Existence and Global Exponential Stability of Periodic Solutions for General Neural Networks with Time-Varying Delays

Abstract

By using the coincidence degree theorem and differential inequality techniques, sufficient conditions are obtained for the existence and global exponential stability of periodic solutions for general neural networks with time-varying (including bounded and unbounded) delays. Some known results are improved and some new results are obtained. An example is employed to illustrate our feasible results.

1. Introduction

In recent years, the delayed cellular neural networks (DCNNs) have been extensively studied because of their immense potentials of application perspective in different areas such as pattern recognition, optimization, and signal and image processing [1–3]. Hence, they have been the object of intensive analysis by numerous authors, and some interesting results on the existence and stability of periodic and almost periodic solutions have been obtained [4–12]. To our knowledge, few authors have considered global stability of periodic solutions for the neural networks with bounded and unbounded time-varying delays. In theory and application, global stability of periodic solutions of DCNNs is of great importance since the global stability of equilibrium points can be considered as a special case of periodic solution with zero period [8]. Hence, in this paper, we will study the existence and global exponential stability of periodic solutions of the following general neural networks with time-varying delays:𝑥^′_𝑖(𝑡)=−𝑎_𝑖(𝑡,𝑥_𝑖(𝑡))+^𝑛_𝑗=1[𝑎_𝑖𝑗(𝑡)𝑓_𝑗(𝑥_𝑗(𝑡))+𝑏_𝑖𝑗(𝑡)𝑓_𝑗(𝑥_𝑗(𝑡−𝜏_𝑖𝑗(𝑡)))+𝑐_𝑖𝑗(𝑡)^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑔_𝑗(𝑥_𝑗(𝑠))d𝑠]+𝐼_𝑖(𝑡),𝑖=1,2,…,𝑛, (1.1)where 𝑥_𝑖(𝑡) is the state of neuron, 𝑖=1,2,…,𝑛,𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛,𝐵=(𝑏_𝑖𝑗)_𝑛×𝑛, and 𝐶=(𝑐_𝑖𝑗)_𝑛×𝑛 are connection matrices, 𝐼=(𝐼₁,𝐼₂,…,𝐼_𝑛)^𝑇 is the input function, and 𝑓(𝑥)=(𝑓₁(𝑥₁),𝑓₂(𝑥₂),…,𝑓_𝑛(𝑥_𝑛))^𝑇 is the activation function of the neurons.

DCNNs in [4–12] and the references cited therein are special cases of (1.1). In particular, when 𝑎_𝑖(𝑡,𝑥_𝑖(𝑡))=𝑎_𝑖𝑥_𝑖(𝑡),𝑎_𝑖𝑗(𝑡)=𝑎_𝑖𝑗,𝑏_𝑖𝑗(𝑡)=𝑏_𝑖𝑗,𝑐_𝑖𝑗(𝑡)=𝑐_𝑖𝑗,𝐼_𝑖(𝑡)=𝐼_𝑖,𝑎_𝑖,𝑎_𝑖𝑗,𝑏_𝑖𝑗,𝑐_𝑖𝑗,𝐼_𝑖 are constants, the authors of [9] considered the existence and global exponential stability for (1.1) with periodic impulses. The methods used in [9] are Mawhin's coincidence degree theorem [13] and Lyapunov functions. In [14], by using Mawhin's coincidence degree theorem [13], the authors investigated the global existence of positive periodic solutions of mutualism systems with bounded and unbounded time-varying delays, and some sufficient conditions are obtained. In [12], the authors considered (1.1) when 𝑎_𝑖,𝑎_𝑖𝑗,𝑏_𝑖𝑗,𝑐_𝑖𝑗 are constants.

We assume what follows.

(H₁)𝑎_𝑖𝑗(𝑡),𝑏_𝑖𝑗(𝑡),𝑐_𝑖𝑗(𝑡),𝜏_𝑖𝑗(𝑡),𝐼_𝑖(𝑡) are continuous 𝜔-periodic functions, and 𝑎_𝑖(𝑡,𝑥_𝑖(𝑡)) are continuous 𝜔-periodic in the first variable. 𝜏_𝑖𝑗(𝑡)≥0,𝑖,𝑗=1,2,…,𝑛.(H₂) There exist positive constants 𝑎_𝑖 and 𝑎_𝑖 such that (𝑎_𝑖(𝑡,𝑢)−𝑎_𝑖(𝑡,𝑣))(𝑢−𝑣)>0, and 𝑎_𝑖|𝑢|≤|𝑎_𝑖(𝑡,𝑢)|≤𝑎_𝑖|𝑢|, for all 𝑡,𝑢,𝑣∈𝑅,𝑖=1,2,…,𝑛.(H₃)𝑓_𝑗,𝑔_𝑗∈𝐶(𝑅,𝑅),𝑗=1,2,…,𝑛. There are positive constants 𝐿^𝑓_𝑗>0, 𝐿^𝑔_𝑗>0, such that |𝑔_𝑗(𝑥)−𝑔_𝑗(𝑦)|≤𝐿^𝑔_𝑗|𝑥−𝑦|,|𝑓_𝑗(𝑥)−𝑓_𝑗(𝑦)|≤𝐿^𝑓_𝑗|𝑥−𝑦|, for all 𝑥,𝑦∈𝑅.(H₄) The delay kernels 𝑘_𝑖𝑗∶[0,+∞)→𝑅 are continuous, integrable, and satisfy₀^+∞|𝑘_𝑖𝑗(𝑠)|d𝑠≤𝛽_𝑖𝑗,𝑖,𝑗=1,2,…,𝑛. (1.2)(H₅) There exists a constant 𝛼₀>0 such that₀^+∞|𝑘_𝑖𝑗(𝑠)|𝑒^𝛼0𝑠d𝑠≤+∞,𝑖,𝑗=1,2,…,𝑛. (1.3)

The organization of this paper is as follows. In Section 2, we introduce some notations and definitions, and state some preliminary results needed in later sections. We then study, in Section 3, the existence of periodic solutions of system (1.1) by using the continuation theorem of coincidence degree theorem proposed by Gains and Mawhin [13]. In Section 4, by constructing Lyapunov function we will derive sufficient conditions for the global exponential stability of the periodic solution of system (1.1). At last, an example is employed to illustrate the feasible results of this paper.

2. Preliminaries

For convenience, we use 𝑎_𝑖𝑗,𝑏_𝑖𝑗,𝑐_𝑖𝑗,𝐼_𝑖 to denote the maximums of |𝑎_𝑖𝑗(𝑡)|,|𝑏_𝑖𝑗(𝑡)|,|𝑐_𝑖𝑗(𝑡)|,|𝐼_𝑖(𝑡)|, respectively. We also use symbols 𝜏=max{𝜏_𝑖𝑗(𝑡)∶𝑡∈[0,𝜔],𝑖,𝑗=1,2,…,𝑛},𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇 to denote a column vector, in which the symbol (𝑇) denotes the transpose of a vector. 𝐸_𝑛 denotes the identity matrix of size 𝑛. A matrix or vector 𝐷≥0 means that all entries of 𝐷 are greater than or equal to zero (resp., 𝐷>0). For matrices or vectors 𝐷 and 𝐸, 𝐷≥𝐸 (resp., 𝐷>𝐸) means that 𝐷−𝐸≥0 (resp., 𝐷−𝐸>0).

