Global exponential stability of periodic solution for fuzzy cellular neural networks with distributed delays and variable coefficients

Abstract

In this paper, fuzzy cellular neural networks with distributed delays and variable coefficients are studied. Applying Gaines and Mawhin^,s continuation theorem of coincidence degree theory, the method of Lyapunov function and the elementary inequality 2xy ≤ x² + y², we establish some sufficient conditions which ensure the existence and global exponential stability of periodic solution of such fuzzy cellular neural networks with distributed delays and variable coefficients. These sufficient conditions, which are easily checked in practice by simple algebra methods, have a wide range of application. An example with its numerical simulations is given to show the feasibility of the obtained theoretical predictions. The results obtained in this article are completely new and complement the previously known studies.

Keywords

Fuzzy cellular neural networks exponential stability periodic solution distributed delay topological degree theory variable coefficient

1 Introduction

It is well known that cellular neural networks (CNNs) have been successfully applied in various disciplines such as image processing, analyzing 3D surfaces, solving partial differential equations, reducing non-visual problems to geometric maps and so on [1 –4]. In real world, uncertainty or vagueness is often occur. Thus it is necessary for us to take vagueness into consideration. Fuzzy theory can be regarded as a suitable method. In 1996, fuzzy cellular neural networks (FCNNs) were first introduced by Yang et al. [1 –3]. FCNNs paly an important role in image processing and pattern recognition. Such applications depend heavily on the dynamics of the networks such as stability, oscillation and convergence, see [5]. In particular, global exponential stability of periodic solution for FCNNs plays a key role in the design and application of neural networks [5]. In many cases, to optimize FCNNs, we need the property of global exponential stability of periodic solution for FCNNs. Thus the global exponential stability of periodic solution of FCNNs has gain much attention, and numerous results related to this aspect have been reported. For example, Balasubramaniam et al. [4] considered the global asymptotic stability of BAM fuzzy cellular neural networks with time delay in the leakage term, discrete and unbounded distributed delays, Syed Ali and Balasubramaniam [6] focused on the global asymptotic stability of stochastic fuzzy cellular neural networks with multiple discrete and distributed time-varying delays, Long and Xu [7] studied the global exponential p-stability of stochastic non-autonomous Takagi-Sugeno fuzzy cellular neural networks with time-varying delays and impulses, Song and Wang [8] investigated the dynamical behaviors of fuzzy reaction-diffusion periodic cellular neural networks with variable coefficients and delays, Huang [9] presented the exponential stability of fuzzy cellular neural networks with distributed delay, Yang et al. [10] gave a theoretical study on the exponential stability of impulsive stochastic fuzzy cellular neural networks with mixed delays and reaction-diffusion terms, Yuan et al. [11] made a discussion on the exponential stability and periodic solutions of fuzzy cellular neural networks with time-varying delays. For more details, we refer the readers to [12–27 , 36–38].

Here we would like to point out that neural networks usually have a spatial extent and there will lead to a distribution of propagation delays. Thus, the continuous distributed delays are more suitable for fuzzy cellular neural networks [28 –32]. In addition, we know that the parameters of networks usually will arise change along with time [33], then non-autonomous phenomenon often in many realistic networks. Inspired by the discussion above, in this paper, we consider the following fuzzy cellular neural networks with distributed delays and variable coefficients

$\begin{matrix} {\dot{x}}_{i} (t) & = & - c_{i} x_{i} (t) + \sum_{j = 1}^{n} a_{ij} (t) f_{j} (x_{j} (t)) \\ + \sum_{j = 1}^{n} b_{ij} (t) u_{j} (t) + I_{i} (t) \\ + ⋀_{j = 1}^{n} α_{ij} (t) g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (t - s) ds) \\ + ⋁_{j = 1}^{n} β_{ij} (t) g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (t - s) ds) \\ + ⋀_{j = 1}^{n} T_{ij} (t) u_{j} (t) + ⋀_{j = 1}^{n} H_{ij} (t) u_{j} (t), \end{matrix}$ (1) where i = 1, 2, ⋯ , n, c_i > 0 denotes the passive decay rates to the state of ith unit at time t. α_ij (t) , β_ij (t) , T_ij (t) and H_ij (t) denote elements of fuzzy feedback MIN template and fuzzy feedback MAX template, fuzzy feed forward MIN template and Fuzzy feed forward MAX template, respectively. a_ij (t) and B_ij (t) represent elements of feedback template and feed forward template. ⋀ and ⋁ are the fuzzy AND and OR operation, respectively. x_i (t) , u_j (t) and I_i (t) stands for state, input and bias of the ith neurons, respectively. f_j (.) and g_j (.) denote the activation functions. k_ij (s) stands for delay kernels.

The principle object of this article is to explore the dynamics of system (1). That is, we will apply the Mawhin^,s continuous theorem [34], the method of Lyapunov function and elementary inequality technique to study the existence and global asymptotic stability of periodic solutions of system (1).

The rest of the paper is organized as follows. In Section 2, some notations, definitions and lemmas are given. In Section 3, applying the coincidence degree and the related continuation theorem, some sufficient conditions which ensure the existence of periodic solution of the FCNNs are established. Using the method of Lyapunov function, a series of sufficient conditions which guarantee the globally exponentially stability of the FCNNs are derived in Section 4. In Section 5, we give an example to show the feasibility of the main results. A brief conclusion is drawn in Section 6.

2 Preliminaries

For convenience, we introduce some notations and recall several basic definitions.

Let f (t) be a continuous ω-periodic function defined on R, where ω > 0. Denote $f^{+} = max_{0 \leq t \leq ω} | f (t) |,$ $\bar{f} = \frac{1}{ω} \int_{0}^{ω} f (t) dt, | | f | |_{2} = {(\int_{0}^{ω} | f (t) |^{2} ds)}^{\frac{1}{2}} .$ Let x = (x₁, x₂, ⋯ , x_n) ^T ∈ Rⁿ denote a column vector (here “T " stands for the transpose of a vector). B = (b_ij) _n×n is a n × n matrix and E_n is the identity matrix of size n. For a matrix B ≥ 0 means that all entries of B are greater than or equal to zero. B > 0 can be defined similarly. For a matrix B ≥ E (respectively, B > E) means that B - E ≥ 0 (respectively, B - E > 0).

The initial conditions associated with system (1) take the form $x_{i} (s) = φ_{i} (s), s \in [- \infty, 0], i = 1, 2, \dots, n,$ (2) where φ = (φ₁ (s) , φ₂ (s) , ⋯ , φ_n (s)) ^T ∈ C ([- ∞ , 0] , Rⁿ) . Throughout this paper, we always make the following assumptions:

For i = 1, 2, ⋯ , n, a_ij (t) , b_ij (t) , α_ij (t) , β_ij (t) , T_ij (t) , H_ij (t) , u_ij (t) , I_i (t) are continuous ω-periodic functions.

For i = 1, 2, ⋯ , n,f_j (.) and g_j (.) are Lipschitz continuous on R with Lipschitz constants $L_{j}^{f}$ and $L_{j}^{g}$ and f_j (0) = g_i (0) =0, i.e., for all x, y ∈ R, one has $\begin{matrix} | f_{j} (x) - f_{j} (y) | \leq L_{j}^{f} | x - y |, \\ | g_{j} (x) - g_{j} (y) | \leq L_{j}^{g} | x - y | . \end{matrix}$

For i, j = 1, 2, ⋯ , n, the delay kernel k_ij : [0, ∞) → [0, ∞) are continuous functions and satisfy $\int_{0}^{\infty} k_{ij} (s) ds = 1, \int_{0}^{\infty} k_{ij} (s) e^{λ s} ds < \infty,$ where λ > 0 and ${\bar{k}}_{ij} > 0$ are constants.

Definition 2.1. The periodic solution $y^{*} (t) = (x_{1}^{*} (t),$ $x_{2}^{*} (t), \dots, x_{n}^{*} (t))^{T}$ of system (1) with the initial value $φ^{*} = (φ_{1}^{*}, φ_{1}^{*}, \dots, φ_{n}^{*})^{T} \in C ([- \infty, 0], R^{n})$ is said to be globally exponentially stable if there exist constants λ > 0 and M > 0 such that for i = 1, 2, ⋯ , n, $| x_{i} (t) - x_{i}^{*} (t) | \leq M | | φ - φ^{*} | | e^{- λ t}, t > 0,$ for every solution y (t) = (x₁ (t) , x₂ (t) , ⋯ , x_n (t)) ^T of system (1) with the initial value φ ∈ C ([- ∞ , 0] , Rⁿ).

