Open Access

Global exponential stability and existence of periodic solutions for delayed reaction-diffusion BAM neural networks with Dirichlet boundary conditions

Boundary Value Problems20132013:105

DOI: 10.1186/1687-2770-2013-105

Received: 18 October 2012

Accepted: 12 April 2013

Published: 26 April 2013

Abstract

In this paper, both global exponential stability and periodic solutions are investigated for a class of delayed reaction-diffusion BAM neural networks with Dirichlet boundary conditions. By employing suitable Lyapunov functionals, sufficient conditions of the global exponential stability and the existence of periodic solutions are established for reaction-diffusion BAM neural networks with mixed time delays and Dirichlet boundary conditions. The derived criteria extend and improve previous results in the literature. A numerical example is given to show the effectiveness of the obtained results.

Keywords

neural networks reaction-diffusion mixed time delays global exponential stability Poincaré mapping Lyapunov functional

1 Introduction

Neural networks (NNs) have been extensively studied in the past few years and have found many applications in different areas such as pattern recognition, associative memory, combinatorial optimization, etc. Delayed versions of NNs were also proved to be important for solving certain classes of motion-related optimization problems. Various results concerning the dynamical behavior of NNs with delays have been reported during the last decade (see, e.g., [17]). Recently, the authors in [1] and [2] considered the problem of exponential passivity analysis for uncertain NNs with time-varying delays and passivity-based controller design for Hopfield NNs, respectively.

Since NNs related to bidirectional associative memory (BAM) were proposed by Kosko [8], the BAM NNs have been one of the most interesting research topics and have attracted the attention of researchers. In the design and applications of networks, the stability of the designed BAM NNs is one of the most important issues (see, e.g., [912]). Many important results concerning mainly the existence and stability of equilibrium of BAM NNs have been obtained (see, e.g., [915]).

However, strictly speaking, diffusion effects cannot be avoided in the NNs when electrons are moving in asymmetric electromagnetic fields. So, we must consider that the activations vary in space as well as in time. In [1634], the authors considered the stability of NNs with diffusion terms which were expressed by partial differential equations. In particular, the existence and attractivity of periodic solutions for non-autonomous reaction-diffusion Cohen-Grossberg NNs with discrete time delays were investigated in [20]. The authors derived sufficient conditions on the stability and periodic solutions of delayed reaction-diffusion NNs (RDNNs) with Neumann boundary conditions in [2125]. In these works, due to the divergence theorem employed, a negative integral term with gradient was removed in their deduction. Therefore, the stability criteria acquired by them do not contain diffusion terms; that is to say, the diffusion terms do not have any effect on their deduction and results. Meanwhile, some conditions dependent on the diffusion coefficients were given in [30, 3234] to ensure the global exponential stability and periodicity of RDNNs with Dirichlet boundary conditions based on 2-norm.

To the best of our knowledge, there are few reports about global exponential stability and periodicity of RDNNs with mixed time delays and Dirichlet boundary conditions, which are very important in theories and applications and also are a very challenging problem. In this paper, by employing suitable Lyapunov functionals, we shall apply inequality techniques to establish global exponential stability criteria of the equilibrium and periodic solutions for RDNNs with mixed time delays and Dirichlet boundary conditions. The derived criteria extend and improve previous results in the literature [22, 29].

Throughout this paper, we need the following notations. R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq1_HTML.gif denotes the n-dimensional Euclidean space. We denote
u ( t , x ) u = Ω i = 1 m | u i u i | r d x , φ u ( s , x ) u = sup s 0 [ Ω i = 1 m | φ u i ( s , x ) u i | r d x ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equa_HTML.gif
and
v ( t , x ) v = Ω j = 1 n | v j v j | r d x , φ v ( s , x ) v = sup s 0 [ Ω j = 1 n | φ v j ( s , x ) v j | r d x ] , r 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equb_HTML.gif

Let u i = u i ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq2_HTML.gif, v j = v j ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq3_HTML.gif.

The remainder of this paper is organized as follows. In Section 2, the basic notations, model description and assumptions are introduced. In Sections 3 and 4, criteria are proposed to determine global exponential stability, and periodic solutions are considered for reaction-diffusion recurrent neural networks with mixed time delays, respectively. An illustrative example is given to illustrate the effectiveness of the obtained results in Section 5. We also conclude this paper in Section 6.