The initial condition 𝜙=(𝜙₁,…,𝜙_𝑛)^𝑇 of (1.1) is of the form𝑥_𝑖(𝑠)=𝜙_𝑖(𝑠),𝑠∈(−∞,0],𝑖=1,…,𝑛, (2.1)where 𝜙_𝑖(𝑠),𝑖=1,2,…,𝑛, are continuous 𝜔-periodic solutions.

Definition 2.1. Let 𝑥^∗(𝑡) be an 𝜔-periodic solution of (1.1) with initial value 𝜙^∗=(𝜙^∗₁,…,𝜙^∗_𝑛)^𝑇∈𝐶((−∞,0];𝑅^𝑛). If there exist constants 𝛼>0 and 𝑃>1 such that for every solution 𝑥(𝑡) of (1.1) with initial value 𝜙∈𝐶((−∞,0];𝑅^𝑛),|𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)|≤𝑃‖𝜙−𝜙^∗‖𝑒^−𝛼𝑡,∀𝑡>0,𝑖=1,2,…,𝑛, (2.2)where ‖𝜙−𝜙^∗‖=sup_𝑠≤0max_{1≤𝑖≤𝑛}|𝜙_𝑖(𝑠)−𝜙^∗_𝑖(𝑠)|, then 𝑥^∗(𝑡) is said to be globally exponentially stable.

Definition 2.2. (See [15, 16]). A real 𝑛×𝑛 matrix 𝐹=(𝑓_𝑖𝑗)_𝑛×𝑛 is said to be a nonsingular 𝑀-matrix if 𝑓_𝑖𝑗≤0,𝑖,𝑗=1,2,…,𝑛,𝑖≠𝑗, and 𝐹⁻¹≥0, where 𝐹⁻¹ denotes the inverse of 𝐹.

Lemma 2.3 (See [15, 16]). Let 𝐹=(𝑓_𝑖𝑗)_𝑛×𝑛 with 𝑓_𝑖𝑗≤0,𝑖,𝑗=1,2,…,𝑛,𝑖≠𝑗. Then the following statements are equivalent:
(1)𝐹 is a nonsingular 𝑀-matrix,(2) there exists a vector 𝜂^𝑇=(𝜂₁,𝜂₂,…,𝜂_𝑛)>0 such that 𝜂𝐹>0,(3) there exists a vector 𝜉=(𝜉₁,𝜉₂,…,𝜉_𝑛)^𝑇>0 such that 𝐹𝜉>0.

Lemma 2.4 (See [16]). Let 𝐴≥0 be an 𝑛×𝑛 matrix and 𝜌(𝐴)<1, then there exists a vector 𝜉=(𝜉₁,𝜉₂,…,𝜉_𝑛)^𝑇>0 such that (𝐸_𝑛−𝐴)𝜉>0, where 𝜌(𝐴) denotes the spectral radius of 𝐴.

To end this section, we introduce Mawhin's continuation theorem [13, page 40] as follows.

Consider an abstract equation in a Banach space 𝑋, 𝐿𝑥=𝜆𝑁𝑥,𝜆∈(0,1), (2.3)where 𝐿∶Dom𝐿∩𝑋→𝑋 is a Fredholm operator with index-zero, and 𝜆 is a parameter. Let 𝑃 and 𝑄 denote two projectors, 𝑃∶Dom𝐿∩𝑋→Ker𝐿 and 𝑄∶𝑋→𝑋/Im𝐿 such that Im𝑃=Ker𝐿,Ker𝑄=Im𝐿.

Lemma 2.5. Let 𝑋 be a Banach space. Suppose that 𝐿∶Dom𝐿∩𝑋→𝑋 is a Fredholm operator with index-zero, let Ω⊂𝑋 be an open bounded set, and let 𝑁∶𝑋→𝑋 be a continuous operator which is 𝐿-compact on Ω. Moreover, assume that the following conditions are satisfied:
(a) for each 𝜆∈(0,1),𝑥∈𝜕Ω∩Dom𝐿,𝐿𝑥≠𝜆𝑁𝑥,(b) for each 𝑥∈𝜕Ω∩Ker𝐿,𝑄𝑁𝑥≠0,(c)deg(𝑄𝑁,Ω∩Ker𝐿,0)≠0. Then, 𝐿𝑥=𝑁𝑥 has at least one solution in Ω∩Dom𝐿.

3. Existence of Periodic Solutions

Theorem 3.1. Let (H₁)-(H₄) hold. Assume that the following condition is satisfied:
(H₆) there exists a vector 𝜂^𝑇=(𝜂₁,…,𝜂_𝑛)>0 such that𝜂^𝑇=(𝜂₁,…,𝜂_𝑛)=𝜂^𝑇(𝐸_𝑛−Γ)>0, (3.1)where Γ=(𝛾_𝑖𝑗)_𝑛×𝑛,𝛾_𝑖𝑗=𝑎_𝑖⁻¹(𝑎_𝑖𝑗𝐿^𝑓_𝑗+𝑏_𝑖𝑗𝐿^𝑓_𝑗+𝑐_𝑖𝑗𝛽_𝑖𝑗𝐿^𝑔_𝑗),𝑖,𝑗=1,2,…,𝑛.
Then, (1.1) has at least one 𝜔-periodic solution.

