Existence and Exponential Stability of Periodic Solution for a Class of Generalized Neural Networks with Arbitrary Delays

Abstract

By the continuation theorem of coincidence degree and 𝑀-matrix theory, we obtain some sufficient conditions for the existence and exponential stability of periodic solutions for a class of generalized neural networks with arbitrary delays, which are milder and less restrictive than those of previous known criteria. Moreover our results generalize and improve many existing ones.

1. Introduction

Consider the following generalized neural networks with arbitrary delays:

𝑥^′(𝑡)=𝐴(𝑡,𝑥(𝑡))[𝐵(𝑡,𝑥(𝑡))+𝐹(𝑡,𝑥_𝑡)], (1.1) where 𝐴(𝑡,𝑥(𝑡))=diag(𝑎₁(𝑡,𝑥₁(𝑡)),𝑎₂(𝑡,𝑥₂(𝑡)),…,𝑎_𝑛(𝑡,𝑥_𝑛(𝑡)),𝐵(𝑡,𝑥(𝑡))=𝑏₁(𝑡,𝑥₁(𝑡)),𝑏₂(𝑡,𝑥₂(𝑡)),…,𝑏_𝑛(𝑡,𝑥_𝑛(𝑡)))^𝑇,   𝐹(𝑡,𝑥_𝑡)=(𝑓₁(𝑡,𝑥_𝑡),𝑓₂(𝑡,𝑥_𝑡),…,𝑓_𝑛(𝑡,𝑥_𝑡))^𝑇, 𝑓_𝑖(𝑡,𝑥_𝑡)=𝑓_𝑖(𝑡,𝑥_1𝑡, 𝑥_2𝑡,…,𝑥_𝑛𝑡),   𝑥_𝑡=(𝑥_1𝑡,𝑥_2𝑡,…,𝑥_𝑛𝑡)^𝑇 is defined by 𝑥_𝑡(𝜃)=𝑥(𝑡+𝜃) = (𝑥₁(𝑡+𝜃),𝑥₂(𝑡+𝜃), …,𝑥_𝑛(𝑡+𝜃))^𝑇,𝜃∈𝐸, and 𝐸 is a subset of 𝑅⁻=(−∞,0].

System (1.1) contains many neural networks, for examples, the higher-order Cohen-Grossberg type neural networks with delays (see [1])

𝑥^′_𝑖(𝑡)=−𝑎_𝑖(𝑥_𝑖(𝑡))[𝑏_𝑖(𝑥_𝑖(𝑡))−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑔_𝑗(𝑥_𝑗(𝑡))−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑔_𝑗(𝑥_𝑗(𝑡−𝜏_𝑗−(𝑡)))_{𝑛𝑛𝑗=1𝑙=1}𝑏_𝑖𝑗𝑙(𝑡)𝑔_𝑗(𝑥_𝑗(𝑡−𝜏_𝑗(𝑡)))𝑔_𝑙(𝑥_𝑙(𝑡−𝜏_𝑙(𝑡)))+𝐼_𝑖(𝑡)],𝑖=1,2,…,𝑛, (1.2) the Cohen-Grossberg neural networks with bounded and unbounded delays (see [2])

𝑥^′_𝑖(𝑡)=−𝑎_𝑖(𝑡,𝑥_𝑖(𝑡))[𝑏_𝑖(𝑡,𝑥_𝑖(𝑡))−^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)𝑓_𝑗(𝑥_𝑗(𝑡−𝜏_𝑖𝑗−(𝑡)))_𝑛𝑗=1𝑑_𝑖𝑗(𝑡)𝑔_𝑗(^∞₀𝐾_𝑖𝑗(𝑢)𝑥_𝑗(𝑡−𝑢)d𝑢)+𝐼_𝑖(𝑡)],𝑖=1,2,…,𝑛, (1.3) the Cohen-Grossberg neural networks with time-varying delays (see [3])

𝑥^′_𝑖(𝑡)=−𝑎_𝑖(𝑡,𝑥_𝑖(𝑡))[𝑏_𝑖(𝑡,𝑥_𝑖(𝑡))−^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)𝑓_𝑗(𝑥_𝑗−(𝑡))_𝑛𝑗=1𝑑_𝑖𝑗(𝑡)𝑓_𝑗(𝑥_𝑗(𝑡−𝜏_𝑖𝑗(𝑡)))+𝐼_𝑖(𝑡)],𝑖=1,2,…,𝑛, (1.4) the celluar neural networks (see [4, Page 193]):

𝑥^′_𝑖(𝑡)=−𝑟_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑔_𝑗(𝑥_𝑗+(𝑡))_𝑛𝑗=1𝑏_𝑖𝑗(𝑡)𝑔_𝑗(𝑥_𝑗(𝑡−𝜏_𝑖𝑗(𝑡)))+𝐼_𝑖(𝑡),𝑖=1,2,…,𝑛, (1.5) and so on.

Since the model of Cohen-Grossberg neural networks was first introduced by Cohen and Grossberg in [5], the dynamical characteristics (including stable, unstable, and periodic oscillatory) of Cohen-Grossberg neural networks have been widely investigated for the sake of theoretical interest as well as application considerations. Many good results have already been obtained by some authors in [6–15] and the references cited therein. Moreover, the existing results are based on the assumption that demanding either the activation functions, the behaved functions, or delays is bounded in the above-mentioned literature. However, to the best of our knowledge, few authors have discussed the existence and exponential stability of periodic solutions of (1.1). In this paper, by using the continuation theorem of coincidence degree and 𝑀-matrix theory, we study model (1.1), and get some sufficient conditions for the existence and exponential stability of the periodic solution of system (1.1); our results generalize and improve many existing ones.

Let 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛,𝐵=(𝑏_𝑖𝑗)_𝑛×𝑛∈𝑅^𝑛×𝑛 be two matrices, 𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑛)^𝑇∈𝑅^𝑛,𝑣=(𝑣₁,𝑣₂,…,𝑣_𝑛)^𝑇∈𝑅^𝑛 be two vectors. For convenience, we introduce the following notations.

(i)𝐴≥0(𝐴>0) means that each element 𝑎_𝑖𝑗 is nonnegative (positive) respectively,(ii)𝐴≥𝐵(>𝐵) means 𝐴−𝐵≥0(>0),(iii)𝑢≥0(𝑢>0) means each element 𝑢_𝑖≥0(𝑢_𝑖>0),(iv)𝑢≤𝑣(𝑢<𝑣) means 𝑣−𝑢≥0(𝑣−𝑢>0),(v)|𝑢|=(|𝑢₁|,|𝑢₂|,…,|𝑢_𝑛|)^𝑇.