Definition 2.2. A real matrix A = (a_ij) _n×n is said to be an M-matrix if a_ij ≤ 0, i, j = 1, 2, ⋯ , n, i ≠ j, and A^-1 ≥ 0.

Lemma 2.1. [35] Let A = (a_ij) be an n × n matrix with non-positive off-diagonal elements. Then the following statements are equivalent:

A is an M-matrix.

The real parts of all eigenvalues of A arepositive.

There exists a vector η > 0 such that Aη > 0.

There exists a vector ξ > 0 such that ξ^TA > 0.

Lemma 2.2. [1] Let x and y be two states of system (1). Then $\begin{matrix} | ⋀_{j = 1}^{n} α_{ij} (t) g_{j} (x) - ⋀_{j = 1}^{n} α_{ij} (t) g_{j} (y) | \\ \leq \sum_{j = 1}^{n} | α_{ij} (t) | | g_{j} (x) - g_{j} (y) |, \\ | ⋁_{j = 1}^{n} β_{ij} (t) g_{j} (x) - ⋁_{j = 1}^{n} β_{ij} (t) g_{j} (y) | \\ \leq \sum_{j = 1}^{n} | β_{ij} (t) | | g_{j} (x) - g_{j} (y) | . \end{matrix}$

In order to obtain the existence of periodic solutions of (1), we still make the following preparations.

Denote $| x_{i} |_{\infty} = max_{t \in [0, ω]} | x_{i} (t) |,$ $| | u | |_{X} = max {| x_{1} |_{\infty}, | x_{2} |_{\infty}, \dots, | x_{n} |_{\infty}},$ where u (t) = (x₁ (t) , x₂ (t) , ⋯ , x_n (t)) ^T and X denotes the set of all continuously ω-periodic solutions u (t) defined on R.

Let X, Y be normed vector spaces, L : DomL ⊂ X → Y be a linear mapping, N : X → Y be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dimKerL = codimImL< + ∞ and ImL is closed in Y. If L is a Fredholm mapping of index zero and there exist continuous projectors P : X → X and Q : Y → Y such that ImP = KerL, ImL = KerQ = Im (I - Q) , It follows that L ∣ DomL ∩ KerP : (I - P) X → ImL is invertible. We denote the inverse of that map by K_P. If Ω is an open bounded subset of X, the mapping N will be called L-compact on $\bar{Ω}$ if $QN (\bar{Ω})$ is bounded and $K_{P} (I - Q) N : \bar{Ω} \to X$ is compact. Since ImQ is isomorphic to KerL, there exists an isomorphism J : ImQ → KerL .

Lemma 2.3. ([34] Continuation Theorem) Let L be a Fredholm mapping of index zero and N be L-compact on $\bar{Ω} .$ Suppose

For each λ ∈ (0, 1), every solution x of Lx = λNx is such that x∉ ∂Ω ;

QNx ≠ 0 for each x ∈ KerL ⋂ ∂Ω, and deg {JQN, Ω ⋂ KerL, 0} ≠0;

Then the equation Lx = Nx has at least one solution lying in

Dom L ⋂ \bar{Ω} .

3 Existence of periodic solutions

In the section, we will ready to establish our result.

Theorem 3.1. In addition to the conditions (H1) and (H2), assume that the following condition holds: (H3) E_n - D is an M-matrix, where D = (d_ij) _n×n and $d_{ij} = \frac{1}{c_{i}} [\sum_{j = 1}^{n} a_{ij}^{+} L_{j}^{f} + ⋀_{j = 1}^{n} α_{ij}^{+} L_{j}^{g} + ⋁_{j = 1}^{n} β_{ij}^{+} L_{j}^{g}],$ where i, j = 1, 2, ⋯ , n . Then system (1) has at least one ω-periodic solution.

Proof. Let

$\begin{matrix} (Nz)_{i} (t) & = & - c_{i} x_{i} (t) + \sum_{j = 1}^{n} a_{ij} (t) f_{j} (x_{j} (t)) \\ + \sum_{j = 1}^{n} b_{ij} (t) u_{j} (t) + I_{i} (t) \\ + ⋀_{j = 1}^{n} α_{ij} (t) g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (t - s) ds) \\ + ⋁_{j = 1}^{n} β_{ij} (t) g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (t - s) ds) \\ + ⋀_{j = 1}^{n} T_{ij} (t) u_{j} (t) + ⋀_{j = 1}^{n} H_{ij} (t) u_{j} (t), \end{matrix}$ (3) where i = 1, 2, ⋯ , n . $(L z) (t) = z^{'} (t) = {(x_{1}^{'} (t), x_{2}^{'} (t), \dots, x_{n}^{'} (t))}^{T},$ (4) $Dom L = {z (t) : z (t) \in X, z' (t) \in X},$ (5)

$\begin{matrix} Pz & = & Qz = \frac{1}{ω} \int_{0}^{ω} z (t) dt \\ = & {(\frac{1}{ω} \int_{0}^{ω} x_{1} (t) dt, \dots, \frac{1}{ω} \int_{0}^{ω} x_{n} (t) dt)}^{T} \end{matrix}$ (6) for z (t) = (x₁ (t) , x₂ (t) , ⋯ , x_n (t)) ^T ∈ X ∩ DomL . One can easily prove that L is a Fredholm mapping of index zero, P : X ∩ DomL → KerL and Q : X → X/ImL are two projectors, and N is L compact on $\bar{Ω}$ for any given open bounded set.

Now we will search for an appropriate open, bounded subset Ω for the application of the continuation theorem. Corresponding to the operator equation Lz = λNz, λ ∈ (0, 1) , we get

$\begin{array}{l} x_{i}^{'} (t) = λ [- c_{i} x_{i} (t) + \sum_{j = 1}^{n} a_{i j} (t) f_{j} (x_{j} (t)) \\ + \sum_{j = 1}^{n} b_{i j} (t) u_{j} (t) + I_{i} (t) \\ + \land_{j = 1}^{n} α_{i j} (t) g_{j} (\int_{0}^{\infty} k_{i j} (s) x_{j} (t - s) d s) \\ + \lor_{j = 1}^{n} β_{i j} (t) g_{j} (\int_{0}^{\infty} k_{i j} (s) x_{j} (t - s) d s) \\ + \begin{array}{l} \land_{j = 1}^{n} T_{i j} (t) u_{j} (t) \\ + \land_{j = 1}^{n} H_{i j} (t) u_{j} (t) \end{array}], \end{array}$ (7) where i = 1, 2, ⋯ , n . Suppose that z (t) = (x₁ (t) , x₂ (t) , ⋯ , x_n (t)) ^T ∈ X is an arbitrary solution of system (7) for a certain λ ∈ (0, 1), Then x_i (t) (i = 1, 2, ⋯ , n) is continuously differentiable. Hence there exists t_i ∈ [0, ω] such that $| x_{i} (t_{i}) | = max_{t \in [0, ω]} | x_{i} (t) | .$ Thus. By (7), for i = 1, 2, ⋯ , n, we have

$\begin{matrix} c_{i} x_{i} (t_{i}) \\ = \sum_{j = 1}^{n} a_{ij} (t_{i}) f_{j} (x_{j} (t_{i})) + \sum_{j = 1}^{n} b_{ij} (t_{i}) u_{j} (t_{i}) \\ + ⋀_{j = 1}^{n} α_{ij} (t_{i}) g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (t_{i} - s) ds) \\ + ⋁_{j = 1}^{n} β_{ij} (t_{i}) g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (t_{i} - s) ds) \\ + ⋀_{j = 1}^{n} T_{ij} (t_{i}) u_{j} (t_{i}) + ⋀_{j = 1}^{n} H_{ij} (t_{i}) u_{j} (t_{i}) + I_{i} (t_{i}), \end{matrix}$ (8) Then for i = 1, 2, ⋯ , n, we have