Proof. For convenience, we introduce the following notations:𝐺_𝑖(𝑡)=−𝑎_𝑖(𝑡,𝑥_𝑖(𝑡))+^𝑛_𝑗=1[𝑎_𝑖𝑗(𝑡)𝑓_𝑗(𝑥_𝑗(𝑡))+𝑏_𝑖𝑗(𝑡)𝑓_𝑗(𝑥_𝑗(𝑡−𝜏_𝑖𝑗(𝑡)))+𝑐_𝑖𝑗(𝑡)^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑔_𝑗(𝑥_𝑗(𝑠))d𝑠]+𝐼_𝑖(𝑡),𝑖=1,2,…,𝑛. (3.2) In order to use Lemma 2.5, we take 𝑋={𝑥(𝑡)∈𝐶(𝑅,𝑅^𝑛)∶𝑥(𝑡+𝜔)=𝑥(𝑡),𝑡∈𝑅}; then 𝑋 is a Banach space with the norm‖𝑥‖=max{|𝑥₁|₀,|𝑥₂|₀,…,|𝑥_𝑛|₀},|𝑥_𝑖|₀=sup_{𝑡∈[0,𝜔]}|𝑥_𝑖(𝑡)|,𝑖=1,2,…,𝑛. (3.3) Set𝐿𝑥=𝑥^′1,𝑃𝑥=𝑄𝑥=𝜔^𝜔₀𝑥(𝑡)d𝑡,𝑥∈𝑋,𝑁𝑥=(𝐺₁(𝑡),𝐺₂(𝑡),…,𝐺_𝑛(𝑡))^𝑇,𝑥∈𝑋. (3.4) Obviously, Ker𝐿={𝑥∣𝑥∈𝑋,𝑥=ℎ,ℎ∈𝑅^𝑛},Im𝐿={𝑥∣𝑥∈𝑋,∫^𝜔₀𝑥(𝑠)d𝑠=0}, and dimKer𝐿=𝑛=codimIm𝐿. So, Im𝐿 is closed in 𝑋. It is easy to show that 𝑃 and 𝑄 are continuous projectors satisfying Im𝑃=Ker𝐿,Im𝐿=Ker𝑄=Im(𝐼−𝑄). Hence, 𝐿 is a Fredholm mapping of index-zero. Furthermore, through an easy computation, we find that the generalized inverse 𝐾_𝑃⁻¹∶Im𝐿→Ker𝑃∩dom𝐿 of 𝐿 has the form𝐾_𝑃⁻¹(𝑥)=^𝑡₀1𝑥(𝑠)d𝑠−𝜔^𝜔₀^𝑡₀𝑥(𝑠)d𝑠d𝑡. (3.5)Thus,1𝑄𝑁𝑥=(𝜔^𝜔₀𝐺₁1(𝑡)d𝑡,…,𝜔^𝜔₀𝐺_𝑛(𝑡)d𝑡)^𝑇𝐾,𝑥∈𝑋,_𝑝−1(𝐼−𝑄)𝑁𝑥=^𝑡0𝐺₁⋮(𝑠)d𝑠_𝑡0𝐺_𝑗⋮(𝑠)d𝑠_𝑡0𝐺_𝑛−1(𝑠)d𝑠𝜔^𝜔₀^𝑡₀𝐺₁⋮1(𝑠)d𝑠d𝑡𝜔^𝜔₀^𝑡₀𝐺_𝑗⋮1(𝑠)d𝑠d𝑡𝜔^𝜔₀^𝑡₀𝐺_𝑛−(𝑡(𝑠)d𝑠d𝑡𝜔−12)^𝜔₀𝐺₁⋮(𝑡(𝑠)d𝑠𝜔−12)^𝜔₀𝐺_𝑗⋮(𝑡(𝑠)d𝑠𝜔−12)^𝜔₀𝐺_𝑛.(𝑠)d𝑠 (3.6) Clearly, 𝑄𝑁 and 𝐾_𝑃⁻¹(𝐼−𝑄)𝑁 are continuous. Using the Arzela-Ascoli theorem, it is not difficult to show that 𝑄𝑁(Ω), 𝐾_𝑃⁻¹(𝐼−𝑄)𝑁(Ω) are relatively compact for any open bounded set Ω⊂𝑋. Therefore, 𝑁 is 𝐿-compact on Ω for any open bounded set Ω⊂𝑋.
Now, we reach the position to search for an appropriate open bounded subset Ω for the application of Lemma 2.5. Corresponding to the operator equation (2.3), we have𝑥^′_𝑖(𝑡)=𝜆{−𝑎_𝑖(𝑡,𝑥_𝑖(𝑡))+^𝑛_𝑗=1[𝑎_𝑖𝑗(𝑡)𝑓_𝑗(𝑥_𝑗(𝑡))+𝑏_𝑖𝑗(𝑡)𝑓_𝑗(𝑥_𝑗(𝑡−𝜏_𝑖𝑗(𝑡)))+𝑐_𝑖𝑗(𝑡)^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑔_𝑗(𝑥_𝑗(𝑠))d𝑠]+𝐼_𝑖(𝑡)},𝑖=1,2,…,𝑛,𝜆∈(0,1). (3.7)Let 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇∈𝑋 be a solution of system (3.7) for some 𝜆∈(0,1). Then, for any 𝑖=1,2,…,𝑛,𝑥_𝑖 are all continuously differentiable. Thus, there exist 𝑡_𝑖∈[0,𝜔],𝑖=1,2,…,𝑛, such that |𝑥_𝑖(𝑡_𝑖)|=max_{𝑡∈[0,𝜔]}|𝑥_𝑖(𝑡)|. Hence, 𝑥^′_𝑖(𝑡_𝑖)=0,𝑖=1,2,…,𝑛. From (3.7), we have𝑎_𝑖(𝑡_𝑖,𝑥_𝑖(𝑡_𝑖))=^𝑛_𝑗=1[𝑎_𝑖𝑗(𝑡_𝑖)𝑓_𝑗(𝑥_𝑗(𝑡_𝑖))+𝑏_𝑖𝑗(𝑡_𝑖)𝑓_𝑗(𝑥_𝑗(𝑡_𝑖−𝜏_𝑖𝑗(𝑡_𝑖)))+𝑐_𝑖𝑗(𝑡_𝑖)^𝑡𝑖_−∞𝑘_𝑖𝑗(𝑡_𝑖−𝑠)𝑔_𝑗(𝑥_𝑗(𝑠))d𝑠]+𝐼_𝑖(𝑡_𝑖),𝑖=1,2,…,𝑛. (3.8)In view of (H₁)-(H₄), we