For continuous 𝜔-periodic function 𝑔∶𝑅→𝑅, we denote |𝑔|=max_{0≤𝑡≤𝜔}|𝑔(𝑡)|,𝐶_𝐸=𝐶[𝐸,𝑅^𝑛] is the family of continuous functions 𝜙=(𝜙₁,𝜙₂,…,𝜙_𝑛)^𝑇 from 𝐸⊂(−∞,0] to 𝑅^𝑛. Clearly, it is a Banach space with the norm ‖𝜙‖=max_{0≤𝑖≤𝑛}|𝜙_𝑖|, where |𝜙_𝑖|=sup_𝜃∈𝐸|𝜙_𝑖(𝜃)|. The initial conditions of system (1.1) are of the form

𝑥₀=𝜙,thatis,𝑥_𝑖(𝜃)=𝜙_𝑖(𝜃),𝜃∈𝐸,𝑖=1,2,…,𝑛, (1.6) where 𝜙=(𝜙₁,𝜙₂,…,𝜙_𝑛)^𝑇∈𝐶_𝐸. For 𝑉(𝑡)∈𝐶((𝑎,+∞),𝑅), let

𝐷⁻𝑉(𝑡)=limsup_ℎ→0−𝐷(𝑡+ℎ)−𝐷(𝑡)ℎ,𝐷⁻𝑉(𝑡)=liminf_ℎ→0−𝐷(𝑡+ℎ)−𝐷(𝑡)ℎ,𝑡∈(𝑎,+∞). (1.7)

Throughout this paper, we assume the following:

(𝐻₁) For 𝑖=1,2,…,𝑛,𝑎_𝑖,𝑏_𝑖∈𝐶[𝑅²,𝑅],𝑓_𝑖∈𝐶[𝑅×𝐶_𝐸,𝑅] and are 𝜔-periodic for their first arguments, respectively, that is, 𝑎_𝑖(𝑡+𝜔,𝑢)=𝑎_𝑖(𝑡,𝑢),𝑏_𝑖(𝑡+𝜔,𝑢)=𝑏_𝑖(𝑡,𝑢),𝑓_𝑖(𝑡+𝜔,𝜙)=𝑓_𝑖(𝑡,𝜙) so 𝐴(𝑡+𝜔,𝑢)=𝐴(𝑡,𝑢),𝐵(𝑡+𝜔,𝑢)=𝐵(𝑡,𝑢),𝐹(𝑡+𝜔,𝜙)=𝐹(𝑡,𝜙), for all 𝑡∈𝑅,𝑢∈𝑅^𝑛,𝜙∈𝐶_𝐸.(𝐻₂) There exists a positive diagonal matrix 𝐴=diag(𝑎₁,𝑎₂,…,𝑎_𝑛) such that 𝐴(𝑡,𝑢)≥𝐴, for all (𝑡,𝑢)∈𝑅^𝑛+1.(𝐻₃) There is a positive diagonal matrix 𝐵=diag(𝑏₁,𝑏₂,…,𝑏_𝑛) such that |𝐵(𝑡,𝑢)|≥𝐵|𝑢|, and 𝑏_𝑖(𝑡,𝑢_𝑖)𝑢_𝑖>0 or 𝑏_𝑖(𝑡,𝑢_𝑖)𝑢_𝑖<0 for all (𝑡,𝑢)∈𝑅^𝑛+1,𝑖=1,2,…,𝑛.(𝐻₄) There exist a nonnegative matrix 𝐶=(𝑐_𝑖𝑗)_𝑛×𝑛∈𝑅^𝑛×𝑛 and a nonnegative vector 𝐷=(𝐷₁,𝐷₂,…,𝐷_𝑛)^𝑇 such that |𝐹(𝑡,𝜙)|≤𝐶|𝜙|+𝐷, for all (𝑡,𝜙)∈𝑅×𝐶_𝐸, where 𝜙=(𝜙₁,𝜙₂,…,𝜙_𝑛)^𝑇∈𝐶_𝐸,|𝜙|=(|𝜙₁|,|𝜙₂|,…,|𝜙_𝑛|)^𝑇.

2. Preliminaries

In this section, we first introduce some definitions and lemmas which play an important role in the proof of our main results in this paper.

Definition 2.1. Let 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇 be an 𝜔-periodic solution of system (1.1) with initial value 𝜙∈𝐶_𝐸, if there exist two constants 𝛼>0 and 𝑀>1 such that for every solution 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛)^𝑇 of system (1.1) with initial value (1.6), |𝑥_𝑖(𝑡)−𝑥_𝑖(𝑡)|≤𝑀‖𝜙−𝜙‖𝑒^−𝛼𝑡,∀𝑡>0,𝑖=1,2,…,𝑛. (2.1) Then 𝑥(𝑡) is said to be globally exponential stable.

Definition 2.2. A real matrix 𝑊=(𝑤_𝑖𝑗)_𝑛×𝑛∈𝑅^𝑛×𝑛 is said to be an 𝑀-matrix if 𝑤_𝑖𝑗≤0,𝑖,𝑗=1,2,…,𝑛,𝑖≠𝑗, and 𝑊⁻¹≥0.

Lemma 2.3 (see [15, 16]). Assume that 𝐴 is an 𝑀-matrix and 𝐴𝑢≤𝑑,𝑢,𝑑∈𝑅^𝑛, then 𝑢≤𝐴⁻¹𝑑.

Lemma 2.4 (see [15, 16]). Let 𝑊=(𝑤_𝑖𝑗)_𝑛×𝑛 with 𝑤_𝑖𝑗≤0,𝑖,𝑗=1,2,…,𝑛,𝑖≠𝑗, then the following statements are equivalent.(i)𝑊 is an 𝑀-matrix.(ii)There exists a positive vector 𝜂=(𝜂₁,𝜂₂,…,𝜂_𝑛)>0 such that 𝜂𝑊>0.(iii)There exists a positive vector 𝜉=(𝜉₁,𝜉₂,…,𝜉_𝑛)^𝑇>0 such that 𝑊𝜉>0.

Lemma 2.5 (see [15, 16]). Let 𝐴≥0 be an 𝑛×𝑛 matrix and 𝜌(𝐴)<1, then (𝐸_𝑛−𝐴)⁻¹≥0, where 𝐸_𝑛 denotes the identity matrix of size 𝑛, so 𝐸_𝑛−𝐴 is an 𝑀-matrix.

Now we introduce Mawhin's continuation theorem which will be fundamental in this paper.

Lemma 2.6 (see [17]). Let 𝑋 and 𝑌 be two Banach spaces and 𝐿 a Fredholm mapping of index zero. Assume that Ω⊂𝑋 is an open bounded set and 𝑁∶𝑋→𝑍 is a continuous operator which is 𝐿-compact on Ω. Then 𝐿𝑥=𝑁𝑥 has at least one solution in Dom𝐿∩Ω, if the following conditions are satisfied:(1)𝐿𝑥≠𝜆𝑁𝑥, for all (𝑥,𝜆)∈(Dom𝐿∩𝜕Ω)×(0,1),(2)𝑄𝑁𝑥≠0, for all 𝑥∈𝜕Ω∩Ker𝐿,(3)deg{𝐽𝑄𝑁|_Ker𝐿∩Ω,Ω∩Ker𝐿,0}≠0.