$\begin{matrix} | x_{i} (t_{i}) | & = & | \frac{1}{c_{i}} [\sum_{j = 1}^{n} a_{ij} (t_{i}) f_{j} (x_{j} (t_{i})) \\ + \sum_{j = 1}^{n} b_{ij} (t_{i}) u_{j} (t_{i}) + I_{i} (t_{i}) \\ + ⋀_{j = 1}^{n} α_{ij} (t_{i}) g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (t_{i} - s) ds) \\ + ⋁_{j = 1}^{n} β_{ij} (t_{i}) g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (t_{i} - s) ds) \\ + ⋀_{j = 1}^{n} T_{ij} (t_{i}) u_{j} (t_{i}) + ⋀_{j = 1}^{n} H_{ij} (t_{i}) u_{j} (t_{i})] | \\ \leq \frac{1}{c_{i}} [\sum_{j = 1}^{n} | a_{ij} (t_{i}) | | f_{j} (x_{j} (t_{i})) | \\ + \sum_{j + 1}^{n} | b_{ij} (t_{i}) | | u_{j} (t_{i}) | + | I_{i} (t_{i}) | \\ + ⋀_{j = 1}^{n} | α_{ij} (t_{i}) | | g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (t_{i} - s) ds) | \\ + ⋁_{j = 1}^{n} | β_{ij} (t_{i}) | | g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (t_{i} - s) ds) | \\ + ⋀_{j = 1}^{n} | T_{ij} (t_{i}) | | u_{j} (t_{i}) | + ⋀_{j = 1}^{n} | H_{ij} (t_{i}) | | u_{j} (t_{i}) |] \\ \leq & \frac{1}{c_{i}} [\sum_{j = 1}^{n} a_{ij}^{+} L_{j}^{f} | x_{j} (t_{j}) | + ⋀_{j = 1}^{n} α_{ij}^{+} L_{j}^{g} | x_{j} (t_{j}) | \\ + \sum_{j + 1}^{n} b_{ij}^{+} u_{j}^{+} + I_{i}^{+} + ⋁_{j = 1}^{n} β_{ij}^{+} L_{j}^{g} | x_{j} (t_{j}) | \\ + ⋀_{j = 1}^{n} T_{ij}^{+} u_{j}^{+} + ⋀_{j = 1}^{n} H_{ij}^{+} u_{j}^{+}] \\ \leq & \sum_{j = 1}^{n} d_{ij} | x_{j} (t_{j}) | + h_{i}, \end{matrix}$ (9) where $\begin{matrix} d_{ij} & = & \frac{1}{c_{i}} [\sum_{j = 1}^{n} a_{ij}^{+} L_{j}^{f} + ⋀_{j = 1}^{n} α_{ij}^{+} L_{j}^{g} + ⋁_{j = 1}^{n} β_{ij}^{+} L_{j}^{g}], \\ h_{i} & = & \frac{1}{c_{i}} [\sum_{j + 1}^{n} b_{ij}^{+} u_{j}^{+} + ⋀_{j = 1}^{n} T_{ij}^{+} u_{j}^{+} \\ + ⋀_{j = 1}^{n} H_{ij}^{+} u_{j}^{+} + I_{i}^{+}] . \end{matrix}$ In view of (3.7), we have

$\begin{matrix} (E_{n} - D) (| x_{1} (t) |, | x_{2} (t) |, \dots, | x_{n} (t) |)^{T} \\ \leq (h_{1}, h_{2}, \dots, h_{n})^{T} : = h . \end{matrix}$ (10) It follows from (H3) and Lemma 2.1 that there exists a vector η = (η₁, η₂, ⋯ , η_n) > (0, 0, ⋯ , 0) such that

$\begin{matrix} η^{*} & = & (η_{1}^{*}, η_{2}^{*}, \dots, η_{n}^{*}) \\ = & η (E_{n} - D) > (0, 0, \dots, 0) . \end{matrix}$ (11) By (10) and (11), we get

$\begin{matrix} min {η_{1}^{*}, η_{2}^{*}, \dots, η_{n}^{*}} \\ \times (| x_{1} (t) | + | x_{2} (t) | + \dots + | x_{n} (t) |) \\ \leq η_{1}^{*} | x_{1} (t) | + η_{2}^{*} | x_{2} (t) | + \dots + η_{n}^{*} | x_{n} (t) | \\ = η (E_{n} - D) (| x_{1} (t) |, | x_{2} (t) |, \dots, | x_{n} (t) |)^{T} \\ \leq (h_{1}, h_{2}, \dots, h_{n})^{T} \\ = η_{1} h_{1} + η_{2} h_{2} + \dots + η_{n} h_{n} . \end{matrix}$ (12) Thus we obtain $| x_{i} |_{\infty} = max_{t \in [0, ω]} | x_{i} (t) | = | x_{i} (t_{i}) | \leq σ,$ (13) where $σ = \frac{1}{{min}_{1 \leq i \leq n} {η_{i}^{*}}} (η_{1} h_{1} + η_{2} h_{2} + \dots + η_{n} h_{n}) .$ By (H3) and Lemma 2.1 again, there exists a vector ξ = (ξ₁, ξ₂, ⋯ , ξ_n) ^T > (0, 0, ⋯ , 0) ^T such that (E_n - D) ξ > 0. Then we can choose a positive constant c > 0 such that $ξ^{*} = (ξ_{1}^{*}, ξ_{2}^{*}, \dots, ξ_{n}^{*})^{T} = (c ξ_{1}, c ξ_{2}, \dots, c ξ_{n})^{T} = c ξ$ and $ξ_{i}^{*} = c ξ_{i} > σ, (E_{n} - D) ξ^{*} > h .$ Take $Ω = {z : z (t) \in X, | | z (t) | | \leq | ξ^{*} |, t \in R},$ (14) which satisfies the condition (a) of Lemma 2.3. It z (t) = (x₁ (t) , x₂ (t) , ⋯ , x_n (t)) ∈ ∂Ω ∩ KerL, then z (t) is a constant vector Rⁿ. Then there exists i ∈ {1, 2, ⋯ , n} such that $| x_{i} | = ξ_{i}^{*}$ . Thus for i = 1, 2, ⋯ , n, we have

$\begin{matrix} (QNz)_{i} (t) \\ = - c_{i} x_{i} + \sum_{j = 1}^{n} \frac{1}{ω} \int_{0}^{ω} a_{ij} (t) dt f_{j} (x_{j}) \\ + \sum_{j = 1}^{n} \frac{1}{ω} \int_{0}^{ω} b_{ij} (t) u_{j} (t) dt + + \frac{1}{ω} \int_{0}^{ω} I_{i} (t) dt \\ + ⋀_{j = 1}^{n} \frac{1}{ω} \int_{0}^{ω} α_{ij} (t) dt g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (t - s) ds) \\ + ⋁_{j = 1}^{n} \frac{1}{ω} \int_{0}^{ω} β_{ij} (t) dt g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (t - s) ds) \\ + ⋀_{j = 1}^{n} \frac{1}{ω} \int_{0}^{ω} T_{ij} (t) u_{j} (t) dt \\ + ⋀_{j = 1}^{n} \frac{1}{ω} \int_{0}^{ω} H_{ij} (t) u_{j} (t) dt . \end{matrix}$ (15) Next we prove that $| (QNz)_{i} | > 0, i = 1, 2, \dots, n .$ (16) Assume, by way of contradiction, that (16) does not hold. Then | (QNz) _i|=0, i.e.,

$\begin{matrix} 0 = - c_{i} x_{i} + \sum_{j = 1}^{n} \frac{1}{ω} \int_{0}^{ω} a_{ij} (t) dt f_{j} (x_{j}) \\ + \sum_{j = 1}^{n} \frac{1}{ω} \int_{0}^{ω} b_{ij} (t) u_{j} (t) dt + \frac{1}{ω} \int_{0}^{ω} I_{i} (t) dt \\ + ⋀_{j = 1}^{n} \frac{1}{ω} \int_{0}^{ω} α_{ij} (t) dt g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (t - s) ds) \\ + ⋁_{j = 1}^{n} \frac{1}{ω} \int_{0}^{ω} β_{ij} (t) dt g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (t - s) ds) \\ + ⋀_{j = 1}^{n} \frac{1}{ω} \int_{0}^{ω} T_{ij} (t) u_{j} (t) dt \\ + ⋀_{j = 1}^{n} \frac{1}{ω} \int_{0}^{ω} H_{ij} (t) u_{j} (t) dt . \end{matrix}$ (17) Then there exists t₀ ∈ [0, ω] such that

$\begin{matrix} 0 & = & - c_{i} x_{i} + \sum_{j = 1}^{n} a_{ij} (t_{0}) f_{j} (x_{j}) + \sum_{j = 1}^{n} b_{ij} (t_{0}) u_{j} (t_{0}) \\ + ⋀_{j = 1}^{n} α_{ij} (t_{0}) g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (t - s) ds) \\ + ⋁_{j = 1}^{n} β_{ij} (t_{0}) g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (t - s) ds) \\ + ⋀_{j = 1}^{n} T_{ij} (t_{0}) u_{j} (t_{0}) + ⋀_{j = 1}^{n} H_{ij} (t_{0}) u_{j} (t_{0}) \\ + I_{i} (t_{0}) . \end{matrix}$ (18) For i = 1, 2, ⋯ , n, we have