2 Model description and preliminaries

In this paper, the RDNNs with mixed time delays are described as follows:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ1_HTML.gif
(1)

The RDNNs model given in (1) can be regarded as RDNNs with two layers, where m is the number of neurons in the first layer and n is the number of neurons in the second layer. x = ( x 1 , x 2 , , x l ) T Ω R l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq4_HTML.gif, Ω is a compact set with smooth boundary Ω and mes Ω > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq5_HTML.gif in the space R l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq6_HTML.gif; u = ( u 1 , u 2 , , u m ) T R m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq7_HTML.gif, v = ( v 1 , v 2 , , v n ) T R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq8_HTML.gif. u i ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq9_HTML.gif and v j ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq10_HTML.gif represent the state of the i th neuron in the first layer and the j th neuron in the second layer at time t and in the space x, respectively. b j i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq11_HTML.gif, b ˜ j i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq12_HTML.gif, b ¯ j i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq13_HTML.gif, d i j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq14_HTML.gif, d ¯ i j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq15_HTML.gif and d ˜ i j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq16_HTML.gif are known constants denoting the synaptic connection strengths between the neurons in the two layers, respectively; f j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq17_HTML.gif, f ˜ j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq18_HTML.gif, f ¯ j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq19_HTML.gif, g i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq20_HTML.gif, g ˜ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq21_HTML.gif and g ¯ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq22_HTML.gif denote the activation functions of the neurons and the signal propagation functions, respectively. I i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq23_HTML.gif and J j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq24_HTML.gif denote the external inputs on the i th neuron and j th neuron, respectively; p i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq25_HTML.gif and q j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq26_HTML.gif are differentiable real functions with positive derivatives defining the neuron charging time; τ i j ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq27_HTML.gif and θ j i ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq28_HTML.gif represent continuous time-varying delay and discrete delay, respectively; D i k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq29_HTML.gif and D j k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq30_HTML.gif, i = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq31_HTML.gif, k = 1 , 2 , , l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq32_HTML.gif and j = 1 , 2 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq33_HTML.gif, stand for the transmission diffusion coefficient along the i th neuron and j th neuron, respectively.

System (1) is supplemented with the following boundary conditions and initial values:
u i ( t , x ) = 0 , v j ( t , x ) = 0 , t 0 , x Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ2_HTML.gif
(2)
u i ( s , x ) = φ u i ( s , x ) , v j ( s , x ) = φ v j ( s , x ) , ( s , x ) ( , 0 ] × Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ3_HTML.gif
(3)
for any i = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq31_HTML.gif and j = 1 , 2 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq33_HTML.gif, where n ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq34_HTML.gif is the outer normal vector of Ω, φ = ( φ u φ v ) = ( φ u 1 , , φ u m , φ v 1 , , φ v n ) T C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq35_HTML.gif are bounded and continuous, where C = { φ | φ = ( φ u φ v ) , φ : ( ( , 0 ] × R m ( , 0 ] × R n ) R m + n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq36_HTML.gif. It is the Banach space of continuous functions which maps ( ( , 0 ] ( , 0 ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq37_HTML.gif into R m + n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq38_HTML.gif with the topology of uniform convergence for the norm
φ = ( φ u φ v ) = sup s 0 [ Ω i = 1 m | φ u i | r d x ] + sup s 0 [ Ω j = 1 n | φ v j | r d x ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equc_HTML.gif

Remark 1 Some famous NN models became a special case of system (1). For example, when D i k = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq39_HTML.gif and D j k = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq40_HTML.gif ( i = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq31_HTML.gif, k = 1 , 2 , , l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq41_HTML.gif), the special case of model (1) is the model which has been studied in [1315]. When b ˜ j i = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq42_HTML.gif and d ˜ i j = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq43_HTML.gif, i = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq31_HTML.gif, j = 1 , 2 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq33_HTML.gif, system (1) became NNs with distributed delays and reaction-diffusion terms [18, 22, 29].

Throughout this paper, we assume that the following conditions are made.