Let

𝑋=𝑌={𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇∈𝐶(𝑅,𝑅^𝑛)∶𝑥(𝑡+𝜔)=𝑥(𝑡),𝑡∈𝑅} (2.2) with the norm defined by ‖𝑥‖=max_{1≤𝑖≤𝑛}|𝑥_𝑖|, where |𝑥_𝑖|=max_{𝑡∈[0,𝜔]}|𝑥_𝑖(𝑡)|. Clearly, 𝑋 and 𝑌 are two Banach spaces. Let 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇∈𝑋=𝑌, we define the linear operator 𝐿∶Dom𝐿⊂𝑋→𝑌 as

(𝐿𝑥)(𝑡)=𝑥^′(𝑡)=(𝑥^′₁(𝑡),𝑥^′₂(𝑡),…,𝑥^′_𝑛(𝑡))^𝑇,Dom𝐿={𝑥∈𝑋∶𝑥^′∈𝑌}, (2.3) and the operators 𝑁∶𝑋→𝑋,𝑃∶𝑋→𝑋,𝑄∶𝑌→𝑌 as

(𝑁𝑥)(𝑡)=𝐴(𝑡,𝑥(𝑡))[𝐵(𝑡,𝑥(𝑡))+𝐹(𝑡,𝑥_𝑡)]∶=Δ(𝑡,𝑥_𝑡1),𝑃𝑥=𝑄𝑥=𝜔^𝜔₀1𝑥(𝑡)d𝑡=(𝜔^𝜔₀𝑥₁1(𝑡)d𝑡,𝜔^𝜔₀𝑥₂1(𝑡)d𝑡,…,𝜔^𝜔₀𝑥_𝑛(𝑡)d𝑡)^𝑇. (2.4) It is not difficult to show that 𝑃 and 𝑄 are continuous projectors and the following conditions are satisfied:

Ker𝐿=𝑅^𝑛=Im𝑃=Im𝑄,Im𝐿={𝑦∈𝑌=𝑋∶^𝜔₀𝑦(𝑡)d𝑡=0}=Ker𝑄=Im(𝐼−𝑄),Im𝐿isclosedin𝑌,dimKer𝐿=𝑛=codimIm𝐿. (2.5) Thus, the mapping 𝐿 is a Fredholm mapping of index zero and the isomorphism 𝐽∶Im𝑄→Ker𝐿 is the identity operator; the generalized inverse (of 𝐿|_{Dom𝐿∩Ker𝑃})𝐾_𝑃∶Im𝐿→Ker𝑃∩Dom𝐿 exists, which has the form

(𝐾_𝑃𝑥)(𝑡)=^𝑡₀1𝑥(𝑠)d𝑠−𝜔^𝜔₀^𝑡₀𝑥(𝑠)d𝑠d𝑡,∀𝑥∈Im𝐿. (2.6) Therefore

1(𝑄𝑁𝑥)(𝑡)=𝜔^𝜔₀Δ(𝑡,𝑥_𝑡)d𝑡,(𝐾_𝑃(𝐼−𝑄)𝑁𝑥)(𝑡)=^𝑡₀Δ(𝑠,𝑥_𝑠1)d𝑠−𝜔^𝜔₀^𝑡₀Δ(𝑠,𝑥_𝑠1)d𝑠d𝑡+(2−𝑡𝜔)^𝜔₀Δ(𝑠,𝑥_𝑠)d𝑠. (2.7)

3. Existence of Periodic Solutions

In this section, we shall use Lemma 2.6 to study the existence of at least one periodic solution of system (1.1).

Theorem 3.1. Let (𝐻₁)–(𝐻₄) hold. Moveover, suppose that (𝐻₅)𝐸−𝐾 is a 𝑀-matrix, where the matrix 𝐾=(𝑘_𝑖𝑗)_𝑛×𝑛=𝐵⁻¹𝐶.then (i)system (1.1) has at least one 𝜔-periodic solution;(ii)there exists a nonnegative constant 𝛿 such that for all 𝜔-periodic solution 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇 of system (1.1), |𝑥_𝑖(𝑡)|≤𝛿,𝑖=1,2,…,𝑛.