$\begin{matrix} ξ_{i}^{*} & = & | x_{i} | \\ = & | \frac{1}{c_{i}} [\sum_{j = 1}^{n} a_{ij} (t_{0}) f_{j} (x_{j}) + \sum_{j = 1}^{n} b_{ij} (t_{0}) u_{j} (t_{0}) \\ + ⋀_{j = 1}^{n} α_{ij} (t_{0}) g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (t_{0} - s) ds) \\ + ⋁_{j = 1}^{n} β_{ij} (t_{0}) g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (t_{0} - s) ds) \\ + ⋀_{j = 1}^{n} T_{ij} (t_{0}) u_{j} (t_{0}) + ⋀_{j = 1}^{n} H_{ij} (t_{0}) u_{j} (t_{0}) \\ + I_{i} (t_{0})] | \\ \leq & \frac{1}{c_{i}} [\sum_{j = 1}^{n} | a_{ij} (t_{0}) | | f_{j} (x_{j}) | + \sum_{j = 1}^{n} | b_{ij} (t_{0}) | | u_{j} (t_{0}) | \\ + ⋀_{j = 1}^{n} | α_{ij} (t_{0}) | | g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (t_{0} - s) ds) | \\ + ⋁_{j = 1}^{n} | β_{ij} (t_{0}) | | g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (t_{0} - s) ds) | \\ + ⋀_{j = 1}^{n} | T_{ij} (t_{0}) | | u_{j} (t_{0}) | + ⋀_{j = 1}^{n} | H_{ij} (t_{0}) | | u_{j} (t_{0}) | \\ + | I_{i} (t_{0}) |] \\ \leq & \frac{1}{c_{i}} [\sum_{j = 1}^{n} a_{ij}^{+} L_{j}^{f} | x_{j} | + \sum_{j = 1}^{n} b_{ij}^{+} u_{j}^{+} \\ + ⋀_{j = 1}^{n} α_{ij}^{+} L_{j}^{g} {\bar{k}}_{ij} | x_{j} | + ⋁_{j = 1}^{n} β_{ij}^{+} L_{j}^{g} {\bar{k}}_{ij} | x_{j} | \\ + ⋀_{j = 1}^{n} T_{ij}^{+} u_{j}^{+} + ⋀_{j = 1}^{n} H_{ij}^{+} u_{j}^{+} + I_{i}^{+}] \\ \leq & \sum_{j = 1}^{n} d_{ij} | x_{j} | + Θ_{i} \leq \sum_{j = 1}^{n} d_{ij} ξ_{j}^{*} + h_{i} . \end{matrix}$ (19) Then ((E_n - D) ξ^*) _i ≤ h_i, which contradicts (E_n - D) ξ^* > h. Therefore (16) holds, namely, the condition QNx ≠ 0 for each x ∈ KerL ⋂ ∂Ω in (b) of Lemma 2.3 is fulfilled.

Now let us define a continuous function Ψ : Ω ∩ KerL × [0, 1] → X by

$\begin{matrix} Ψ (z, μ) & = & μ diag (- c_{1}, c_{2}, \dots, - c_{n}) z \\ + (1 - μ) QNz, \end{matrix}$ (20) where z = (x₁ (t) , x₂ (t) , ⋯ , x_n (t)) ^T ∈ Ω ∩ KerL = Ω ∩ Rⁿ and μ ∈ [0, 1]. If z (t) = (x₁ (t) , x₂ (t) , ⋯, x_n (t)) ^T ∈ Ω ∩ KerL, then z (t) is a constant vector in Rⁿ and there exists i ∈ {1, 2, ⋯ , n} such that $| x_{i} | = ξ_{i}^{*}$ . Then we have

$\begin{matrix} (Ψ (z, μ))_{i} = - c_{i} x_{i} + (1 - μ) \\ \times [\sum_{j = 1}^{n} \frac{1}{ω} \int_{0}^{ω} a_{ij} (t) dt f_{j} (x_{j}) \\ + \sum_{j = 1}^{n} \frac{1}{ω} \int_{0}^{ω} b_{ij} (t) u_{j} (t) dt + \frac{1}{ω} \int_{0}^{ω} I_{i} (t) dt \\ + ⋀_{j = 1}^{n} \frac{1}{ω} \int_{0}^{ω} α_{ij} (t) dt g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (t - s) ds) \\ + ⋁_{j = 1}^{n} \frac{1}{ω} \int_{0}^{ω} β_{ij} (t) dt g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (t - s) ds) \\ + ⋀_{j = 1}^{n} \frac{1}{ω} \int_{0}^{ω} T_{ij} (t) u_{j} (t) dt \\ + ⋀_{j = 1}^{n} \frac{1}{ω} \int_{0}^{ω} H_{ij} (t) u_{j} (t) dt] . \end{matrix}$ (21) Now we claim that $| (Ψ (z, μ)_{i} | > 0, i = 1, 2, \dots, n .$ (22) Assume, by way of contradiction, that (22) does not hold. Then | (Ψ (z, μ) _i|=0, i.e.,

$\begin{matrix} 0 = - c_{i} x_{i} + (1 - μ) [\sum_{j = 1}^{n} \frac{1}{ω} \int_{0}^{ω} a_{ij} (t) dt f_{j} (x_{j}) \\ + \sum_{j = 1}^{n} \frac{1}{ω} \int_{0}^{ω} b_{ij} (t) u_{j} (t) dt \\ + ⋀_{j = 1}^{n} \frac{1}{ω} \int_{0}^{ω} α_{ij} (t) dt g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (t - s) ds) \\ + ⋁_{j = 1}^{n} \frac{1}{ω} \int_{0}^{ω} β_{ij} (t) dt g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (t - s) ds) \\ + ⋀_{j = 1}^{n} \frac{1}{ω} \int_{0}^{ω} T_{ij} (t) u_{j} (t) dt \\ + ⋀_{j = 1}^{n} \frac{1}{ω} \int_{0}^{ω} H_{ij} (t) u_{j} (t) dt + \frac{1}{ω} \int_{0}^{ω} I_{i} (t) dt] . \end{matrix}$ (23) There exists $\bar{t} \in [0, ω]$ such that

$\begin{matrix} 0 = - c_{i} x_{i} + (1 - μ) [\sum_{j = 1}^{n} a_{ij} (\bar{t}) f_{j} (x_{j}) \\ + \sum_{j = 1}^{n} b_{ij} (\bar{t}) u_{j} (\bar{t}) + I_{i} (\bar{t}) \\ + ⋀_{j = 1}^{n} α_{ij} (\bar{t}) g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (\bar{t} - s) ds) \\ + ⋁_{j = 1}^{n} β_{ij} (\bar{t}) g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (\bar{t} - s) ds) \\ + ⋀_{j = 1}^{n} T_{ij} (\bar{t}) u_{j} (\bar{t}) + ⋀_{j = 1}^{n} H_{ij} (\bar{t}) u_{j} (\bar{t})] . \end{matrix}$ (24) For i = 1, 2, ⋯ , n, we get $\begin{matrix} ξ_{i}^{*} & = & | x_{i} | \\ = & | \frac{1}{c_{i}} (1 - μ) [\sum_{j = 1}^{n} a_{ij} (\bar{t}) f_{j} (x_{j}) + \sum_{j = 1}^{n} b_{ij} (\bar{t}) u_{j} (\bar{t}) \\ + ⋀_{j = 1}^{n} α_{ij} (\bar{t}) g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (\bar{t} - s) ds) \\ + ⋁_{j = 1}^{n} β_{ij} (t_{0}) g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (\bar{t} - s) ds) \\ + ⋀_{j = 1}^{n} T_{ij} (\bar{t}) u_{j} (\bar{t}) + ⋀_{j = 1}^{n} H_{ij} (\bar{t}) u_{j} (\bar{t}) + I_{i} (\bar{t})] | \\ \leq & \frac{1}{c_{i}} [\sum_{j = 1}^{n} | a_{ij} (\bar{t}) | | f_{j} (x_{j}) | + \sum_{j = 1}^{n} | b_{ij} (\bar{t}) | | u_{j} (\bar{t}) | \\ + ⋀_{j = 1}^{n} | α_{ij} (\bar{t}) | | g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (\bar{t} - s) ds) | \\ + ⋁_{j = 1}^{n} | β_{ij} (\bar{t}) | | g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (\bar{t} - s) ds) | \\ + ⋀_{j = 1}^{n} | T_{ij} (\bar{t}) | | u_{j} (\bar{t}) | + ⋀_{j = 1}^{n} | H_{ij} (\bar{t}) | | u_{j} (\bar{t}) | + | I_{i} (\bar{t}) |] \\ \leq & \frac{1}{c_{i}} [\sum_{j = 1}^{n} a_{ij}^{+} L_{j}^{f} | x_{j} | + \sum_{j = 1}^{n} b_{ij}^{+} u_{j}^{+} + ⋀_{j = 1}^{n} α_{ij}^{+} L_{j}^{g} | x_{j} | \\ + ⋁_{j = 1}^{n} β_{ij}^{+} L_{j}^{g} | x_{j} | + ⋀_{j = 1}^{n} T_{ij}^{+} u_{j}^{+} + ⋀_{j = 1}^{n} H_{ij}^{+} u_{j}^{+} + I_{i}^{+}] \\ \leq & \sum_{j = 1}^{n} d_{ij} | x_{j} | + h_{i} \leq \sum_{j = 1}^{n} d_{ij} ξ_{j}^{*} + h_{i} . \end{matrix}$ (25)