have𝑎_𝑖|𝑥_𝑖(𝑡_𝑖)|≤|𝑎_𝑖(𝑡_𝑖,𝑥_𝑖(𝑡_𝑖≤))|_𝑛𝑗=1[|𝑎_𝑖𝑗(𝑡_𝑖)|(|𝑓_𝑗(𝑥_𝑗(𝑡_𝑖))−𝑓_𝑗(0)|+|𝑓_𝑗(0)|)+|𝑏_𝑖𝑗(𝑡_𝑖)|(|𝑓_𝑗(𝑥_𝑗(𝑡_𝑖−𝜏_𝑖𝑗(𝑡_𝑖)))−𝑓_𝑗(0)|+|𝑓_𝑗(0)|)+|𝑐_𝑖𝑗(𝑡_𝑖)|^𝑡𝑖_−∞|𝑘_𝑖𝑗(𝑡_𝑖−𝑠)|(|𝑔_𝑗(𝑥_𝑗(𝑠))−𝑔_𝑗(0)|+|𝑔_𝑗(0)|)d𝑠]+|𝐼_𝑖(𝑡_𝑖≤)|_𝑛𝑗=1[𝑎_𝑖𝑗𝐿^𝑓_𝑗|𝑥_𝑗(𝑡_𝑖)|+𝑏_𝑖𝑗𝐿^𝑓_𝑗|𝑥_𝑗(𝑡_𝑖−𝜏_𝑖𝑗(𝑡_𝑖+))|𝑐_𝑖𝑗^𝑡𝑖_−∞|𝑘_𝑖𝑗(𝑡_𝑖−𝑠)|(𝐿^𝑔_𝑗|𝑥_𝑗(𝑠)|+|𝑔_𝑗(0)|)d𝑠+(𝑎_𝑖𝑗+𝑏_𝑖𝑗)|𝑓_𝑗(0)|]+𝐼_𝑖≤^𝑛_𝑗=1(𝑎_𝑖𝑗𝐿^𝑓_𝑗+𝑏_𝑖𝑗𝐿^𝑓_𝑗+𝑐_𝑖𝑗𝛽_𝑖𝑗𝐿^𝑔_𝑗)|𝑥_𝑗(𝑡_𝑗+)|_𝑛𝑗=1(𝑎_𝑖𝑗|𝑓_𝑗(0)|+𝑏_𝑖𝑗|𝑓_𝑗(0)|+𝑐_𝑖𝑗𝛽_𝑖𝑗|𝑔_𝑗(0)|)+𝐼_𝑖,𝑖=1,2,…,𝑛. (3.9) Set 𝐻_𝑖=∑^𝑛_𝑗=1𝑎_𝑖⁻¹(𝑎_𝑖𝑗|𝑓_𝑗(0)|+𝑏_𝑖𝑗|𝑓_𝑗(0)|+𝑐_𝑖𝑗𝛽_𝑖𝑗|𝑔_𝑗(0)|)+𝑎_𝑖⁻¹𝐼_𝑖. Clearly, (3.9) implies that|𝑥_𝑖(𝑡_𝑖)|≤^𝑛_𝑗=1𝛾_𝑖𝑗|𝑥_𝑗(𝑡_𝑗)|+𝐻_𝑖,𝑖=1,2,…,𝑛. (3.10)Thus,(𝐸_𝑛−Γ)(|𝑥₁(𝑡₁)|,|𝑥₂(𝑡₂)|,…,|𝑥_𝑛(𝑡_𝑛)|)^𝑇≤(𝐻₁,𝐻₂,…,𝐻_𝑛)^𝑇∶=𝐻; (3.11)together with (H₆), we havemin{𝜂₁,𝜂₂,…,𝜂_𝑛}(|𝑥₁(𝑡₁)|+|𝑥₂(𝑡₂)|+⋯+|𝑥_𝑛(𝑡_𝑛≤)|)𝜂₁|𝑥₁(𝑡₁)|+𝜂₂|𝑥₂(𝑡₂)|+⋯+𝜂_𝑛|𝑥_𝑛(𝑡_𝑛)|=𝜂^𝑇(𝐸_𝑛−Γ)(|𝑥₁(𝑡₁)|,|𝑥₂(𝑡₂)|,…,|𝑥_𝑛(𝑡_𝑛)|)^𝑇≤𝜂^𝑇𝐻. (3.12)Therefore,|𝑥_𝑖|₀=|𝑥_𝑖(𝑡_𝑖𝜂)|≤^𝑇𝐻min{𝜂₁,𝜂₂,…,𝜂_𝑛}∶𝜈,𝑖=1,2,…,𝑛. (3.13)Again from (H₆), it follows from Lemma 2.3 that 𝐸_𝑛−Γ is a nonsingular 𝑀-matrix, and there exists a vector 𝜉=(𝜉₁,𝜉₂,…,𝜉_𝑛)^𝑇>0 such that (𝐸_𝑛−Γ)𝜉>0, which implies that we can choose a constant 𝑑>1 such that 𝜉=(𝜉₁,𝜉₂,…,𝜉_𝑛)^𝑇=𝑑𝜉 and 𝜉_𝑖=𝑑𝜉_𝑖>𝜈,𝑖=1,2,…,𝑛, and(𝐸_𝑛−Γ)𝜉=𝑑(𝐸_𝑛−Γ)𝜉>𝐻. (3.14) We takeΩ={𝑥(𝑡)∈𝑋∶−𝜉<𝑥(𝑡)<𝜉,∀𝑡∈𝑅}, (3.15)which satisfies Condition (a) of Lemma 2.5. If 𝑥(𝑡)∈𝜕Ω∩ker𝐿, then 𝑥(𝑡) is a constant vector in 𝑅^𝑛, and there exists some 𝑖∈{1,2,…,𝑛} such that |𝑥_𝑖|=𝜉_𝑖. We claim that|(𝑄𝑁𝑥)_𝑖|>0,𝑥∈𝜕Ω∩ker𝐿. (3.16)By way of contradiction, suppose that |(𝑄𝑁𝑥)_𝑖|=0; then there exists some 𝑡^∗∈[0,𝜔] such that−𝑎_𝑖(𝑡^∗,𝑥_𝑖)+∑^𝑛_𝑗=1[𝑎_𝑖𝑗(𝑡^∗)𝑓_𝑗(𝑥_𝑗)+𝑏_𝑖𝑗(𝑡^∗)𝑓_𝑗(𝑥_𝑗)+𝑐_𝑖𝑗(𝑡^∗)∫^𝑡∗_−∞𝑘_𝑖𝑗(𝑡^∗−𝑠)𝑔_𝑗(𝑥_𝑗(𝑠))d𝑠]+𝐼_𝑖(𝑡^∗)=0, (3.17)which implies that𝑎_𝑖𝜉_𝑖=𝑎_𝑖|𝑥_𝑖|≤|𝑎_𝑖(𝑡^∗,𝑥_𝑖)|=|^𝑛_𝑗=1[𝑎_𝑖𝑗(𝑡^∗)𝑓_𝑗(𝑥_𝑗)+𝑏_𝑖𝑗(𝑡^∗)𝑓_𝑗(𝑥_𝑗)+𝑐_𝑖𝑗(𝑡^∗)^𝑡∗_−∞𝑘_𝑖𝑗(𝑡^∗−𝑠)𝑔_𝑗(𝑥_𝑗(𝑠))d𝑠]+𝐼_𝑖(𝑡^∗≤)|_𝑛𝑗=1(𝑎_𝑖𝑗𝐿^𝑓_𝑗+𝑏_𝑖𝑗𝐿^𝑓_𝑗+𝑐_𝑖𝑗𝛽_𝑖𝑗𝐿^𝑔_𝑗)|𝑥_𝑗|+^𝑛_𝑗=1(𝑎_𝑖𝑗|𝑓_𝑗(0)|+𝑏_𝑖𝑗|𝑓_𝑗(0)|+𝑐_𝑖𝑗𝛽_𝑖𝑗|𝑔_𝑗(0)|)+𝐼_𝑖≤^𝑛_𝑗=1(𝑎_𝑖𝑗𝐿^𝑓_𝑗+𝑏_𝑖𝑗𝐿^𝑓_𝑗+𝑐_𝑖𝑗𝛽_𝑖𝑗𝐿^𝑔_𝑗)𝜉_𝑗+^𝑛_𝑗=1(𝑎_𝑖𝑗|𝑓_𝑗(0)|+𝑏_𝑖𝑗|𝑓_𝑗(0)|+𝑐_𝑖𝑗𝛽_𝑖𝑗|𝑔_𝑗(0)|)+𝐼_𝑖; (3.18)hence𝜉_𝑖≤^𝑛_𝑗=1𝛾_𝑖𝑗𝜉_𝑗+𝐻_𝑖. (3.19)This implies that ((𝐸_𝑛−Γ)𝜉)_𝑖≤𝐻_𝑖, which contradicts (3.14). Therefore, (3.16) holds, and hence, Condition (b) of Lemma 2.5 is satisfied.