(A1) The functions τ i j ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq27_HTML.gif, θ j i ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq28_HTML.gif are piecewise-continuous of class C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq44_HTML.gif on the closure of each continuity subinterval and satisfy
0 τ i j ( t ) τ i j , 0 θ j i ( t ) θ j i , τ ˙ i j ( t ) μ τ < 1 , θ ˙ j i ( t ) μ θ < 1 , τ = max 1 i m , 1 j n { τ i j } , θ = max 1 i m , 1 j n { θ j i } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equd_HTML.gif

with some constants τ i j 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq45_HTML.gif, θ j i 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq46_HTML.gif, τ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq47_HTML.gif, θ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq48_HTML.gif for all t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq49_HTML.gif.

(A2) The functions p i ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq50_HTML.gif and q j ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq51_HTML.gif are piecewise-continuous of class C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq44_HTML.gif on the closure of each continuity subinterval and satisfy
a i = inf ζ R p i ( ζ ) > 0 , p i ( 0 ) = 0 , c j = inf ζ R q j ( ζ ) > 0 , q j ( 0 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Eque_HTML.gif
(A3) The activation functions and the signal propagation functions are bounded and Lipschitz continuous, i.e., there exist positive constants L j f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq52_HTML.gif, L j f ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq53_HTML.gif, L j f ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq54_HTML.gif, L i g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq55_HTML.gif, L i g ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq56_HTML.gif and L i g ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq57_HTML.gif such that for all η 1 , η 2 R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq58_HTML.gif,
| f j ( η 1 ) f j ( η 2 ) | L j f | η 1 η 2 | , | f ˜ j ( η 1 ) f ˜ j ( η 2 ) | L j f ˜ | η 1 η 2 | , | f ¯ j ( η 1 ) f ¯ j ( η 2 ) | L j f ¯ | η 1 η 2 | , | g i ( η 1 ) g i ( η 2 ) | L i g | η 1 η 2 | , | g ˜ i ( η 1 ) g ˜ i ( η 2 ) | L i g ˜ | η 1 η 2 | , | g ¯ i ( η 1 ) g ¯ i ( η 2 ) | L i g ¯ | η 1 η 2 | . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equf_HTML.gif
(A4) The delay kernels K j i ( s ) , K ¯ i j ( s ) : [ 0 , ) [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq59_HTML.gif ( i = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq31_HTML.gif, j = 1 , 2 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq33_HTML.gif) are real-valued non-negative continuous functions that satisfy the following conditions:
  1. (i)

    0 + K j i ( s ) d s = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq60_HTML.gif, 0 + K ¯ j i ( s ) d s = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq61_HTML.gif;

     
  2. (ii)

    0 + s K j i ( s ) d s < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq62_HTML.gif, 0 + s K ¯ i j ( s ) d s < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq63_HTML.gif;

     
  3. (iii)
    There exist a positive μ such that
    0 + s e μ s K j i ( s ) d s < , 0 + s e μ s K ¯ i j ( s ) d s < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equg_HTML.gif
     

Let ( u , v ) = ( u 1 , u 2 , , u n , v 1 , v 2 , , v n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq64_HTML.gif be the equilibrium point of system (1).

Definition 1 The equilibrium point of system (1) is said to be globally exponentially stable if we can find r 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq65_HTML.gif such that there exist constants α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq66_HTML.gif and β 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq67_HTML.gif such that
u ( t , x ) u + v ( t , x ) v β e 2 α t ( φ u ( s , x ) u + φ v ( s , x ) v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ4_HTML.gif
(4)

for all t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq49_HTML.gif.

Remark 2 It is well known that bounded activation functions always guarantee the existence of an equilibrium point for system (1).

Lemma 1 [33]

Let Ω be a cube | x l | < d l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq68_HTML.gif ( l = 1 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq69_HTML.gif), and let h ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq70_HTML.gif be a real-valued function belonging to C 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq71_HTML.gif which vanishes on the boundary Ω of Ω, i.e., h ( x ) | Ω = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq72_HTML.gif. Then
Ω h 2 ( x ) d x d l 2 Ω | h x l | 2 d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ5_HTML.gif
(5)

3 Global exponential stability

Now we are in a position to investigate the global exponential stability of system (1). By constructing a suitable Lyapunov functional, we arrive at the following conclusion.