Proof. Clearly, 𝑄𝑁 and 𝐾_𝑃(𝐼−𝑄)𝑁 are continuous functions and for every bounded subset Ω⊂𝑋,𝑄𝑁(Ω),𝐾_𝑃(𝐼−𝑄)𝑁(Ω), and (𝐾_𝑃(𝐼−𝑄)𝑁𝑥)^′,𝑥∈Ω are bounded. By using the Arzela-Ascoli theorem, 𝑄𝑁(Ω) and 𝐾_𝑃(𝐼−𝑄)𝑁(Ω) are compact, therefore 𝑁 is 𝐿-compact on Ω. Consider the following operator equation: 𝐿𝑥=𝜆𝑁𝑥,𝜆∈(0,1). (3.1) That is, 𝑥^′(𝑡)=𝜆𝐴(𝑡,𝑥(𝑡))[𝐵(𝑡,𝑥(𝑡))+𝐹(𝑡,𝑥_𝑡)], (3.2) or 𝑥^′_𝑖(𝑡)=𝜆𝑎_𝑖(𝑡,𝑥_𝑖(𝑡))[𝑏_𝑖(𝑡,𝑥_𝑖(𝑡))+𝑓_𝑖(𝑡,𝑥_𝑡)],𝑖=1,2,…,𝑛. (3.3) Assume that 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇∈𝑋 is a solution of (3.3) for some 𝜆∈(0,1). Then, for any 𝑖=1,2,…,𝑛,𝑥_𝑖(𝑡) are all continuous 𝜔-periodic functions, and there exist 𝑡_𝑖∈[0,𝜔], such that |𝑥_𝑖(𝑡_𝑖)|=max_{𝑡∈[0,𝜔]}|𝑥_𝑖(𝑡)|=|𝑥_𝑖|,𝑥^′_𝑖(𝑡_𝑖)=0,𝑖=1,2,…,𝑛, (3.4) from (𝐻₂), we have 𝑏_𝑖(𝑡_𝑖,𝑥_𝑖(𝑡_𝑖))+𝑓_𝑖(𝑡_𝑖,𝑥_𝑡𝑖)=0,𝑖=1,2,…,𝑛. (3.5) It follows from (𝐻₃) that 𝑏_𝑖|𝑥_𝑖|≤|𝑏_𝑖(𝑡_𝑖,𝑥_𝑖(𝑡_𝑖))|=|𝑓_𝑖(𝑡_𝑖,𝑥_𝑡𝑖≤)|_𝑛𝑗=1𝑐_𝑖𝑗|𝑥_𝑗𝑡𝑖|+𝐷_𝑖≤^𝑛_𝑗=1𝑐_𝑖𝑗|𝑥_𝑗|+𝐷_𝑖,𝑖=1,2,…,𝑛. (3.6) Thus |𝑥_𝑖|≤𝑏_𝑖^𝑛−1_𝑗=1𝑐_𝑖𝑗|𝑥_𝑗|+𝑏_𝑖⁻¹𝐷_𝑖,𝑖=1,2,…,𝑛, (3.7) we denote the vector 𝑑=(𝑑₁,𝑑₂,…,𝑑_𝑛)^𝑇,|𝑥|=(|𝑥₁|,|𝑥₂|,…,|𝑥_𝑛|)^𝑇, where 𝑑_𝑖=𝑏_𝑖⁻¹(𝐷_𝑖+1)>0,𝑖=1,2,…,𝑛. It follows from (3.7) that (𝐸−𝐾)|𝑥|<𝑑. (3.8) Since (𝐻₅), and application of Lemma 2.3 yields |𝑥|<(𝐸−𝐾)⁻¹𝑑=(ℎ₁,ℎ₂,…,ℎ_𝑛)^𝑇=ℎ, (3.9) where ℎ satisfies the equation ℎ=𝐾ℎ+𝑑, that is, ℎ_𝑖=∑^𝑛_𝑗=1𝑘_𝑖𝑗ℎ_𝑗+𝑑_𝑖>0.
Take Ω={𝑥∈𝑋∶|𝑥_𝑖|<ℎ_𝑖,𝑖=1,2,…,𝑛}. (3.10) It is easy to see that Ω satisfies condition (1) in Lemma 2.6.
For all 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇∈𝜕Ω∩Ker𝐿, 𝑥 is a constant vector in 𝑅^𝑛 and there exists some 𝑖∈{1,2,…,𝑛} such that |𝑥_𝑖|=|𝑥_𝑖|=ℎ_𝑖, we claim that |(𝑄𝑁𝑥)_𝑖|>0,sothat𝑄𝑁𝑥≠0. (3.11) We firstly claim that (1)if 𝑏_𝑖(𝑡,𝑢_𝑖)𝑢_𝑖>0, then 𝑥_𝑖(𝑄𝑁𝑥)_𝑖>0,(2)if 𝑏_𝑖(𝑡,𝑢_𝑖)𝑢_𝑖<0, then 𝑥_𝑖(𝑄𝑁𝑥)_𝑖<0.
We only prove (1), since the proof of (2) is similar. If 𝑏_𝑖(𝑡,𝑢_𝑖)𝑢_𝑖>0, we have 𝑥_𝑖[𝑏_𝑖(𝑡,𝑥_𝑖(𝑡))+𝑓_𝑖(𝑡,𝑥_𝑡)]≥𝑏_𝑖𝑥²_𝑖−|𝑥_𝑖|[^𝑛_𝑗=1𝑐_𝑖𝑗|𝑥_𝑗|+𝐷_𝑖]>𝑏_𝑖ℎ_𝑖[ℎ_𝑖−(^𝑛_𝑗=1𝑏_𝑖⁻¹𝑐_𝑖𝑗ℎ_𝑗+𝑑_𝑖)]=𝑏_𝑖ℎ_𝑖[ℎ_𝑖−(^𝑛_𝑗=1𝑘_𝑖𝑗ℎ_𝑗+𝑑_𝑖)]=0. (3.12) Therefore 𝑥_𝑖(𝑄𝑁𝑥)_𝑖=𝜔⁻¹𝑥_𝑖^𝜔₀𝑎_𝑖(𝑡,𝑥_𝑖(𝑡))[𝑏_𝑖(𝑡,𝑥_𝑖(𝑡))+𝑓_𝑖(𝑡,𝑥_𝑡)]d𝑡>0. (3.13) Thus (3.11) is valid.
Next, we define continuous functions 𝐻_𝑖∶(Ω∩Ker𝐿)×[0,1]→Ω∩Ker𝐿,𝑖=1,2, by 𝐻₁𝐻(𝑥,𝑡)=𝑡𝑥+(1−𝑡)𝑄𝑁𝑥,∀(𝑥,𝑡)∈(Ω∩Ker𝐿)×[0,1],₂(𝑥,𝑡)=−𝑡𝑥+(1−𝑡)𝑄𝑁𝑥,∀(𝑥,𝑡)∈(Ω∩Ker𝐿)×[0,1], (3.14) respectively. If 𝑏_𝑖(𝑡,𝑢_𝑖)𝑢_𝑖>0, from (i) we have 𝐻₁(𝑥,𝑡)≠0,∀(𝑥,𝑡)∈Ker𝐿𝜕Ω×[0,1], (3.15) If 𝑏_𝑖(𝑡,𝑢_𝑖)𝑢_𝑖<0, from (2) we can get 𝐻₂(𝑥,𝑡)≠0,∀(𝑥,𝑡)∈Ker𝐿𝜕Ω×[0,1]. (3.16) Using the homotopy invariance theorem, we obtain if 𝑏_𝑖(𝑡,𝑢_𝑖)𝑢_𝑖>0, deg{𝐽𝑄𝑁|_Ω∩Ker𝐿,Ω∩Ker𝐿,0}=deg{𝐻₁(⋅,0),Ω∩Ker𝐿,0}=deg{𝐻₁(⋅,1),Ω∩Ker𝐿,0}=deg{𝑥,Ω∩Ker𝐿,0}=1, (3.17) or if 𝑏_𝑖(𝑡,𝑢_𝑖)𝑢_𝑖<0, deg{𝐽𝑄𝑁|_Ω∩Ker𝐿,Ω∩Ker𝐿,0}=deg{𝐻₂(⋅,0),Ω∩Ker𝐿,0}=deg{𝐻₂(⋅,1),Ω∩Ker𝐿,0}=deg{−𝑥,Ω∩Ker𝐿,0}=(−1)^𝑛. (3.18) To summarize, Ω satisfies all the conditions of Lemma 2.6. This completes the proof of (i).
For all 𝜔-periodic solution 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇 of system (1.1), from (3.3)–(3.7) we have |𝑥_𝑖|≤𝑏_𝑖^𝑛−1_𝑗=1𝑐_𝑖𝑗|𝑥_𝑗|+𝑏_𝑖⁻¹𝐷_𝑖,|𝑥|≤(𝐸−𝐾)⁻¹𝜐=𝜈=(𝜈₁,𝜈₂,…,𝜈_𝑛)^𝑇, (3.19) where 𝜐=(𝜐₁,𝜐₂,…,𝜐_𝑛)^𝑇,|𝑥|=(|𝑥₁|,|𝑥₂|,…,|𝑥_𝑛|)^𝑇, 𝜐_𝑖=𝑏_𝑖⁻¹𝐷_𝑖≥0,𝑖=1,2,…,𝑛. Notes 𝛿=max_{1≤𝑖≤𝑛}𝜈_𝑖≥0, thus |𝑥_𝑖(𝑡)|≤𝛿, for all 𝑖=1,2,…,𝑛. This completes the proof of (ii).