Furthermore, we define a continuous function 𝑊∶Ω∩Ker𝐿×[0,1]→𝑋 by𝑊(𝑥,𝜇)=𝜇diag(−𝑎₁,−𝑎₂,…,−𝑎_𝑛)𝑥+(1−𝜇)𝑄𝑁𝑥, (3.20)for all 𝑥∈Ω∩Ker𝐿=Ω∩𝑅^𝑛 and 𝜇∈[0,1]. It follows that(𝑊(𝑥,𝜇))_𝑖=−𝜇𝑎_𝑖𝑥_𝑖1+(1−𝜇)𝜔^𝜔₀𝐺_𝑖(𝑡)d𝑡. (3.21) If 𝑥(𝑡)∈𝜕Ω∩Ker𝐿, then 𝑥(𝑡) is a constant vector in 𝑅^𝑛, and there exists some 𝑖∈{1,2,…,𝑛} such that |𝑥_𝑖|=𝜉_𝑖. We claim that|(𝑊(𝑥,𝜇))_𝑖|>0,𝑥∈𝜕Ω∩ker𝐿,𝜇∈[0,1]. (3.22)By way of contradiction, suppose that |(𝑊(𝑥,𝜇))_𝑖|=0; then there exists some ^0𝑥0𝑡^02𝑑𝑐∈[0,𝜔] such that−𝜇𝑎_𝑖𝑥_𝑖+(1−𝜇){−𝑎_𝑖(𝑡^{0𝑥002𝑑𝑐},𝑥_𝑖)+^𝑛_𝑗=1[𝑎_𝑖𝑗(𝑡^{0𝑥002𝑑𝑐})𝑓_𝑗(𝑥_𝑗)+𝑏_𝑖𝑗(𝑡^{0𝑥002𝑑𝑐})𝑓_𝑗(𝑥_𝑗)+𝑐_𝑖𝑗(𝑡^{0𝑥002𝑑𝑐})^{0𝑥002𝑑𝑐𝑡}_−∞𝑘_𝑖𝑗(𝑡^{0𝑥002𝑑𝑐}−𝑠)𝑔_𝑗(𝑥_𝑗(𝑠))d𝑠]+𝐼_𝑖(𝑡^{0𝑥002𝑑𝑐})}=0, (3.23)that is,𝜇(𝑎_𝑖(𝑡^{0𝑥002𝑑𝑐},𝑥_𝑖)−𝑎_𝑖𝑥_𝑖)−𝑎_𝑖(𝑡^{0𝑥002𝑑𝑐},𝑥_𝑖)=−(1−𝜇){^𝑛_𝑗=1[𝑎_𝑖𝑗(𝑡^{0𝑥002𝑑𝑐})𝑓_𝑗(𝑥_𝑗)+𝑏_𝑖𝑗(𝑡^{0𝑥002𝑑𝑐})𝑓_𝑗(𝑥_𝑗)+𝑐_𝑖𝑗(𝑡^{0𝑥002𝑑𝑐})^{0𝑥002𝑑𝑐𝑡}_−∞𝑘_𝑖𝑗(𝑡^{0𝑥002𝑑𝑐}−𝑠)𝑔_𝑗(𝑥_𝑗(𝑠))d𝑠]+𝐼_𝑖}. (𝑡^{0𝑥002𝑑𝑐})(3.24)Now, we will consider the following two cases.
Case 1. If 𝑥_𝑖>0, from (H₂), we have𝑎_𝑖(𝑡^{0𝑥002𝑑𝑐},𝑥_𝑖)>0,𝑎_𝑖(𝑡^{0𝑥002𝑑𝑐},𝑥_𝑖)−𝑎_𝑖𝑥_𝑖≤0. (3.25)Then, from (3.24), we have𝑎_𝑖(𝑡^{0𝑥002𝑑𝑐},𝑥_𝑖)≤(1−𝜇){^𝑛_𝑗=1[𝑎_𝑖𝑗(𝑡^{0𝑥002𝑑𝑐})𝑓_𝑗(𝑥_𝑗)+𝑏_𝑖𝑗(𝑡^{0𝑥002𝑑𝑐})𝑓_𝑗(𝑥_𝑗)+𝑐_𝑖𝑗(𝑡^{0𝑥002𝑑𝑐})^{0𝑥002𝑑𝑐𝑡}_−∞𝑘_𝑖𝑗(𝑡^{0𝑥002𝑑𝑐}−𝑠)𝑔_𝑗(𝑥_𝑗d𝑠]+𝐼_𝑖}, (𝑠))(𝑡^{0𝑥002𝑑𝑐})(3.26)which implies that𝑎_𝑖𝜉_𝑖=𝑎_𝑖|𝑥_𝑖|≤|𝑎_𝑖(𝑡^{0𝑥002𝑑𝑐},𝑥_𝑖)|≤|^𝑛_𝑗=1[𝑎_𝑖𝑗(𝑡^{0𝑥002𝑑𝑐})𝑓_𝑗(𝑥_𝑗)+𝑏_𝑖𝑗(𝑡^{0𝑥002𝑑𝑐})𝑓_𝑗(𝑥_𝑗)+𝑐_𝑖𝑗(𝑡^{0𝑥002𝑑𝑐})^{0𝑥002𝑑𝑐𝑡}_−∞𝑘_𝑖𝑗(𝑡^{0𝑥002𝑑𝑐}−𝑠)𝑔_𝑗(𝑥_𝑗(𝑠))d𝑠]+𝐼_𝑖(𝑡^{0𝑥002𝑑𝑐}≤)|_𝑛𝑗=1(𝑎_𝑖𝑗𝐿^𝑓_𝑗+𝑏_𝑖𝑗𝐿^𝑓_𝑗+𝑐_𝑖𝑗𝛽_𝑖𝑗𝐿^𝑔_𝑗)|𝑥_𝑗|+^𝑛_𝑗=1(𝑎_𝑖𝑗|𝑓_𝑗(0)|+𝑏_𝑖𝑗|𝑓_𝑗(0)|+𝑐_𝑖𝑗𝛽_𝑖𝑗|𝑔_𝑗(0)|)+𝐼_𝑖≤^𝑛_𝑗=1(𝑎_𝑖𝑗𝐿^𝑓_𝑗+𝑏_𝑖𝑗𝐿^𝑓_𝑗+𝑐_𝑖𝑗𝛽_𝑖𝑗𝐿^𝑔_𝑗)𝜉_𝑗+^𝑛_𝑗=1(𝑎_𝑖𝑗|𝑓_𝑗(0)|+𝑏_𝑖𝑗|𝑓_𝑗(0)|+𝑐_𝑖𝑗𝛽_𝑖𝑗|𝑔_𝑗(0)|)+𝐼_𝑖. (3.27) Hence,𝜉_𝑖≤^𝑛_𝑗=1𝛾_𝑖𝑗𝜉_𝑗+𝐻_𝑖; (3.28)this implies that ((𝐸_𝑛−Γ)𝜉)_𝑖≤𝐻_𝑖, which contradicts (3.14). Therefore, (3.22) holds.
Case 2. If 𝑥_𝑖<0, from (H₂), we have𝑎_𝑖(𝑡^{0𝑥002𝑑𝑐},𝑥_𝑖)<0,𝑎_𝑖(𝑡^{0𝑥002𝑑𝑐},𝑥_𝑖)−𝑎_𝑖𝑥_𝑖≥0. (3.29)Then, from (3.24), we have𝑎_𝑖(𝑡^{0𝑥002𝑑𝑐},𝑥_𝑖)≥(1−𝜇){^𝑛_𝑗=1[𝑎_𝑖𝑗(𝑡^{0𝑥002𝑑𝑐})𝑓_𝑗(𝑥_𝑗)+𝑏_𝑖𝑗(𝑡^{0𝑥002𝑑𝑐})𝑓_𝑗(𝑥_𝑗)+𝑐_𝑖𝑗(𝑡^{0𝑥002𝑑𝑐})^{0𝑥002𝑑𝑐𝑡}_−∞𝑘_𝑖𝑗(𝑡^{0𝑥002𝑑𝑐}−𝑠)𝑔_𝑗(𝑥_𝑗)d𝑠]+𝐼_𝑖}. (𝑡^{0𝑥002𝑑𝑐})(3.30)The later proof is similar to that of Case 1. We can also show that (3.22) holds. It follows that 𝑊(𝑥,𝜇)≠0 for 𝑥∈𝜕Ω∩Ker𝐿, 𝜇∈[0,1]. Hence, by homotopy invariance theorem and 𝑎_𝑖>0, 𝑖=1,2,…,𝑛, we obtain deg{𝑄𝑁,Ω∩ker𝐿,0}=deg{(−𝑎₁𝑥₁,−𝑎₂𝑥₂,…,−𝑎_𝑛𝑥_𝑛)^𝑇,Ω∩ker𝐿,0}≠0 for 𝑥∈𝜕Ω∩Ker𝐿. Till now, we have proved that Ω satisfies all conditions of Lemma 2.5. Therefore, (1.1) has a periodic solution 𝑥^∗(𝑡). This completes the proof.