Theorem 1 Let (A1)-(A4) be in force. If there exist w i > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq73_HTML.gif ( i = 1 , 2 , , n + m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq74_HTML.gif), r 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq65_HTML.gif, γ i j > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq75_HTML.gif, β j i > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq76_HTML.gif such that
w i ( r n a i r 1 D i l r n a i r + 2 ( r 1 ) j = 1 n a i r + ( r 1 ) j = 1 n a i r β j i r r 1 ) + j = 1 n w m + j m r ( | d i j | r ( L i g ) r + | d ˜ i j | r e τ 1 μ τ ( L i g ˜ ) r + | d ¯ i j | r γ i j r ( L i g ¯ ) r ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equh_HTML.gif
and
w m + j ( r m c j r 1 D j l r m c j r + 2 ( r 1 ) i = 1 m c j r + ( r 1 ) i = 1 m c j r γ i j r r 1 ) + i = 1 m w i n r ( | b j i | r ( L j f ) r + | b ˜ j i | r e θ 1 μ θ ( L j f ˜ ) r + | b ¯ j i | r β j i r ( L j f ¯ ) r ) < 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ6_HTML.gif
(6)

in which i = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq31_HTML.gif, j = 1 , 2 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq77_HTML.gif, L j f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq52_HTML.gif, L j f ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq53_HTML.gif, L j f ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq54_HTML.gif, L i g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq55_HTML.gif, L i g ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq56_HTML.gif and L i g ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq57_HTML.gif are Lipschitz constants, D i = min 1 k l { D i k / d k 2 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq78_HTML.gif, D j = min 1 k l { D j k / d k 2 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq79_HTML.gif, then the equilibrium point ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq80_HTML.gif of system (1) is unique and globally exponentially stable.

Proof If (6) holds, we can always choose a positive number δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq81_HTML.gif (may be very small) such that
w i ( r n a i r 1 D i l r n a i r + 2 ( r 1 ) j = 1 n a i r + ( r 1 ) j = 1 n a i r β j i r r 1 ) + j = 1 n w m + j m r ( | d i j | r ( L i g ) r + | d ˜ i j | r e τ 1 μ τ ( L i g ˜ ) r + | d ¯ i j | r γ i j r ( L i g ¯ i ) r ) + δ < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equi_HTML.gif
and
w m + j ( r m c j r 1 D j l r m c j r + 2 ( r 1 ) i = 1 m c j r + ( r 1 ) i = 1 m c j r γ i j r r 1 ) + i = 1 m w i n r ( | b j i | r ( L j f ) r + | b ˜ j i | r e θ 1 μ θ ( L j f ˜ ) r + | b ¯ j i | r β j i r ( L j f ¯ ) r ) + δ < 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ7_HTML.gif
(7)

where i = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq31_HTML.gif, j = 1 , 2 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq33_HTML.gif.

Let us consider the functions
F i ( x i ) = w i ( r n a i r 1 D i l r n a i r + 2 ( r 1 ) j = 1 n a i r + ( r 1 ) j = 1 n a i r β j i r r 1 0 + k j i ( s ) d s + 2 x i n a i r 1 ) + j = 1 n w m + j m r ( | d i j | r ( L i g ) r + | d ˜ i j | r e τ 1 μ τ ( L i g ˜ ) r + | d ¯ i j | r γ i j r ( L i g ¯ i ) r 0 + e 2 x i s k ¯ i j ( s ) d s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equj_HTML.gif
and
G j ( y j ) = w m + j [ r m c j r 1 D j l r m c j r + 2 ( r 1 ) i = 1 m c j r + ( r 1 ) i = 1 m c j r γ i j r r 1 0 + k ¯ i j ( s ) d s + 2 y j m c j r 1 ] + i = 1 m w i n r ( | b j i | r ( L j f ) r + | b ˜ j i | r e θ 1 μ θ ( L j f ˜ ) r + | b ¯ j i | r β j i r ( L j f ¯ ) r 0 + e 2 y j s k j i ( s ) d s ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ8_HTML.gif
(8)

where x i , y j [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq82_HTML.gif, i = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq31_HTML.gif, j = 1 , 2 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq33_HTML.gif.