Then ((E_n - D) ξ^*) _i ≤ h_i, which contradicts(E_n - D) ξ^* > h. Therefore (22) holds, namely, which implies that for all (x₁, x₂, ⋯ , x_n) ∈ ∂Ω ∩ KerL, μ ∈ [0, 1] , Ψ (x₁, x₂, ⋯ , x_n, μ) ≠ (0, 0, ⋯ , 0) ^T . Applying the homotopy invariancetheorem, we have deg {QN, Ω ∩ KerL, (0, 0, ⋯ , 0) ^T} = sgn {(-1) ⁿc₁c₂ ⋯c_n)} ≠0 . By now, we have proved that Ω verifies all requirements of Lemma 2.3, then it follows that Lz = Nz has at least one solution in $Dom L \cap \bar{Ω}$ , namely, system (1) has at least one ω-periodic solution. The proof is complete.

4 Global exponential stability of periodic solution

In this section, we shall establish sufficient conditions for the global exponential stability of periodic solution of system (1).

Theorem 4.1. Suppose that (H1)-(H3) and the following condition (H4) $\begin{matrix} - c_{i} + \frac{1}{2} \sum_{j = 1}^{n} a_{ij}^{+} L_{j} + \frac{1}{2} \sum_{j = 1}^{n} \frac{ϱ_{j}}{ϱ_{i}} a_{ji}^{+} L_{i} \\ + \frac{1}{2} \sum_{j = 1}^{n} (α_{ij}^{+} + β_{ij}^{+}) L_{j}^{g} + \frac{1}{2} \sum_{j = 1}^{n} \frac{ϱ_{j}}{ϱ_{i}} (α_{ji}^{+} + β_{ji}^{+}) L_{i}^{g} < 0 \end{matrix}$ holds, then system (1) has exactly one ω-periodic solution which is globally exponentially stable.

Proof. In view of Theorem 3.1, system (1) has an ω-periodic solution $u^{*} (t) = (x_{1}^{*} (t), x_{2}^{*} (t), \dots, x_{n}^{*} (t))^{T} .$ Let u (t) = (x₁ (t) , x₂ (t) , ⋯ , x_n (t)) ^T be an arbitrary solution of system (1). By (1), we have

$\begin{matrix} \frac{d}{dt} (x_{i} (t) - x_{i}^{*} (t)) = - c_{i} (x_{i} (t) - x_{i}^{*} (t)) \\ + \sum_{j = 1}^{n} a_{ij} (t) [f_{j} (x_{j} (t)) - f_{j} (x_{j}^{*} (t))] \\ + ⋀_{j = 1}^{n} α_{ij} (t) g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (t - s) ds) \\ - ⋀_{j = 1}^{n} α_{ij} (t) g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j}^{*} (t) ds) \\ + ⋁_{j = 1}^{n} β_{ij} (t) g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (t - s) ds) \\ - ⋁_{j = 1}^{n} β_{ij} (t) g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j}^{*} (t) ds) . \end{matrix}$ (26) Then we get

$\begin{matrix} \frac{d^{+}}{dt} | x_{i} (t) - x_{i}^{*} (t) | \leq - c_{i} | x_{i} (t) - x_{i}^{*} (t) | \\ + \sum_{j = 1}^{n} | a_{ij} (t) | | f_{j} (x_{j} (t)) - f_{j} (x_{j}^{*} (t)) | \\ + | ⋀_{j = 1}^{n} α_{ij} (t) g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (t - s) ds) \\ - ⋀_{j = 1}^{n} α_{ij} (t) g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j}^{*} (t) ds) | \\ + | ⋁_{j = 1}^{n} β_{ij} (t) g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j} (t - s) ds) \\ - ⋁_{j = 1}^{n} β_{ij} (t) g_{j} (\int_{0}^{\infty} k_{ij} (s) x_{j}^{*} (t) ds) |, \end{matrix}$ where $\frac{d^{+}}{dt}$ denotes the upper right derivative. Let $u_{i} (t) = x_{i} (t) - x_{i}^{*} (t) (i = 1, 2, \dots, n)$ . In view of (H2) and Lemma 2.2, we get

$\begin{matrix} \frac{d^{+}}{dt} | u_{i} (t) | \leq - c_{i} | u_{i} (t) | + \sum_{j = 1}^{n} | a_{ij} (t) | L_{j} | u_{j} (t) | \\ + \sum_{j = 1}^{n} (| α_{ij} (t) | + | β_{ij} (t) |) \\ \times L_{j}^{g} \int_{0}^{\infty} k_{ij} (s) u_{j} (t - s) ds . \end{matrix}$ (27) In view of (H4), we can choose a small constant λ > 0 such that

$\begin{matrix} (λ - c_{i}) + \frac{1}{2} \sum_{j = 1}^{n} a_{ij}^{+} L_{j} + \frac{1}{2} \sum_{j = 1}^{n} \frac{ϱ_{j}}{ϱ_{i}} a_{ji}^{+} L_{i} \\ + \frac{1}{2} \sum_{j = 1}^{n} (α_{ij}^{+} + β_{ij}^{+}) L_{j}^{g} \int_{0}^{\infty} k_{ij} (s) e^{λ s} ds \\ + \frac{1}{2} \sum_{j = 1}^{n} \frac{ϱ_{j}}{ϱ_{i}} (α_{ji}^{+} + β_{ji}^{+}) L_{i}^{g} \int_{0}^{\infty} k_{ji} (s) e^{λ s} ds \\ \leq 0 . \end{matrix}$ (28) Set $w_{i} (t) = e^{λ t} | u_{i} (t) |, i = 1, 2, \dots, n, t > 0 .$ (29) Then we have

$\begin{matrix} \frac{d^{+} w_{i} (t)}{dt} = λ e^{λ t} | u_{i} (t) | + e^{λ t} \frac{d^{+} u_{i} (t)}{dt} \\ \leq λ e^{λ t} | u_{i} (t) | - c_{i} e^{λ t} | u_{i} (t) | \\ + \sum_{j = 1}^{n} | a_{ij} (t) | L_{j} e^{λ t} | u_{j} (t) | \\ + \sum_{j = 1}^{n} (| α_{ij} (t) | + | β_{ij} (t) |) L_{j}^{g} \\ \times \int_{0}^{\infty} k_{ij} (s) e^{λ t} u_{j} (t - s) ds \\ = (λ - c_{i}) w_{i} (t) + \sum_{j = 1}^{n} a_{ij}^{+} L_{j} w_{j} (t) \\ + \sum_{j = 1}^{n} (α_{ij}^{+} + β_{ij}^{+}) L_{j}^{g} \\ \times \int_{0}^{\infty} k_{ij} (s) e^{λ s} w_{j} (t - s) ds . \end{matrix}$ (30) Define a Lyapunov functional V by

$\begin{matrix} V (t) = \sum_{i = 1}^{n} ϱ_{i} [w_{i}^{2} (t) + \sum_{j = 1}^{n} (α_{ij}^{+} + β_{ij}^{+}) L_{j}^{g} \\ \int_{0}^{\infty} k_{ij} (s) e^{λ s} (\int_{t - s}^{t} w_{j}^{2} (θ) d θ) ds] . \end{matrix}$ (31) Calculating the upper right derivative $\frac{d^{+}}{dt} V (t)$ along the solution of (1), we have