Corollary 3.2. Let (H₁)-(H₄) hold. Assume that 𝐸_𝑛−Γ is a nonsingular 𝑀-matrix, then (1.1) has at least one 𝜔-periodic solution, where Γ is defined as above.

Corollary 3.3. Let (H₁)-(H₄) hold. Assume that 𝜌(Γ)<1, then (1.1) has at least one 𝜔-periodic solution, where Γ is defined as above.

Proof. Obviously, Γ≥0. By 𝜌(Γ)<1 and from Lemma 2.4, there exists a vector 𝜉=(𝜉₁,𝜉₂,…,𝜉_𝑛)^𝑇>0 such that (𝐸_𝑛−𝐴)𝜉>0. The remaining part of the proof is the same as that of Theorem 3.1.

4. Global Exponential Stability of Periodic Solutions

In this section, we will construct some suitable Lyapunov functionals to derive sufficient conditions which ensure that (1.1) has a unique 𝜔-periodic solution, and all solutions of (1.1) exponentially converge to its unique 𝜔-periodic solution.

Theorem 4.1. Assume that (H₁)-(H₆) hold and
(H₇)|𝑎_𝑖(𝑡,𝑢)−𝑎_𝑖(𝑡,𝑣)|≥𝑎_𝑖|𝑢−𝑣|, for all 𝑡,𝑢,𝑣∈𝑅,𝑖=1,2,…,𝑛, where 𝑎_𝑖,𝑖=1,2,…,𝑛, are the positive constants of Hypothesis (H₂). Then, (1.1) has exactly one 𝜔-periodic solution, which is globally exponentially stable.