From (8) and (A4), we derive F i ( 0 ) < δ < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq83_HTML.gif, G j ( 0 ) < δ < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq84_HTML.gif; F i ( x i ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq85_HTML.gif and G j ( y j ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq86_HTML.gif are continuous for x i , y j [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq87_HTML.gif. Moreover, F i ( x i ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq88_HTML.gif as x i + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq89_HTML.gif and G j ( y j ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq90_HTML.gif as y j + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq91_HTML.gif. Thus there exist constants ε i , σ j [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq92_HTML.gif such that
F i ( ε i ) = w i ( r n a i r 1 D i l r n a i r + 2 ( r 1 ) j = 1 n a i r F i ( ε i ) = + ( r 1 ) j = 1 n a i r β j i r r 1 0 + k j i ( s ) d s + 2 ε i n a i r 1 ) + j = 1 n w m + j m r ( | d i j | r ( L i g ) r F i ( ε i ) = + | d ˜ i j | r e τ 1 μ τ ( L i g ˜ ) r + | d ¯ i j | r γ i j r ( L i g ¯ i ) r 0 + e 2 ε i s k ¯ i j ( s ) d s ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equk_HTML.gif
and
G j ( σ j ) = w m + j ( r m c j r 1 D j l r m c j r + 2 ( r 1 ) i = 1 m c j r + ( r 1 ) i = 1 m c j r γ i j r r 1 0 + k ¯ i j ( s ) d s + 2 σ j m c j r 1 ) + i = 1 m w i n r ( | b j i | r ( L j f ) r + | b ˜ j i | r e θ 1 μ θ ( L j f ˜ ) r + | b ¯ j i | r β j i r ( L j f ¯ ) r 0 + e 2 σ j s k j i ( s ) d s ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ9_HTML.gif
(9)

where i = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq31_HTML.gif, j = 1 , 2 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq33_HTML.gif.

By using α = min 1 i m , 1 j n { ε i , σ j } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq93_HTML.gif, obviously, we get
F i ( α ) = w i ( r n a i r 1 D i l r n a i r + 2 ( r 1 ) j = 1 n a i r F i ( α ) = + ( r 1 ) j = 1 n a i r β j i r r 1 0 + k j i ( s ) d s + 2 α n a i r 1 ) + j = 1 n w m + j m r ( | d i j | r ( L i g ) r F i ( α ) = + | d ˜ i j | r e τ 1 μ τ ( L i g ˜ ) r + | d ¯ i j | r γ i j r ( L i g ¯ i ) r 0 + e 2 α s k ¯ i j ( s ) d s ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equl_HTML.gif
and
G j ( α ) = w m + j ( r m c j r 1 D j l r m c j r + 2 ( r 1 ) i = 1 m c j r + ( r 1 ) i = 1 m c j r γ i j r r 1 0 + k ¯ i j ( s ) d s + 2 α m c j r 1 ) + i = 1 m w i n r ( | b j i | r ( L j f ) r + | b ˜ j i | r e θ 1 μ θ ( L j f ˜ ) r + | b ¯ j i | r β j i r ( L j f ¯ ) r 0 + e 2 α s k j i ( s ) d s ) 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ10_HTML.gif
(10)

where i = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq31_HTML.gif, j = 1 , 2 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq33_HTML.gif.