$\begin{matrix} \frac{d^{+}}{dt} V (t) = \sum_{i = 1}^{n} ϱ_{i} [2 w_{i} (t) \frac{d^{+}}{dt} w_{i} (t) \\ + \sum_{j = 1}^{n} (α_{ij}^{+} + β_{ij}^{+}) L_{j}^{g} \\ \times \int_{0}^{\infty} k_{ij} (s) e^{λ s} [w_{j}^{2} (t) - w_{j}^{2} (t - s)] ds] \\ \leq \sum_{i = 1}^{n} ϱ_{i} [2 (λ - c_{i}) w_{i}^{2} (t) + \sum_{j = 1}^{n} 2 a_{ij}^{+} L_{j} w_{i} (t) w_{j} (t) \\ + \sum_{j = 1}^{n} (α_{ij}^{+} + β_{ij}^{+}) L_{j}^{g} \int_{0}^{\infty} k_{ij} (s) e^{λ s} \\ \times 2 w_{i} (t) w_{j} (t - s) ds + \sum_{j = 1}^{n} (α_{ij}^{+} + β_{ij}^{+}) L_{j}^{g} \\ \times \int_{0}^{\infty} k_{ij} (s) e^{λ s} (w_{j}^{2} (t) - w_{j}^{2} (t - s)) ds] \\ \leq \sum_{i = 1}^{n} ϱ_{i} [2 (λ - c_{i}) w_{i}^{2} (t) + \sum_{j = 1}^{n} a_{ij}^{+} L_{j} \\ \times (w_{i}^{2} (t) + w_{j}^{2} (t)) + \sum_{j = 1}^{n} (α_{ij}^{+} + β_{ij}^{+}) L_{j}^{g} \\ \times \int_{0}^{\infty} k_{ij} (s) e^{λ s} (w_{i}^{2} (t) + w_{j}^{2} (t - s)) ds \\ + \sum_{j = 1}^{n} (α_{ij}^{+} + β_{ij}^{+}) L_{j}^{g} \\ \times \int_{0}^{\infty} k_{ij} (s) e^{λ s} (w_{j}^{2} (t) - w_{j}^{2} (t - s)) ds] \\ = & \sum_{i = 1}^{n} ϱ_{i} [2 (λ - c_{i}) w_{i}^{2} (t) + \sum_{j = 1}^{n} a_{ij}^{+} L_{j} w_{i}^{2} (t) \\ + \sum_{j = 1}^{n} \frac{ϱ_{j}}{ϱ_{i}} a_{ji}^{+} L_{i} w_{i}^{2} (t) \\ + \sum_{j = 1}^{n} (α_{ij}^{+} + β_{ij}^{+}) L_{j}^{g} \int_{0}^{\infty} k_{ij} (s) e^{λ s} ds w_{i}^{2} (t) \\ + \sum_{j = 1}^{n} \frac{ϱ_{j}}{ϱ_{i}} (α_{ji}^{+} + β_{ji}^{+}) L_{i}^{g} \int_{0}^{\infty} k_{ji} (s) e^{λ s} ds w_{i}^{2} (t)] \\ = & \sum_{i = 1}^{n} 2 ϱ_{i} [(λ - c_{i}) + \frac{1}{2} \sum_{j = 1}^{n} a_{ij}^{+} L_{j} + \frac{1}{2} \sum_{j = 1}^{n} \frac{ϱ_{j}}{ϱ_{i}} a_{ji}^{+} L_{i} \\ + \frac{1}{2} \sum_{j = 1}^{n} (α_{ij}^{+} + β_{ij}^{+}) L_{j}^{g} \int_{0}^{\infty} k_{ij} (s) e^{λ s} ds \\ + \frac{1}{2} \sum_{j = 1}^{n} \frac{ϱ_{j}}{ϱ_{i}} (α_{ji}^{+} + β_{ji}^{+}) L_{i}^{g} \int_{0}^{\infty} k_{ji} (s) e^{λ s} ds] \\ \times w_{i}^{2} (t) . \end{matrix}$ (32) It follows from (28) that $\frac{d^{+}}{dt} V (t) \leq 0 (t > 0)$ which leads to V (t) ≤ V (0) (t > 0). Notice that

$\begin{matrix} V (0) = \sum_{i = 1}^{n} ϱ_{i} [w_{i}^{2} (0) + \sum_{j = 1}^{n} (α_{ij}^{+} + β_{ij}^{+}) L_{j}^{g} \\ \times \int_{0}^{\infty} k_{ij} (s) e^{λ s} (\int_{- s}^{0} w_{j}^{2} (θ) d θ) ds] \\ = \sum_{i = 1}^{n} ϱ_{i} [w_{i}^{2} (0) + \sum_{j = 1}^{n} \frac{ϱ_{j}}{ϱ_{i}} (α_{ji}^{+} + β_{ji}^{+}) L_{i}^{g} \\ \int_{0}^{\infty} k_{ji} (s) e^{λ s} (\int_{- s}^{0} w_{i}^{2} (θ) d θ) ds] \\ \leq \sum_{i = 1}^{n} ϱ_{i} [1 + \sum_{j = 1}^{n} \frac{ϱ_{j}}{ϱ_{i}} (α_{ji}^{+} + β_{ji}^{+}) L_{i}^{g} \\ \times \int_{0}^{\infty} k_{ji} (s) e^{λ s} ds sup_{- \infty < s \leq 0} w_{i}^{2} (s)] . \end{matrix}$ (33) Since $sup_{- \infty < s \leq 0} u_{i}^{2} (s) < \infty (i = 1, 2, \dots, n)$ , and in view of (H4), we have V (0)< ∞. Then V (t)< V (0) < ∞ for all t > 0. By (31), we get

$\begin{matrix} \sum_{i = 1}^{n} ϱ_{i} w_{i}^{2} (t) \leq V (t) \\ \leq \sum_{i = 1}^{n} ϱ_{i} [1 + \sum_{j = 1}^{n} \frac{ϱ_{j}}{ϱ_{i}} (α_{ji}^{+} + β_{ji}^{+}) L_{i}^{g} \\ \int_{0}^{\infty} k_{ji} (s) e^{λ s} ds sup_{- \infty < s \leq 0} w_{i}^{2} (s)], \end{matrix}$ (34) for t > 0. That is

$\begin{matrix} \sum_{i = 1}^{n} ϱ_{i} w_{i}^{2} (t) \leq \sum_{i = 1}^{n} [ϱ_{i} + \sum_{j = 1}^{n} ϱ_{j} (α_{ji}^{+} + β_{ji}^{+}) L_{i}^{g} \\ \times \int_{0}^{\infty} k_{ji} (s) e^{λ s} ds sup_{- \infty < s \leq 0} w_{i}^{2} (s)], \end{matrix}$ (35) where t > 0 . By (29), we have

$\begin{matrix} \sum_{i = 1}^{n} ϱ_{i} e^{2 λ t} | x_{i} (t) - x_{i}^{*} (t) |^{2} \\ \leq \sum_{i = 1}^{n} [ϱ_{i} + \sum_{j = 1}^{n} ϱ_{j} (α_{ji}^{+} + β_{ji}^{+}) L_{i}^{g} \\ \times \int_{0}^{\infty} k_{ji} (s) e^{λ s} ds \\ \times sup_{- \infty < s \leq 0} | x_{i} (x) - x_{i}^{*} (x) |^{2}] \end{matrix}$ (36) for t > 0. Hence

$\begin{matrix} \sum_{i = 1}^{n} | x_{i} (t) - x_{i}^{*} (t) |^{2} \\ \leq K e^{- 2 λ t} sup_{- \infty < s \leq 0} | x_{i} (x) - x_{i}^{*} (x) |^{2}, \end{matrix}$ (37) where $K = \frac{χ}{{min}_{1 \leq i \leq n} {ϱ_{i}}} .$ where $χ = max_{1 \leq i, j \leq n}$ ${\sum_{i = 1}^{n} ϱ_{i} + \sum_{j = 1}^{n} ϱ_{j} (α_{ji}^{+} + β_{ji}^{+}) L_{i}^{g} \int_{0}^{\infty} k_{ji} (s) e^{λ s} ds} .$ Noticing that x_i (s) = φ_i (s) , - ∞ ≤ s ≤ 0 and by (37), we have $| | x (t) - x^{*} | | \leq M e^{- λ t} | | φ - φ^{*} | |$ (38) for t > 0, where $M = \sqrt{K} .$ Thus, the periodic solution of system (1) is globally exponentiallystable.

Remark 4.1. In [25 –27], Cao established the sufficient conditions for the globally exponential stability of delayed cellular neural networks by constructing a suitable Lyapunov functional and combining with the elementary inequality technique. All the coefficients of cellular neural networks are constants and there is no fuzzy logic. In this paper, we consider the existence and globally exponential stability of cellular neural networks with distributed delays with varying coefficients and fuzzy logic by the coincidence degree theory, Lyapunov function. (1) is more general than the systems in [25 –27]. Moreover, the results in [25 –27] cannot be applicable to system (1) to obtain the existence and exponential stability of periodic solutions. In addition, one also can observe that all the results in [25 –27] and references therein cannot be applicable to system (1) to obtain the existence and exponential stability of periodic solutions. This implies that the results of this paper are essentially new.