From the proof of Theorem 3.1, we can easily obtain the following corollary.

Corollary 3.2. Suppose that (𝐻₁)–(𝐻₅) hold, and 𝐷=0 in (𝐻₄), then system (1.1) has only one 𝜔-periodic solution 𝑥(𝑡)=0.

Some special cases of Theorem 3.1 are in what follows.

Corollary 3.3. Equation (1.3) has at least one 𝜔-periodic solution, if the following conditions are satisfied.(𝐴₁) For 𝑖,𝑗=1,2,…,𝑛,𝑎_𝑖,𝑏_𝑖,𝑎_𝑖𝑗,𝑏_𝑖𝑗,𝜏_𝑗,𝐼_𝑖∶𝑅→𝑅 are continuous 𝜔-periodic (𝜔>0) functions.(𝐴₂) For 𝑖=1,2,…,𝑛, 𝑎_𝑖(𝑥) are positive, and there exist 𝑎_𝑖>0 such that 𝑎_𝑖(𝑥)≥𝑎_𝑖>0.(𝐴₃) For 𝑖=1,2,…,𝑛, there exist 𝑏_𝑖>0 such that |𝑏_𝑖(𝑥)|≥𝑏_𝑖|𝑥|,𝑏_𝑖(𝑥)𝑥>0,or𝑏_𝑖(𝑥)𝑥<0,∀𝑥∈𝑅. (3.20)(𝐴₄) For 𝑖=1,2,…,𝑛, there exist 𝐺_𝑖,𝑝_𝑖,𝑞_𝑖≥0 such that |𝑔_𝑖(𝑥)|≤𝐺_𝑖,|𝑔_𝑖(𝑥)|≤𝑝_𝑖|𝑥|+𝑞_𝑖. (3.21)(𝐴₅)𝜌(𝐾)<1,𝐾=(𝑘_𝑖𝑗)_𝑛×𝑛,and𝑘_𝑖𝑗=𝑏_𝑖⁻¹(|𝑎_𝑖𝑗|+|𝑏_𝑖𝑗|+∑^𝑛_𝑗=1|𝑏_𝑖𝑗𝑙|𝐺_𝑙)𝑝_𝑗,𝑖,𝑗=1,2,…,𝑛.

Proof. It is clear that 𝐴(𝑡,𝑥)=diag(𝑎₁(𝑥₁),𝑎₂(𝑥₂),…,𝑎_𝑛(𝑥_𝑛))≥diag(𝑎₁,𝑎₂,…,𝑎_𝑛)=𝐴,|𝐵(𝑡,𝑥)|=(|𝑏₁(𝑥₁)|,|𝑏₂(𝑥₂)|,…,|𝑏_𝑛(𝑥_𝑛)|)^𝑇≥(𝑏₁|𝑥₁|,𝑏₂|𝑥₂|,…,𝑏_𝑛|𝑥_𝑛|)^𝑇=diag(𝑏₁,𝑏₂,…,𝑏_𝑛)(|𝑥₁|,|𝑥₂|,…,|𝑥_𝑛|)^𝑇=𝐵|𝑥|,|𝑓_𝑖(𝑡,𝜙)|=|^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑔_𝑗(𝜙_𝑗(0))+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑔_𝑗(𝜙_𝑗(−𝜏_𝑗+(𝑡)))_{𝑛𝑛𝑗=1𝑙=1}𝑏_𝑖𝑗𝑙(𝑡)𝑔_𝑗(𝜙_𝑗(−𝜏_𝑗(𝑡)))𝑔_𝑙(𝜙_𝑙(−𝜏_𝑙(𝑡)))−𝐼_𝑖≤(𝑡)|_𝑛𝑗=1|𝑎_𝑖𝑗|[𝑝_𝑗|𝜙_𝑗|+𝑞_𝑗]+^𝑛_𝑗=1|𝑏_𝑖𝑗|[𝑝_𝑗|𝜙_𝑗|+𝑞_𝑗]+^𝑛𝑛_𝑗=1𝑙=1𝐺_𝑙|𝑏_𝑖𝑗𝑙|(𝑝_𝑗|𝜙_𝑗|+𝑞_𝑗)+|𝐼_𝑖|=^𝑛_𝑗=1[|𝑎_𝑖𝑗|+|𝑏_𝑖𝑗|+^𝑛_𝑙=1|𝑏_𝑖𝑗𝑙|𝐺_𝑙]𝑝_𝑗|𝜙_𝑗|+^𝑛_𝑗=1[|𝑎_𝑖𝑗|+|𝑏_𝑖𝑗|+^𝑛_𝑙=1|𝑏_𝑖𝑗𝑙|𝐺_𝑙]𝑞_𝑗+|𝐼_𝑖|=^𝑛_𝑗=1𝑐_𝑖𝑗|𝜙_𝑗|+𝐷_𝑖,𝑖=1,2,…,𝑛. (3.22) Thus |𝑓(𝑡,𝜙)|≤𝐶|𝜙|+𝐷, (3.23) where 𝐶=(𝑐_𝑖𝑗)_𝑛×𝑛∈𝑅^𝑛×𝑛,𝐷=(𝐷₁,𝐷₂,…,𝐷_𝑛)^𝑇,𝑐_𝑖𝑗=[|𝑎_𝑖𝑗|+|𝑏_𝑖𝑗|+∑^𝑛_𝑙=1|𝑏_𝑖𝑗𝑙|𝐺_𝑙]𝑝_𝑗≥0,𝐷_𝑖=∑^𝑛_𝑗=1[|𝑎_𝑖𝑗|+|𝑏_𝑖𝑗|+∑^𝑛_𝑙=1|𝑏_𝑖𝑗𝑙|𝐺_𝑙]𝑞_𝑗+|𝐼_𝑖|≥0,𝑖,𝑗=1,2,…,𝑛. Therefore, by using Lemma 2.5 and Theorem 3.1, we know that (1.3) has an 𝜔-periodic solution. The proof is complete.

Remark 3.4. For [1, Equation (1.2)], 𝜏_𝑗(𝑡),𝑗=1,2,…,𝑛 are continuous differentiable 𝜔-periodic solutions and 0≤𝜏^′_𝑗(𝑡)≤1, this implies that 𝜏_𝑗(𝑡),𝑗=1,2,…,𝑛 are constant functions, thus 𝜉_𝑗=1,𝑗=1,2,…,𝑛. It is not difficult to verify that all of conditions of Corollary 3.3 are satisfied under the conditions of [1, Theorem 1] moreover the other requirements of [1, Theorem 1] are more restrictive than ours. Therefore, Corollary 3.3 improves the corresponding result obtained in [1].