Suppose ( u , v ) = ( u 1 , u 2 , , u n , v 1 , v 2 , , v n ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq94_HTML.gif is any solution of model (1). Rewrite model (1) as
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ11_HTML.gif
(11)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ12_HTML.gif
(12)
Multiplying (11) by u i u i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq95_HTML.gif and integrating over Ω yield
1 2 d d t Ω ( u i u i ) 2 d x = Ω k = 1 l ( u i u i ) x k ( D i k ( u i u i ) x k ) d x p i ( ξ i ) Ω ( u i u i ) 2 d x + Ω j = 1 n ( b j i ( u i u i ) ( f j ( v j ) f j ( v j ) ) ) d x + j = 1 n Ω ( b ˜ j i ( u i u i ) ( f ˜ j ( v j ( t θ j i ( t ) , x ) ) f ˜ j ( v j ) ) ) d x + j = 1 n Ω [ ( b ¯ j i ( u i u i ) t k j i ( t s ) ( f ¯ j ( v j ( s , x ) ) f ¯ j ( v j ) ) ) d s ] d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ13_HTML.gif
(13)
According to Green’s formula and the Dirichlet boundary condition, we get
Ω k = 1 l ( u i u i ) x k ( D i k ( u i u i ) x k ) d x = k = 1 l Ω D i k ( ( u i u i ) x k ) 2 d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ14_HTML.gif
(14)
Moreover from Lemma 1, we have
k = 1 l Ω D i k ( ( u i u i ) x k ) 2 d x Ω k = 1 l D i k d k 2 ( u i u i ) 2 d x D i l u i u i 2 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ15_HTML.gif
(15)
From (11)-(15), (A2), (A3) and the Holder integral inequality, we obtain that
d d t Ω | u i u i | 2 d x 2 D i l Ω | u i u i | 2 d x 2 a i Ω | u i u i | 2 d x + 2 Ω j = 1 n ( | b j i | | u i u i | L j f | v j v j | ) d x + 2 j = 1 n Ω ( | b ˜ j i | | u i u i | | f ˜ ( v j ( t θ j i ( t ) , x ) ) f ˜ ( v j ) | ) d x + 2 j = 1 n Ω [ ( | b ¯ j i | t k j i ( t s ) | u i u i | | f ¯ j ( v j ( s , x ) ) f ¯ j ( v j ) | ) d s ] d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ16_HTML.gif
(16)
Multiplying both sides of (12) by v j v j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq96_HTML.gif, similarly, we also have
d d t Ω | v j v j | 2 d x 2 D j l Ω | v j v j | 2 d x 2 c j Ω | v j v j | 2 d x + 2 Ω i = 1 m ( | d i j | L i g | u i u i | | v j v j | ) d x + 2 i = 1 m Ω ( | d ˜ i j | | g ˜ ( u i ( t τ i j ( t ) , x ) ) g ˜ ( u i ) | | v j v j | ) d x + 2 i = 1 m Ω [ | d ¯ i j | t k ¯ i j ( t s ) | g ¯ i ( u i ( s , x ) ) g ¯ i ( u i ) | | v j v j | d s ] d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ17_HTML.gif
(17)
Choose a Lyapunov functional as follows:
V ( t ) = Ω i = 1 m w i [ n a i r 1 | u i u i | r e 2 α t + j = 1 n | b ˜ j i | r n r e θ 1 μ θ t θ j i ( t ) t e 2 α ξ | f ˜ j ( v j ( ξ , x ) ) f ˜ j ( v j ) | r d ξ + j = 1 n | b ¯ j i | r n r β j i r 0 + k j i ( s ) t s t e 2 α ( s + ξ ) | f ¯ j ( v j ( ξ , x ) ) f ¯ j ( v j ) | r d ξ d s ] d x + Ω j = 1 n w m + j [ m c j r 1 | v j v j | r e 2 α t + i = 1 m | d ˜ i j | r m r e τ 1 μ τ t τ i j ( t ) t e 2 α ξ | g ˜ i ( u i ( ξ , x ) ) g ˜ i ( u i ) | r d ξ + i = 1 m | d ¯ i j | r m r γ i j r 0 + k ¯ i j ( s ) t s t e 2 α ( s + ξ ) | g ¯ i ( u i ( ξ , x ) ) g ¯ i ( u i ) | r d ξ d s ] d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equm_HTML.gif