5 An illustrate example

In this section, we present numerical examples to illustrate the effectiveness of the obtained results. Consider the following fuzzy cellular neural network with distributed delays and variable coefficients ${\begin{matrix} {\dot{x}}_{1} (t) = - c_{1} x_{1} (t) + \sum_{j = 1}^{2} a_{1 j} (t) f_{j} (x_{j} (t)) \\ + \sum_{j = 1}^{2} b_{1 j} (t) u_{j} (t) + I_{1} (t) \\ + ⋀_{j = 1}^{2} α_{1 j} (t) g_{j} (\int_{0}^{\infty} k_{1 j} (s) x_{j} (t - s) ds) \\ + ⋁_{j = 1}^{2} β_{1 j} (t) g_{j} (\int_{0}^{\infty} k_{1 j} (s) x_{j} (t - s) ds) \\ + ⋀_{j = 1}^{2} T_{1 j} (t) u_{j} (t) + ⋀_{j = 1}^{2} H_{1 j} (t) u_{j} (t), \\ {\dot{x}}_{2} (t) = - c_{2} x_{2} (t) + \sum_{j = 1}^{2} a_{2 j} (t) f_{j} (x_{j} (t)) \\ + \sum_{j = 1}^{2} b_{2 j} (t) u_{j} (t) + I_{2} (t) \\ + ⋀_{j = 1}^{2} α_{2 j} (t) g_{j} (\int_{0}^{\infty} k_{2 j} (s) x_{j} (t - s) ds) \\ + ⋁_{j = 1}^{2} β_{2 j} (t) g_{j} (\int_{0}^{\infty} k_{2 j} (s) x_{j} (t - s) ds) \\ + ⋀_{j = 1}^{2} T_{2 j} (t) u_{j} (t) + ⋀_{j = 1}^{2} H_{2 j} (t) u_{j} (t), \end{matrix}$ (39) where $\begin{matrix} [\begin{matrix} a_{11} (t) & a_{12} (t) \\ a_{21} (t) & a_{22} (t) \end{matrix}] = [\begin{matrix} \frac{1}{3} cos t & \frac{1}{4} sin t \\ \frac{1}{4} sin t & \frac{1}{4} cos t \end{matrix}], \\ [\begin{matrix} b_{11} (t) & b_{12} (t) \\ b_{21} (t) & b_{22} (t) \end{matrix}] = [\begin{matrix} \frac{1}{5} sin t & \frac{1}{5} cos t \\ \frac{1}{8} cos t & \frac{1}{6} sin t \end{matrix}], \\ [\begin{matrix} α_{11} (t) & α_{12} (t) \\ α_{21} (t) & α_{22} (t) \end{matrix}] = [\begin{matrix} \frac{1}{4} sin t & \frac{1}{3} cos t \\ \frac{1}{9} sin t & \frac{1}{10} cos t \end{matrix}], \\ [\begin{matrix} β_{11} (t) & β_{12} (t) \\ β_{21} (t) & β_{22} (t) \end{matrix}] = [\begin{matrix} \frac{1}{6} cos t & \frac{1}{8} sin t \\ \frac{1}{4} cos t & \frac{1}{4} sin t \end{matrix}], \\ [\begin{matrix} T_{11} (t) & T_{12} (t) \\ T_{21} (t) & T_{22} (t) \end{matrix}] = [\begin{matrix} \frac{1}{4} sin t & \frac{1}{5} cos t \\ \frac{1}{6} cos t & \frac{1}{6} sin t \end{matrix}], \\ [\begin{matrix} I_{1} (t) & I_{2} (t) \\ c_{1} & c_{2} \end{matrix}] = [\begin{matrix} \frac{1}{2} cos t & \frac{1}{3} cos t \\ 2 & 2 \end{matrix}], \\ [\begin{matrix} k_{11} (t) & k_{12} (t) \\ k_{21} (t) & k_{22} (t) \end{matrix}] = [\begin{matrix} \frac{2}{π (1 + s^{2})} & \frac{2}{π (1 + s^{2})} \\ \frac{2}{π (1 + s^{2})} & \frac{2}{π (1 + s^{2})} \end{matrix}] . \end{matrix}$ Choose $f_{j} (t) = g_{j} (x) = \frac{1}{2} (| x + 1 | - | x - 1 |) (j = 1, 2), ϱ_{i} = 1 (i = 1, 2) .$ Then $L_{j}^{f} = L_{j}^{g} = 1$ . By direct computation, we have $\begin{matrix} [\begin{matrix} a_{11}^{+} & a_{12}^{+} \\ a_{21}^{+} & a_{22}^{+} \end{matrix}] = [\begin{matrix} \frac{1}{3} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} \end{matrix}], [\begin{matrix} b_{11}^{+} & b_{12}^{+} \\ b_{21}^{+} & b_{22}^{+} \end{matrix}] = [\begin{matrix} \frac{1}{5} & \frac{1}{5} \\ \frac{1}{8} & \frac{1}{6} \end{matrix}], \\ [\begin{matrix} α_{11}^{+} & α_{12}^{+} \\ α_{21}^{+} & α_{22}^{+} \end{matrix}] = [\begin{matrix} \frac{1}{4} & \frac{1}{3} \\ \frac{1}{9} & \frac{1}{10} \end{matrix}], [\begin{matrix} β_{11}^{+} & β_{12}^{+} \\ β_{21}^{+} & β_{22}^{+} \end{matrix}] = [\begin{matrix} \frac{1}{6} & \frac{1}{8} \\ \frac{1}{4} & \frac{1}{4} \end{matrix}] . \end{matrix}$ Then $\begin{matrix} D & = & (d_{ij})_{2 \times 2} = {(\frac{1}{c_{i}} [a_{ij}^{+} L_{j}^{f} + α_{ij}^{+} L_{j}^{g} + β_{ij}^{+} L_{j}^{g}])}_{2 \times 2} \\ = & [\begin{matrix} \frac{3}{8} & \frac{23}{48} \\ \frac{11}{36} & \frac{3}{10} \end{matrix}] . \end{matrix}$ Hence $E_{2} - D = [\begin{matrix} \frac{5}{8} & - \frac{23}{48} \\ - \frac{11}{36} & \frac{7}{10} \end{matrix}],$ which implies that E₂ - D is an M-matrix. Moreover, $\begin{matrix} - c_{1} + \frac{1}{2} \sum_{j = 1}^{2} a_{1 j}^{+} L_{j} + \frac{1}{2} \sum_{j = 1}^{2} \frac{ϱ_{j}}{ϱ_{1}} a_{j 1}^{+} L_{1} \\ + \frac{1}{2} \sum_{j = 1}^{2} (α_{1 j}^{+} + β_{1 j}^{+}) L_{j}^{g} + \frac{1}{2} \sum_{j = 1}^{2} \frac{ϱ_{j}}{ϱ_{1}} (α_{j 1}^{+} + β_{j 1}^{+}) L_{1}^{g} \\ = - \frac{7}{8} < 0 \\ - c_{2} + \frac{1}{2} \sum_{j = 1}^{2} a_{2 j}^{+} L_{j} + \frac{1}{2} \sum_{j = 1}^{2} \frac{ϱ_{j}}{ϱ_{2}} a_{j 2}^{+} L_{2} \\ + \frac{1}{2} \sum_{j = 1}^{2} (α_{2 j}^{+} + β_{2 j}^{+}) L_{j}^{g} + \frac{1}{2} \sum_{j = 1}^{2} \frac{ϱ_{j}}{ϱ_{2}} (α_{j 2}^{+} + β_{j 2}^{+}) L_{2}^{g} \\ = - \frac{251}{180} < 0 . \end{matrix}$ Thus all the assumptions in Theorems 3.1 and 4.1 are fulfilled. Thus we can conclude that system (39) has one 2π-periodic solution, which is globally exponentially stable. The results are illustrated inFig. 1.