Corollary 3.5. If the following conditions are satisfied:(𝐵₁) for 𝑖,𝑗=1,2,…,𝑛,𝑐_𝑖𝑗,𝑑_𝑖𝑗,𝜏_𝑖𝑗,𝐼_𝑖∶𝑅→𝑅 are continuous 𝜔-periodic (𝜔>0) functions, 𝑎_𝑖,𝑏_𝑖 are continuous functions on 𝑅², and are 𝜔-periodic for their first arguments, respectively,(𝐵₂) for 𝑖=1,2,…,𝑛, there exist positive constants 𝑎_𝑖 such that 𝑎_𝑖(𝑡,𝑢)≥𝑎_𝑖, for all 𝑡,𝑢∈𝑅,(𝐵₃) for 𝑖=1,2,…,𝑛, there exist positive constants 𝑏_𝑖 such that |𝑏_𝑖(𝑡,𝑢)|≥𝑏_𝑖|𝑢|,𝑏_𝑖(𝑢)𝑢>0or𝑏_𝑖(𝑢)𝑢<0, for all 𝑡,𝑢∈𝑅,(𝐵₄) there exist nonnegative constants 𝑝^𝑓_𝑗,𝑞^𝑓_𝑗,𝑝^𝑔_𝑗,𝑞^𝑔_𝑗 such that |𝑓_𝑗(𝑢)|≤𝑝^𝑓_𝑗|𝑢|+𝑞^𝑓_𝑗,|𝑔_𝑗(𝑢)|≤𝑝^𝑔_𝑗|𝑢|+𝑞^𝑔_𝑗,∀𝑢∈𝑅,𝑗=1,2,…,𝑛, (3.24)(𝐵₅) the delay kernels 𝐾_𝑖𝑗∶[0,∞]→𝑅 satisfy ^∞₀|𝐾_𝑖𝑗(𝑠)|𝑑𝑠≤𝑘_𝑖𝑗,𝑖,𝑗=1,2,…,𝑛, (3.25)(𝐵₆)𝜌(𝐾)<1,𝐾=(𝑘_𝑖𝑗)_𝑛×𝑛∈𝑅^𝑛×𝑛, where 𝑘_𝑖𝑗=𝑏_𝑖⁻¹(|𝑐_𝑖𝑗|𝑝^𝑓_𝑗+|𝑑_𝑖𝑗|𝑘_𝑖𝑗𝑝^𝑔_𝑗),𝑖,𝑗=1,2,…,𝑛.then (1.3) has at least one 𝜔-periodic solution.

Remark 3.6. In [2, Theorem 3.1], the activation functions 𝑓_𝑗(𝑢),𝑔_𝑗(𝑢),𝑗=1,2,…,𝑛, are required to be Lipschitzian, which implies that condition (𝐵₃) in Corollary 3.5 holds. Therefore, Corollary 3.5 improves Theorem 3. In 2.

Corollary 3.7. Assume that the following conditions are satisfied: (𝐶₁)𝑐_𝑖𝑗,𝑑_𝑖𝑗,𝜏_𝑖𝑗,𝐼_𝑖∶𝑅→𝑅 are continuous 𝜔-periodic (𝜔>0) functions, 𝑎_𝑖,𝑏_𝑖 are continuous functions on 𝑅², and are 𝜔-periodic in the first variable,(𝐶₂) there exist positive constants 𝑎_𝑖 such that 𝑎_𝑖(𝑡,𝑢)≥𝑎_𝑖,∀𝑡,𝑢∈𝑅,𝑖=1,2,…,𝑛, (3.26)(𝐶₃) there exist positive constants 𝑏_𝑖 such that |𝑏_𝑖(𝑡,𝑢)|≥𝑏_𝑖|𝑢|,𝑏_𝑖(𝑢)𝑢>0or𝑏_𝑖(𝑢)𝑢<0,∀𝑡,𝑢∈𝑅,𝑖=1,2,…,𝑛, (3.27)(𝐶₄) There exist nonnegative constants 𝑝_𝑖,𝑞_𝑖 such that |𝑓_𝑖(𝑢)|≤𝑝_𝑖|𝑢|+𝑞_𝑖,∀𝑢∈𝑅,𝑖=1,2,…,𝑛, (3.28)(𝐶₅)𝜌(𝐾)<1, where 𝐾=(𝑘_𝑖𝑗)_𝑛×𝑛∈𝑅^𝑛×𝑛 and 𝑘_𝑖𝑗=𝑏_𝑖⁻¹(|𝑐_𝑖𝑗|+|𝑑_𝑖𝑗|)𝑝_𝑗,𝑖,𝑗=1,2,…,𝑛.Then (1.4) has at least one 𝜔-periodic solution.

Remark 3.8. In [3, Theorem 3.1], the activation functions 𝑓_𝑗(𝑢),𝑗=1,2,…,𝑛, are Lipschitzian (which also implies that condition (𝐶₄) in Corollary 3.7 holds) and the behaved functions 𝑏_𝑖(𝑡,𝑢) are required to satisfy that there exist positive constants 𝑏_𝑖,𝑏_𝑖 such that 0≤𝑢𝑏_𝑖(𝑡,𝑢),𝑏_𝑖|𝑢|≤|𝑏_𝑖(𝑡,𝑢)|≤𝑏_𝑖|𝑢| for all 𝑡,𝑢∈𝑅,𝑖=1,2,…,𝑛, which are more restrictive than that of Corollary 3.7.

Corollary 3.9. Assume that the following conditions are satisfied (𝐷₁) For 𝑖,𝑗=1,2,…,𝑛,𝐼_𝑖,𝑎_𝑖𝑗,𝑏_𝑖𝑗,𝜏_𝑖𝑗∶𝑅→𝑅 are continuous 𝜔-periodic solution (𝜔>0) functions.(𝐷₂) For 𝑗=1,2,…,𝑛,𝑔_𝑗∶𝑅→𝑅 are continuous functions and there exist nonnegative constants 𝑝_𝑗,𝑞_𝑗 such that |𝑔_𝑗(𝑣)|≤𝑝_𝑗|𝑣|+𝑞_𝑗,∀𝑣∈𝑅,𝑗=1,2,…,𝑛, (3.29)(𝐷₃)𝜌(𝐾)<1,𝐾=(𝑘_𝑖𝑗)_𝑛×𝑛∈𝑅^𝑛×𝑛 and 𝑘_𝑖𝑗=𝑟_𝑖⁻¹(|𝑎_𝑖𝑗|+|𝑏_𝑖𝑗|)𝑝_𝑗,𝑖,𝑗=1,2,…,𝑛,then (1.5) has at least one 𝜔-periodic solution.

The proofs of Corollaries 3.5–3.9 are the same as that of Corollary 3.3.

4. Uniqueness and Exponential Stability of Periodic Solution

In this section, we establish some results for the uniqueness and exponential stability of the 𝜔-periodic solution of (1.1).