Proof. By Theorem 3.1, there exists an 𝜔-periodic solution 𝑥^∗(𝑡)=(𝑥^∗₁(𝑡),𝑥^∗₂(𝑡),…,𝑥^∗_𝑛(𝑡))^𝑇 of (1.1). Suppose that 𝑥(𝑡) is an arbitrary solution of (1.1). Set 𝑧(𝑡)=(𝑧₁(𝑡),𝑧₂(𝑡),…,𝑧_𝑛(𝑡))^𝑇=𝑥(𝑡)−𝑥^∗(𝑡). Then,𝑧^′_𝑖(𝑡)=−𝑑_𝑖(𝑡,𝑧_𝑖(𝑡))+^𝑛_𝑗=1[𝑎_𝑖𝑗(𝑡)𝑤_𝑗(𝑧_𝑗(𝑡))+𝑏_𝑖𝑗(𝑡)𝑤_𝑗(𝑧_𝑗(𝑡−𝜏_𝑖𝑗(𝑡)))+𝑐_𝑖𝑗(𝑡)^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑢_𝑗(𝑧_𝑗(𝑠))d𝑠],𝑖=1,2,…,𝑛, (4.1)where𝑑_𝑖(𝑡,𝑧_𝑖(𝑡))=𝑎_𝑖(𝑡,𝑧_𝑖(𝑡)+𝑥^∗_𝑖(𝑡))−𝑎_𝑖(𝑡,𝑥^∗_𝑖𝑤(𝑡)),_𝑗(𝑧_𝑗(𝑡))=𝑓_𝑗(𝑧_𝑗(𝑡)+𝑥^∗_𝑗(𝑡))−𝑓_𝑗(𝑥^∗_𝑗𝑤(𝑡)),_𝑗(𝑧_𝑗(𝑡−𝜏_𝑖𝑗(𝑡)))=𝑓_𝑗(𝑧_𝑗(𝑡−𝜏_𝑖𝑗(𝑡))+𝑥^∗_𝑗(𝑡−𝜏_𝑖𝑗(𝑡)))−𝑓_𝑗(𝑥^∗_𝑗(𝑡−𝜏_𝑖𝑗𝑢(𝑡))),_𝑗(𝑧_𝑗(𝑡))=𝑔_𝑗(𝑧_𝑗(𝑡)+𝑥^∗_𝑗(𝑡))−𝑔_𝑗(𝑥^∗_𝑗(𝑡)). (4.2) From (H₆), it follows from Lemma 2.3 that 𝐸_𝑛−Γ is a nonsingular 𝑀-matrix, and there exists a vector 𝜉=(𝜉₁,𝜉₂,…,𝜉_𝑛)^𝑇>0 such that (𝐸_𝑛−Γ)𝜉>0. Then,−𝑎_𝑖𝜉_𝑖+^𝑛_𝑗=1𝜉_𝑗(𝑎_𝑖𝑗𝐿^𝑓_𝑗+𝑏_𝑖𝑗𝐿^𝑓_𝑗+𝑐_𝑖𝑗𝛽_𝑖𝑗𝐿^𝑔_𝑗)<0,𝑖=1,2,…,𝑛. (4.3)Setℎ_𝑖(𝜆)=(𝜆−𝑎_𝑖)𝜉_𝑖+^𝑛_𝑗=1𝜉_𝑗(𝑎_𝑖𝑗𝐿^𝑓_𝑗+𝑒^𝜆𝜏𝑏_𝑖𝑗𝐿^𝑓_𝑗+𝑐_𝑖𝑗𝐿^𝑔_𝑗0^+∞|𝑘_𝑖𝑗(𝑠)|𝑒^𝜆𝑠d𝑠),𝑖=1,2,…,𝑛. (4.4)Clearly, ℎ_𝑖(𝜆),𝑖=1,2,…,𝑛, are continuous functions on [0,𝛼₀], where 𝛼₀ is the positive constant of Hypothesis (H₅). Sinceℎ_𝑖(0)=−𝑎_𝑖𝜉_𝑖+^𝑛_𝑗=1𝜉_𝑗(𝑎_𝑖𝑗𝐿^𝑓_𝑗+𝑏_𝑖𝑗𝐿^𝑓_𝑗+𝑐_𝑖𝑗𝐿^𝑔_𝑗0^+∞|𝑘_𝑖𝑗(𝑠)|d𝑠)≤−𝑎_𝑖𝜉_𝑖+^𝑛_𝑗=1𝜉_𝑗(𝑎_𝑖𝑗𝐿^𝑓_𝑗+𝑏_𝑖𝑗𝐿^𝑓_𝑗+𝑐_𝑖𝑗𝛽_𝑖𝑗𝐿^𝑔_𝑗)<0, (4.5)we can choose a positive constant 𝛼∈(0,𝛼₀] such thatℎ_𝑖(𝛼)=(𝛼−𝑎_𝑖)𝜉_𝑖+^𝑛_𝑗=1𝜉_𝑗(𝑎_𝑖𝑗𝐿^𝑓_𝑗+𝑒^𝛼𝜏𝑏_𝑖𝑗𝐿^𝑓_𝑗+𝑐_𝑖𝑗𝐿^𝑔_𝑗0^+∞|𝑘_𝑖𝑗(𝑠)|𝑒^𝛼𝑠d𝑠)<0,𝑖=1,2,…,𝑛. (4.6) Now, we choose a positive constant 𝑑 such that𝑑𝜉_𝑖𝑒^−𝛼𝑡>1,for𝑡∈(−∞,0],𝑖=1,2,…,𝑛. (4.7) Define a Lyapunov function 𝑉=(𝑉₁,𝑉₂,…,𝑉_𝑛)^𝑇 by 𝑉_𝑖=𝑒^𝛼𝑡|𝑧_𝑖(𝑡)|,𝑖=1,2,…,𝑛. In view of (4.1), we obtaind⁺𝑉_𝑖(𝑡)d𝑡=𝑒^𝛼𝑡sgn𝑧_𝑖{−𝑑_𝑖(𝑡,𝑧_𝑖(𝑡))+^𝑛_𝑗=1[𝑎_𝑖𝑗(𝑡)𝑤_𝑗(𝑧_𝑗(𝑡))+𝑏_𝑖𝑗(𝑡)𝑤_𝑗(𝑧_𝑗(𝑡−𝜏_𝑖𝑗(𝑡)))+𝑐_𝑖𝑗(𝑡)^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑢_𝑗(𝑧_𝑗(𝑠))d𝑠]}+𝛼𝑒^𝛼𝑡|𝑧_𝑖(𝑡)|≤𝑒^𝛼𝑡{(𝛼−𝑎_𝑖)|𝑧_𝑖(𝑡)|+^𝑛_𝑗=1[𝑎_𝑖𝑗𝐿^𝑓_𝑗|𝑧_𝑗(𝑡)|+𝑏_𝑖𝑗𝐿^𝑓_𝑗|𝑧_𝑗(𝑡−𝜏_𝑖𝑗+(𝑡))|𝑐_𝑖𝑗𝐿^𝑔_𝑗^𝑡_−∞|𝑘_𝑖𝑗(𝑡−𝑠)||𝑧_𝑗(𝑠)|d𝑠]}=(𝛼−𝑎_𝑖)𝑒^𝛼𝑡|𝑧_𝑖(𝑡)|+^𝑛_𝑗=1[𝑎_𝑖𝑗𝐿^𝑓_𝑗𝑒^𝛼𝑡|𝑧_𝑗(𝑡)|+𝑒^{𝛼𝜏𝑖𝑗(𝑡)}𝑏_𝑖𝑗𝐿^𝑓_𝑗𝑒^{𝛼(𝑡−𝜏𝑖𝑗(𝑡))}|𝑧_𝑗(𝑡−𝜏_𝑖𝑗+(𝑡))|𝑐_𝑖𝑗𝐿^𝑔_𝑗^𝑡_−∞|𝑘_𝑖𝑗(𝑡−𝑠)|𝑒^{𝛼(𝑡−𝑠)}𝑒^𝛼𝑠|𝑧_𝑗(𝑠)|d𝑠]≤(𝛼−𝑎_𝑖)𝑉_𝑖(𝑡)+^𝑛_𝑗=1[𝑎_𝑖𝑗𝐿^𝑓_𝑗𝑉_𝑗(𝑡)+𝑒^𝛼𝜏𝑏_𝑖𝑗𝐿^𝑓_𝑗𝑉_𝑗(𝑡−𝜏_𝑖𝑗+(𝑡))𝑐_𝑖𝑗𝐿^𝑔_𝑗^𝑡_−∞|𝑘_𝑖𝑗(𝑡−𝑠)|𝑒^{𝛼(𝑡−𝑠)}𝑉_𝑗(𝑠)d𝑠],𝑖=1,2,…,𝑛. (4.8)From (4.7), we have𝑑𝜉_𝑖‖𝜙−𝜙^∗‖𝑒^−𝛼𝑡>‖𝜙−𝜙^∗‖≥|𝑧_𝑖(𝑡)|,for𝑡∈(−∞,0],𝑖=1,2,…,𝑛, (4.9)which implies that|𝑧_𝑖(𝑡)|<𝑑𝜉_𝑖‖𝜙−𝜙^∗‖𝑒^−𝛼𝑡,for𝑡∈(−∞,0],𝑖=1,2,…,𝑛, (4.10)where ‖𝜙−𝜙^∗‖ is defined as that in Definition 2.1.
We claim that𝑉_𝑖(𝑡)=|𝑧_𝑖(𝑡)|𝑒^𝛼𝑡<𝑑𝜉_𝑖‖𝜙−𝜙^∗‖,𝑖=1,2,…,𝑛,∀𝑡>0. (4.11)Contrarily, there must exist 𝑖₀∈{1,2,…,𝑛} and ^0𝑥0𝑡^02𝑑𝑐>0 such that 𝑉_𝑖0(𝑡^{0𝑥002𝑑𝑐})=𝑑𝜉_𝑖0‖𝜙−𝜙^∗𝑉‖,_𝜃(𝑡^{0𝑥002𝑑𝑐})≤𝑑𝜉_𝜃‖𝜙−𝜙^∗‖,𝜃=1,2,…,𝑖₀−1,𝑖₀𝑉+1,…,𝑛,_𝑖(𝑡)<𝑑𝜉_𝑖‖𝜙−𝜙^∗‖,∀𝑡∈(−∞,𝑡^{0𝑥002𝑑𝑐}),𝑖=1,2,…,𝑛. (4.12)Together with (4.8) and (4.12), we obtaind0≤⁺(𝑉_𝑖0(𝑡^{0𝑥002𝑑𝑐})−𝑑𝜉_𝑖0‖𝜙−𝜙^∗‖)=dd𝑡⁺𝑉_𝑖0(𝑡^{0𝑥002𝑑𝑐})d𝑡≤(𝛼−𝑎_𝑖0)𝑉_𝑖0(𝑡^{0𝑥002𝑑𝑐})+^𝑛_𝑗=1[𝑎_𝑖0𝑗𝐿^𝑓_𝑗𝑉_𝑗(𝑡^{0𝑥002𝑑𝑐})+𝑒^𝛼𝜏𝑏_𝑖0𝑗𝐿^𝑓_𝑗𝑉_𝑗(𝑡^{0𝑥002𝑑𝑐}−𝜏_𝑖0𝑗(𝑡^{0𝑥002𝑑𝑐}))+𝑐_𝑖0𝑗𝐿^𝑔_𝑗^{0𝑥002𝑑𝑐𝑡}_−∞|𝑘_𝑖0𝑗|𝑒^𝛼𝑉_𝑗d𝑠]≤(𝛼−𝑎_𝑖0)𝑑𝜉_𝑖0‖𝜙−𝜙^∗‖+^𝑛_𝑗=1𝑑𝜉_𝑗‖𝜙−𝜙^∗‖(𝑎_𝑖0𝑗𝐿^𝑓_𝑗+𝑒^𝛼𝜏𝑏_𝑖0𝑗𝐿^𝑓_𝑗+𝑐_𝑖0𝑗𝐿^𝑔_𝑗0^+∞|𝑘_𝑖0𝑗(𝑠)|𝑒^𝛼𝑠d𝑠). (𝑡^{0𝑥002𝑑𝑐}−𝑠)^{(0𝑥002𝑑𝑐𝑡−𝑠)}(𝑠)(4.13)Hence,0≤(𝛼−𝑎_𝑖0)𝜉_𝑖0+^𝑛_𝑗=1𝜉_𝑗(𝑎_𝑖0𝑗𝐿^𝑓_𝑗+𝑒^𝛼𝜏𝑏_𝑖0𝑗𝐿^𝑓_𝑗+𝑐_𝑖0𝑗𝐿^𝑔_𝑗0^+∞|𝑘_𝑖0𝑗(𝑠)|𝑒^𝛼𝑠d𝑠), (4.14)which contradicts (4.6). Therefore, (4.11) holds. It follows that|𝑧_𝑖(𝑡)|<𝑑𝜉_𝑖‖𝜙−𝜙^∗‖𝑒^−𝛼𝑡,𝑡>0,𝑖=1,2,…,𝑛. (4.15)Let 𝑀=max_{1≤𝑖≤𝑛}{𝑑𝜉_𝑖+1}. Then, from (4.15), we get|𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)|=|𝑧_𝑖(𝑡)|≤𝑀‖𝜙−𝜙^∗‖𝑒^−𝛼𝑡,∀𝑡>0,𝑖=1,2,…,𝑛. (4.16)In view of Definition 2.1, the 𝜔-periodic solution 𝑥^∗(𝑡) of system (1.1) is globally exponentially stable. This completes the proof.