6 Conclusions

As is known to us, the global exponential stability of periodic solution to fuzzy cellular neural networks plays an important role in describing the behavior of nonlinear differential equations. Thus it has been extensively studied by many scholars in past few decades. In this paper, making use of the continuation theorem of coincidence degree theory, the Lyapunov function methods and the elementary inequality 2xy ≤ x² + y², we investigate the the existence and global exponential stability of periodic solution for fuzzy cellular neural networks with distributed delays and variable coefficients. Some sufficient conditions on the existence and global exponential stability of periodic solutions of the fuzzy cellular neural networks with distributed delays and variable coefficients have been established. Our results show that under some suitable conditions, fuzzy cellular neural networks will keep globally exponentially stable periodic oscillation, which is helpful in design and applications of neural networks. An example and its computer simulations are presented to illustrate the effectiveness of our theoretical findings. In addition, the method of this paper can be applied to some other neural network systems such as fuzzy Hopfield neural networks with distributed delays and variable coefficients and fuzzy BAM neural networks with distributed delays and variable coefficients and so on.

Footnotes

Acknowledgments

This work is supported by National Natural Science Foundation of China (Nos. 61673008, 11261010) and Natural Science and Technology Foundation of Guizhou Province (J[2015]2025).

References

Yang

and Yang

, The global stability of fuzzy cellular neural networks, IEEE Transcations on Circuits and Systems 43(10) (1996), 880–883.

Yang

, Yang

, Wu

C.W.

and Chua

L.O.

, Fuzzy cellular neural networks: Theory, in Proceedings of IEEE Intertional Worksshop on Cellular Neural Network and Applications, 1996, pp. 181–186.

Yang

, Yang

, Wu

C.W.

and Chua

L.O.

, Fuzzy cellular neural networks: Applications, in Proceedings of IEEE Intertional Worksshop on Cellular Neural Network and Applications, 1996, 225–230.

Balasubramaniam

, Kalpana

and Rakkiyappan

, Global asymptotic stability of BAM fuzzy cellular neural networks with time delay in the leakage term, discrete and unbounded distributed delays, Mathematical and Computer Modelling 53(5-6) (2011), 839–853.

Duan

and Huang

L.H.

, Global exponential stability of fuzzy BAM neural networks with distributed delays and time -varying delays in the leakage terms, Neural Computing and Applications 23(1) (2013), 171–178.

Syed Ali

and Balasubramaniam

, Global asymptotic stability of stochastic fuzzy cellular neural networks with multiple discrete and distributed time-varying delays, Communications in Nonlinear Science and Numerical Simulation 16(7) (2011), 2907–2916.

Long

S.J.

and Xu

D.Y.

, Global exponential p-stability of stochastic non-autonomous Takagi-Sugeno fuzzy cellular neural networks with time-varying delays and impulses, Fuzzy Sets and Systems 253 (2014), 82–100.

Song

Q.K.

and Wang

Z.D.

, Dynamical behaviors of fuzzy reactiondiffusion periodic cellular neural networks with variable coefficients and delays, Applied Mathematical Modelling 33(9) (2009), 3533–3545.

Huang

T.W.

, Exponential stability of fuzzy cellular neural networks with distributed delay, Physics Letters A 351(1-2) (2006), 48–52.

10.

Yang

G.W.

, Kao

Y.G.

, Li and W., Sun

X.Q.

, Exponential stability of impulsive stochastic fuzzy cellular neural networks with mixed delays and reaction-diffusion terms, Neural Computing and Applications 23(3-4) (2013), 1109–1121.

11.

Yuan

, Cao

J.D.

and Deng

J.M.

, Exponential stability and periodic solutions of fuzzy cellular neural networks with timevarying delays, Neurocomputing 69(13-15) (2006), 1619–1627.

12.

Liu

Y.Q.

and Tang

W.S.

, Exponential stability of fuzzy cellular neural networks with constant and time-varying delays, Physics Letters A 323(3-4) (2004), 224–233.

13.

Song

Q.K.

and Cao

J.D.

, Dynamical behaviors of a discrete-time fuzzy cellular neural networks with variable delays and impulses, Journal of the Franklin Institute 345(1) (2008), 39–59.

14.

Liu

Z.W.

, Zhang

H.G.

and Wang

Z.S.

, Novel stability criterions of a new fuzzy cellular neural networks with time-varying delays, Neurocomputing 72(4-6) (2009), 1056–1064.

15.

Tan

, Global asymptotic stability of fuzzy cellular neural networks with unbounded distributed delays, Neural Processing Letters 31(2) (2010), 147–157.

16.

, Impulsive effect on global exponential stability of BAM fuzzy cellular neural networkswith time-varying delays, International Journal of Systems Science 41(2) (2010), 131–142.

17.

and Jiang

H.J.

, Global exponential synchronization of fuzzy cellular neural networks with delays and reaction-diffusion terms, Neurocomputing 74(4) (2011), 509–515.

18.

Zhang

Q.H.

and Xiang

R.G.

, Global asymptotic stability of fuzzy cellular neural networks with time-varying delays, Physics Letters A 372(22) (2008), 3971–3977.

19.

Chen

L.P.

, Wu

and Pan

D.H.

, Mean square exponential stability of impulsive stochastic fuzzy cellular neural networks with distributed delays, Expert Systems with Applications 38(5) (2011), 6294–6299.

20.

Rakkiyappan

and Balasubramaniam

, On exponential stability results for fuzzy impulsive neural networks, Fuzzy Sets and Systems 161(13) (2010), 1823–1835.

21.

and Wang

, Existence and global exponential stability of equilibrium for discrete-time fuzzy BAM neural networks with variable delays and impulses, Fuzzy Sets and Systems 217 (2013), 62–79.

22.

, Rakkiyappan

and Balasubramaniam

, Existence and global stability analysis of equilibrium of fuzzy cellular neural networks with time delay in the leakage term under impulsive perturbations, Journal of the Franklin Institute 348(2) (2011), 135–155.

23.

Balasubramaniam

, Kalpana

and Rakkiyappan

, Stationary oscillation of interval fuzzy cellular neural networks with mixed delays under impulsive perturbations, Neural Computing and Applications 22(7-8) (2013), 1645–1654.

24.

Han

, Liu

and Wang

L.S.

, Global exponential stability of delayed fuzzy cellular neural networks with Markovian jumping parameters, Neural Computing and Applications 21(1) (2012), 67–72.

25.

Cao

J.D.

, Global exponential stability and periodic solutions of delayed cellular neural networks, Journal of Computer and Systems Sciences 60(1) (2000), 38–46.

26.

Cao

J.D.

, New results concerning exponential stability and periodic solutions of delayed cellular neural networks, Physics Letters A 307(2-3) (2003), 136–147.

27.

Cao

J.D.

and Li

, On the exponential stability and periodic solutions of delayed cellular neural networks, Journal of Mathematical Analysis and Applications 252(1) (2000), 50–64.

28.

Yan

and Lv

, Exponential syschronization of fuzzy cellular neural networks with mixed delay and general boundary conditions, Communications in Nonlinear Science and Numerical Simulation 17(2) (2012), 1003–1011.

29.

and Yan

, Dynamical behaviors of reaction-diffusion fuzzy neural networks with hybrid delays and general boundary conditions, Communications in Nonlinear Science and Numerical Simulation 16(2) (2011), 993–1001.

30.

Wang

and Lu

, Globally exponential stability of fuzzy cellular neural networks with delays and reaction-diffusion terms, Chaos Soliton and Fractals 38(3) (2008), 878–885.

31.

Huang

, Exponential stability of delayed fuzzy cellular neural networks with diffusion, Chaos Soliton and Fractals 31(3) (2007), 658–664.

32.

Huang

, Exponential stability of fuzzy cellular neural networks with unbounded distributed delay, Physics Letters A 351(1-2) (2006), 44–52.

33.

Niu

S.Y.

, Jiang

H.J.

and Teng

Z.D.

, Periodic oscillation of FCNNs with distributed delays and variable coefficients, Nonlinear Analysis: Real World Applications 10(3) (2009), 1540–1554.

34.

Gaines

R.E.

and Mawhin

J.L.

, Coincidence Degree and Nonlinear Differential Equations, Springer-verlag, Berlin, 1997.

35.

Liao

X.F.

and Yu

J.B.

, Qualitative analysis of bi-directional associative memory with time delay, International Journal of Circuit Theory and Applications 26(3) (1998), 219–229.

36.

Otadi

and Mosleh

, Simulation and evaluation of intervalvalued fuzzy linear Fredholm integral equations with intervalvalued fuzzy neural network, Neurocomputing 205 (2016), 519–528.

37.

Abdurahman

and Jiang

H.J.

, Teng

Z.D.

, Finite-time synchronization for fuzzy cellular neural networks with time-varying delays, Fuzzy Sets and Systems 297 (2016), 96–111.

38.

Mosleh

and Otadi

, Numerical solutions of nonlinear fuzzy Fredholm integro-differential equations of the second kind, Iranian Journal of Fuzzy Systems 12(2) (2015), 117–127.