Theorem 4.1. Assume that 𝐸 is a bounded subset of 𝑅⁻, and (𝐻₁)–(𝐻₃) and (𝐻₅) hold. Suppose also the following conditions are satisfied.(𝐻₄)^′There exists a nonnegative matrix 𝐶=(𝑐_𝑖𝑗)_𝑛×𝑛∈𝑅^𝑛×𝑛 such that |𝐹(𝑡,𝜙)−𝐹(𝑡,𝜑)|≤𝐶|𝜙−𝜑|,∀(𝑡,𝜙),(𝑡,𝜑)∈𝑅×𝐶_𝐸, (4.1) where 𝜙=(𝜙₁,𝜙₂,…,𝜙_𝑛)^𝑇,𝜑=(𝜑₁,𝜑₂,…,𝜑_𝑛)^𝑇∈𝐶_𝐸,|𝜙−𝜑|=(|𝜙₁−𝜑₁|,|𝜙₂−𝜑₂|, …,|𝜙_𝑛−𝜑_𝑛|)^𝑇.(𝐻₆)𝑎_𝑖,𝑖=1,2,…,𝑛, are Lipschitzian with Lipschitz constants 𝐿^𝑎_𝑖>0, and there exist 𝑎_𝑖 such that 𝑎_𝑖(𝑡,𝑢)≤𝑎_i,|𝑎_𝑖(𝑡,𝑢)−𝑎_𝑖(𝑡,𝑣)|≤𝐿^𝑎_𝑖|𝑢−𝑣|,∀(𝑡,𝑢),(𝑡,𝑣)∈𝑅²,𝑖=1,2,…,𝑛. (4.2)(𝐻₇) For all 𝑡,𝑢,𝑣∈𝑅,𝑖=1,2,…,𝑛, there exist positive constants 𝐿_𝑖^𝑎𝑏 such that (𝑢−𝑣)≤0,𝑖=1,2,…,𝑛,|𝑎_𝑖(𝑡,𝑢)𝑏_𝑖(𝑡,𝑢)−𝑎_𝑖(𝑡,𝑣)𝑏_𝑖(𝑡,𝑣)|≥𝐿_𝑖^𝑎𝑏|𝑢−𝑣|,𝑖=1,2,…,𝑛. (4.3)(𝐻₈) For 𝑖=1,2,…,𝑛, set Δ_𝑖=max_{0≤𝑡≤𝜔}|𝑓_𝑖(𝑡,0)|, and assume that 𝐸_𝑛−𝑊 is an 𝑀-matrix, where 𝑊=(𝑤_𝑖𝑗)_𝑛×𝑛∈𝑅^𝑛×𝑛, and𝑤_𝑖𝑗=(𝐿_𝑖^𝑎𝑏−𝐿^𝑎_𝑖Δ_𝑖)⁻¹(𝑎_𝑖+𝐿^𝑎_𝑖𝛿)𝑐_𝑖𝑗,𝐿_𝑖^𝑎𝑏−𝐿^𝑎_𝑖Δ_𝑖>0,𝑖,𝑗=1,2,…,𝑛. (4.4)