Corollary 4.2. Let (H₁)-(H₇) hold. Assume that 𝐸_𝑛−Γ is a nonsingular 𝑀-matrix or 𝜌(Γ)<1. Then, system (1.1) has exactly one 𝜔-periodic solution, which is globally exponentially stable, where Γ is defined as that in (H₆).

Remark 4.3. As a special case, 𝑎_𝑖(𝑡,𝑥_𝑖(𝑡))=𝑎_𝑖(𝑡)𝑥_𝑖(𝑡) and 𝑎_𝑖(𝑡)>0. Let 𝑎_𝑖=min{|𝑎_𝑖(𝑡)|∶𝑡∈[0,𝜔]},𝑎_𝑖=max{|𝑎_𝑖(𝑡)|∶𝑡∈[0,𝜔]}, then 𝐿^𝑎_𝑖=𝑎_𝑖,𝑖=1,2,…,𝑛. Obviously, (H₂) and (H₇) hold.

Remark 4.4. When 𝑎_𝑖(𝑡,𝑥_𝑖(𝑡))=𝑐_𝑖𝑥_𝑖(𝑡),𝑐_𝑖 are constants, and 𝑐_𝑖𝑗≡0, 𝑖,𝑗=1,2,…,𝑛, one can easily know that [11, Theorems 1 and 2] are direct corollaries of Theorems 3.1 and 4.1 of this paper, respectively. Moreover, we need not the following assumption:
(H₀)𝑓_𝑗(0)=0,𝑔_𝑗(0)=0,0≤𝜏^′_𝑖𝑗(𝑡)<1,𝑖,𝑗=1,2,…,𝑛.

Very recently, Zhou and Hu [12] considered the global exponential periodicity and stability of the following cellular neural networks:𝑥^′_𝑖(𝑡)=−𝑑_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1[𝑎_𝑖𝑗𝑓_𝑗(𝑥_𝑗(𝑡))+𝑏_𝑖𝑗𝑓_𝑗(𝑥_𝑗(𝑡−𝜏_𝑖𝑗(𝑡)))+𝑐_𝑖𝑗^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑔_𝑗(𝑥_𝑗(𝑠))d𝑠]+𝐼_𝑖(𝑡),𝑖=1,2,…,𝑛, (4.17)where 𝑑_𝑖>0,𝑎_𝑖𝑗,𝑏_𝑖𝑗,𝑐_𝑖𝑗,𝑖,𝑗=1,2,…,𝑛, are constants, 𝐼_𝑖(𝑡+𝜔)=𝐼_𝑖(𝑡),𝑖,𝑗=1,2,…,𝑛, 𝜏^′_𝑖𝑗(𝑡)<1, and 0≤𝜏_𝑖𝑗(𝑡)≤𝜏. The assumptions in [12] on the delay kernels 𝑘_𝑖𝑗 of (4.17) are as follows.

(F₁) The delay kernels 𝑘_𝑖𝑗∶𝑅⁺→𝑅⁺ are real-valued nonnegative continuous functions and ∫₀^+∞𝑘_𝑖𝑗(𝑠)d𝑠=1,𝑖,𝑗=1,2,…,𝑛.(F₂)∫₀^+∞𝑠𝑘_𝑖𝑗(𝑠)d𝑠≤+∞,𝑖,𝑗=1,2,…,𝑛.(F₃) There exists a positive number 𝜇 such that ∫₀^+∞𝑒^𝜇𝑠𝑘_𝑖𝑗(𝑠)d𝑠≤+∞,𝑖,𝑗=1,2,…,𝑛. Obviously, (4.17) is a special case of (1.1). Moreover, (3.1) of Theorem 3.1 and (3.15) of Theorem 3.2 in [12] are special cases of (4.3) in this paper, that is, 𝜉=(1,1,…,1)^𝑇,𝐿^𝑓_𝑗=𝐿^𝑔_𝑗, and 𝛽_𝑖𝑗=1. So, [12, Theorems 3.1 and 3.2] are special cases of Theorem 4.1 in this paper. To summarize, the results of this paper are completely new and generalize the results of [4–12] and the references cited therein.

5. Application

In this section, we give an example to illustrate that our results are feasible. Consider the following simple DCNNs with time-varying delays:𝑥^′_𝑖(𝑡)=−𝑎_𝑖(𝑡,𝑥_𝑖(𝑡))+²_𝑗=1[𝑎_𝑖𝑗(𝑡)𝑓_𝑗(𝑥_𝑗(𝑡))+𝑏_𝑖𝑗(𝑡)𝑓_𝑗(𝑥_𝑗(𝑡−𝜏_𝑖𝑗(𝑡)))+𝑐_𝑖𝑗(𝑡)^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑔_𝑗(𝑥_𝑗(𝑠))d𝑠]+𝐼_𝑖(𝑡),𝑖=1,2, (5.1)where 𝑓_𝑗(𝑥)=𝑔_𝑗(𝑥)=|𝑥|,𝑎_𝑖(𝑡,𝑥)=2𝑥+sin𝑥,𝐼_𝑖(𝑡)=sin40𝜋𝑡,𝜏_𝑖𝑗(𝑡)=(cos20𝜋𝑡)²,𝑘_𝑖𝑗(𝑡)=sin𝑡𝑒^−𝑡, and 𝑎_𝑖𝑗(𝑡)=𝑏_𝑖𝑗(𝑡)=𝑐_𝑖𝑗(𝑡)=(1/12)cos40𝜋𝑡,𝑖,𝑗=1,2. Then, we have 1𝜔=20,𝐿^𝑓_𝑗=𝐿^𝑔_𝑗=1,𝑎_𝑖=1,𝑎_𝑖=3,𝛽_𝑖𝑗=1,𝑎_𝑖𝑗=𝑏_𝑖𝑗=𝑐_𝑖𝑗=112,𝑖,𝑗=1,2. (5.2) Hence,1Γ=4141414. (5.3) We take 𝜂^𝑇=(𝜂₁,𝜂₂)=(1,1), then)3𝜂=(114−14−14341=(212)>(00). (5.4)It is easy to check that all the conditions needed in Theorem 4.1 are satisfied. Therefore, (5.1) has a unique global exponential 1/20-periodic solution.

Remark 5.1. Because 𝑎_𝑖(𝑡,𝑥_𝑖(𝑡)) is not linear about 𝑥_𝑖(𝑡), thus none of the results in [4–12, 14] can be applied to (5.1). This implies that the results of this paper are new.

Acknowledgments

The author would like to express his gratitude to some anonymous reviewers and the editor for their valuable suggestions and comments. This work was supported by the Scientific Research Fund of Yunnan Provincial Education Department (07Y10085) and the NSF of Honghe University (XSS06009) of China.