvement in this paper. This research is supported by the Chinese Universities Specialized Research Fund for the Doctoral Program (20110185110020), the Research Fund of Kunming University (XJ11L019, XJL11008), and the Fundamental Research Funds for the Central Universities (ZYGX2010J106).