Proof. Obviously, (𝐻₄)^′ implies (𝐻₄), since (𝐻₁)–(𝐻₅) hold, it follows from Theorem 3.1 that system (1.1) has at least one 𝜔-periodic solution 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇 (4.5) with the initial value 𝜙=(𝜙₁,𝜙₂,…,𝜙_𝑛)^𝑇∈𝐶_𝐸. Let 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇 (4.6) be an arbitrary solution of system (1.1) with the initial value (1.6), set 𝑦(𝑡)=𝑥(𝑡)−𝑥(𝑡). Then for 𝑖=1,2,…,𝑛,𝑦^′_𝑖(𝑡)=𝑎_𝑖(𝑡,𝑦_𝑖(𝑡)+𝑥_𝑖(𝑡))𝑏_𝑖(𝑡,𝑦_𝑖(𝑡)+𝑥_𝑖(𝑡))−𝑎_𝑖(𝑡,𝑥_𝑖(𝑡))𝑏_𝑖(𝑡,𝑥_𝑖(𝑡))+𝑎_𝑖(𝑡,𝑦_𝑖(𝑡)+𝑥_𝑖(𝑡))[𝑓_𝑖(𝑡,𝑦_𝑡+𝑥_𝑡)−𝑓_𝑖(𝑡,𝑥_𝑡)]+𝑓_𝑖(𝑡,𝑥_𝑡)[𝑎_𝑖(𝑡,𝑥_𝑖(𝑡))−𝑎_𝑖(𝑡,𝑥_𝑖(𝑡))]. (4.7) Thus, for 𝑖=1,2,…,𝑛, 𝐷⁻|𝑦_𝑖(𝑡)|≤−𝐿_𝑖^𝑎𝑏|𝑦_𝑖(𝑡)|+𝑎_𝑖^𝑛_𝑗=1𝑐_𝑖𝑗|𝑦_𝑖𝑡|+𝐿^𝑎_𝑖|𝑦_𝑖(𝑡)|[^𝑛_𝑗=1𝑐_𝑖𝑗|𝑥_𝑖𝑡|+|𝑓_𝑖(𝑡,0)|]≤−(𝐿_𝑖^𝑎𝑏−𝐿^𝑎_𝑖Δ_𝑖)|𝑦_𝑖(𝑡)|+(𝑎_𝑖+𝐿^𝑎_𝑖𝛿)^𝑛_𝑗=1𝑐_𝑖𝑗|𝑦_𝑖𝑡|, (4.8) for (𝐻₈) and Lemma 2.4, there exist a positive constant 𝜎>0 and a positive constant vector 𝜉=(𝜉₁,𝜉₂,…,𝜉_𝑛)^𝑇>0 such that (𝐸_𝑛−𝑊)𝜉>(𝜎,𝜎,…,𝜎)^𝑇. Hence 𝜉_𝑖−^𝑛_𝑗=1𝑤_𝑖𝑗𝜉_𝑗>𝜎, (4.9) where 𝑤_𝑖𝑗=(𝐿_𝑖^𝑎𝑏−𝐿^𝑎_𝑖Δ_𝑖)⁻¹(𝑎_𝑖+𝐿^𝑎_𝑖𝛿)𝑐_𝑖𝑗,𝑖,𝑗=1,2,…,𝑛. Moreover for all 𝑖=1,2,…,𝑛, −(𝐿_𝑖^𝑎𝑏−𝐿^𝑎_𝑖Δ_𝑖)𝜉_𝑖+(𝑎_𝑖+𝐿^𝑎_𝑖𝛿)^𝑛_𝑗=1𝑐_𝑖𝑗𝜉_𝑗<(𝐿_𝑖^𝑎𝑏−𝐿^𝑎_𝑖Δ_𝑖)𝜎. (4.10) Since, 𝐸 is a bounded subset of 𝑅⁻, we can choose a positive constant 𝛼<1, such that ∀𝜃∈𝐸𝛼𝜉_𝑖+[−(𝐿_𝑖^𝑎𝑏−𝐿^𝑎_𝑖Δ_𝑖)𝜉_𝑖+(𝑎_𝑖+𝐿^𝑎_𝑖𝛿)^𝑛_𝑗=1𝑐_𝑖𝑗𝜉_𝑗𝑒^−𝛼𝜃]<0,𝑖=1,2,…,𝑛, (4.11) and also can choose a positive constant 𝛽>1 such that 𝛽𝜉_𝑖𝑒^−𝛼𝜃>1,∀𝜃∈𝐸,𝑖=1,2,…,𝑛. (4.12) Set, for all 𝜀>0, for all 𝑡∈𝐸, 𝑍_𝑖(𝑡)=𝛽𝜉_𝑖[^𝑛_𝑗=1|𝑦_𝑗0|+𝜀]𝑒^−𝛼𝑡,𝑖=1,2,…,𝑛. (4.13) It follows from (4.11) and (4.13) that 𝐷₋𝑍_𝑖(𝑡)=−𝛼𝛽𝜉_𝑖[^𝑛_𝑗=1|𝑦_𝑗0|+𝜀]𝑒^−𝛼𝑡>[−(𝐿_𝑖^𝑎𝑏−𝐿^𝑎_𝑖Δ_𝑖)𝜉_𝑖+(𝑎_𝑖+𝐿^𝑎_𝑖𝛿)^𝑛_𝑗=1𝑐_𝑖𝑗𝜉_𝑗𝑒^−𝛼𝜃]𝛽[^𝑛_𝑗=1|𝑦_𝑗0|+𝜀]𝑒^−𝛼𝑡=−(𝐿_𝑖^𝑎𝑏−𝐿^𝑎_𝑖Δ_𝑖)𝜉_𝑖𝛽[^𝑛_𝑗=1|𝑦_𝑗0|+𝜀]𝑒^−𝛼𝑡+(𝑎_𝑖+𝐿^𝑎_𝑖𝛿)^𝑛_𝑗=1𝑐_𝑖𝑗𝜉_𝑗𝑒^{−𝛼(𝜃+𝑡)}𝛽[^𝑛_𝑗=1|𝑦_𝑗0|+𝜀],∀𝜃∈𝐸. (4.14) Thus 𝐷₋𝑍_𝑖(𝑡)≥−(𝐿_𝑖^𝑎𝑏−𝐿^𝑎_𝑖Δ_𝑖)𝑍_𝑖(𝑡)+(𝑎_𝑖+𝐿^𝑎_𝑖𝛿)^𝑛_𝑗=1𝑐_𝑖𝑗|𝑍_𝑗𝑡|, (4.15) where |𝑍_𝑗𝑡|=sup_𝜃∈𝐸𝑍_𝑗(𝑡+𝜃), from (4.12) and (4.13), we can get 𝑍_𝑖(𝑡)=𝛽𝜉_𝑖[^𝑛_𝑗=1|𝑦_𝑗0|+𝜀]𝑒^−𝛼𝑡>^𝑛_𝑗=1|𝑦_𝑗0|+𝜀>|𝑦_𝑖(𝑡)|,∀𝑡∈𝐸. (4.16) We claim that |𝑦_𝑖(𝑡)|<𝑍_𝑖(𝑡),∀𝑡>0,𝑖=1,2,…,𝑛. (4.17) Suppose that it is not true, then there exits some 𝑖∈{1,2,…,𝑛} and 𝑡_𝑖>0 such that |𝑦_𝑖(𝑡_𝑖)|=𝑍_𝑖(𝑡_𝑖),|𝑦_𝑗(𝑡)|≤𝑍_𝑗(𝑡),∀𝑡<𝑡_𝑖,𝑗=1,2,…,𝑛. (4.18) Thus 0≤𝐷⁻(|𝑦_𝑖(𝑡_𝑖)|−𝑍_𝑖(𝑡_𝑖))=limsup_ℎ→0−[|𝑦_𝑖(𝑡_𝑖+ℎ)|−𝑍_𝑖(𝑡_𝑖+ℎ)]−[|𝑦_𝑖(𝑡_𝑖)|−𝑍_𝑖(𝑡_𝑖)]ℎ≤limsup_ℎ→0−|𝑦_𝑖(𝑡_𝑖+ℎ)|−|𝑦_𝑖(𝑡_𝑖)|ℎ−liminf_ℎ→0−𝑍_𝑖(𝑡_𝑖+ℎ)−𝑍_𝑖(𝑡_𝑖)ℎ≤𝐷⁻|𝑦_𝑖(𝑡_𝑖)|−𝐷₋𝑍_𝑖(𝑡_𝑖). (4.19) It follows from (4.8), (4.15), and (4.18) that 𝐷⁻|𝑦_𝑖(𝑡_𝑖)|≤−(𝐿_𝑖^𝑎𝑏−𝐿^𝑎_𝑖Δ_𝑖)|𝑦_𝑖(𝑡_𝑖)|+(𝑎_𝑖+𝐿^𝑎_𝑖𝛿)^𝑛_𝑗=1𝑐_𝑖𝑗|𝑦_𝑗𝑡𝑖|≤−(𝐿_𝑖^𝑎𝑏−𝐿^𝑎_𝑖Δ_𝑖)|𝑍_𝑖(𝑡_𝑖)|+(𝑎_𝑖+𝐿^𝑎_𝑖𝛿)^𝑛_𝑗=1𝑐_𝑖𝑗|𝑍_𝑗𝑡𝑖|<𝐷₋𝑍_𝑖(𝑡_𝑖), (4.20) which contradicts to (4.19), thus (4.17) holds. Set 𝜀→0⁺ and 𝑀=𝑛max_{1≤𝑖≤𝑛}{𝛽𝜉_𝑖+1}>1, from (4.17), we have |𝑥_𝑖(𝑡)−𝑥_𝑖(𝑡)|=|𝑦_𝑖(𝑡)|≤𝛽𝜉_𝑖^𝑛_𝑗=1|𝑦_𝑗0|𝑒^−𝛼𝑡≤𝛽𝜉_𝑖𝑛‖𝜙−𝜙‖𝑒^−𝛼𝑡≤𝑀‖𝜙−𝜙‖𝑒^−𝛼𝑡, (4.21) where 𝑖=1,2,…,𝑛. This completes the proof of Theorem 4.1.

5. Conclusion

In this paper, a class of generalized neural networks with arbitrary delays have been studied. Some sufficient conditions for the existence and exponential stability of the periodic solutions have been established. These obtained results are new and they improve and complement previously known results.

Acknowledgment

This work is supported by the National Natural Sciences Foundation of China under Grant 10971183.