Its upper Dini-derivative along the solution to system (1) can be calculated as follows:
D + V ( t ) Ω i = 1 m w i [ r n a i r 1 | u i u i | r 1 | u i u i | t e 2 α t + 2 α e 2 α t n a i r 1 | u i u i | r + e 2 α t j = 1 n | b ˜ j i | r n r e θ 1 μ θ | f ˜ j ( v j ( t , x ) ) f ˜ j ( v j ) | r j = 1 n | b ˜ j i | r n r e θ 1 μ θ ( 1 θ ˙ j i ( t ) ) e 2 α ( t θ j i ( t ) ) | f ˜ j ( v j ( t θ j i ( t ) , x ) ) f ˜ j ( v j ) | r + e 2 α t j = 1 n | b ¯ j i | r n r β j i r 0 + e 2 α s k j i ( s ) | f ¯ j ( v j ( t , x ) ) f ¯ j ( v j ) | r d s e 2 α t j = 1 n | b ¯ j i | r n r β j i r 0 + k j i ( s ) | f ¯ j ( v j ( t s , x ) ) f ¯ j ( v j ) | r d s ] d x + Ω j = 1 n w m + j [ r m c j r 1 | v j v j | r 1 | v j v j | t e 2 α t + 2 α e 2 α t m c j r 1 | v j v j | r + e 2 α t i = 1 m | d ˜ i j | r m r e τ 1 μ τ | g ˜ i ( u i ( t , x ) ) g ˜ i ( u i ) | r i = 1 m | d ˜ i j | r m r e τ 1 μ τ e 2 α ( t τ i j ( t ) ) ( 1 τ ˙ i j ) | g ˜ i ( u i ( t τ i j ( t ) , x ) ) g ˜ i ( u i ) | r + e 2 α t i = 1 m | d ¯ i j | r m r γ i j r 0 + e 2 α s k ¯ i j ( s ) | g ¯ i ( u i ( t , x ) ) g ¯ i ( u i ) | r d s e 2 α t i = 1 m | d ¯ i j | r m r γ i j r 0 + k ¯ i j ( s ) | g ¯ i ( u i ( t s , x ) ) g ¯ i ( u i ) | r d s ] d x Ω i = 1 m w i [ r n a i r 1 | u i u i | r 2 e 2 α t ( D i l | u i u i | 2 a i | u i u i | 2 + j = 1 n ( | b j i | | u i u i | L j f | v j v j | ) + j = 1 n ( | b ˜ j i | | u i u i | | f ˜ ( v j ( t θ j i ( t ) , x ) ) f ˜ ( v j ) | ) + j = 1 n ( | b ¯ j i | t k j i ( t s ) | u i u i | | f ¯ j ( v j ( s , x ) ) f ¯ j ( v j ) | ) d s ) + 2 α e 2 α t n a i r 1 | u i u i | r + e 2 α t j = 1 n | b ˜ j i | r n r e θ 1 μ θ | f ˜ j ( v j ( t , x ) ) f ˜ j ( v j ) | r e 2 α t j = 1 n | b ˜ j i | r n r | f ˜ j ( v j ( t θ j i , x ) ) f ˜ j ( v j ) | r + e 2 α t j = 1 n | b ¯ j i | r n r β j i r 0 + e 2 α s k j i ( s ) | f ¯ j ( v j ( t , x ) ) f ¯ j ( v j ) | r d s e 2 α t j = 1 n | b ¯ j i | r n r β j i r 0 + k j i ( s ) | f ¯ j ( v j ( t s , x ) ) f ¯ j ( v j ) | r d s ] d x + Ω j = 1 n w m + j [ r m c j r 1 | v j v j | r 2 e 2 α t ( D j l | v j v j | 2 c j | v j v j | 2 + i = 1 m ( | d i j | L i g | u i u i | | v j v j | ) + i = 1 m ( | d ˜ i j | | g ˜ ( u i ( t τ i j , x ) ) g ˜ ( u i ) | | v j v j | ) + i = 1 m ( | d ¯ i j | t k ¯ i j ( t s ) | g ¯ i ( u i ( s , x ) ) g ¯ i ( u i ) | | v j v j | ) d s ) + 2 α e 2 α t m c j r 1 | v j v j | r + e 2 α t i = 1 m | d ˜ i j | r m r e τ 1 μ τ | g ˜ i ( u i ( t , x ) ) g ˜ i ( u i ) | r e 2 α t i = 1 m | d ˜ i j | r m r | g ˜ i ( u i ( t τ i j , x ) ) g ˜ i ( u i ) | r + e 2 α t i = 1 m | d ¯ i j | r m r γ i j r 0 + e 2 α s k ¯ j i ( s ) | g ¯ i ( u i ( t , x ) ) g ¯ i ( u i ) | r d s e 2 α t i = 1 m | d ¯ i j | r m r γ i j r 0 + k ¯ i j ( s ) | g ¯ i ( u i ( t s , x ) ) g ¯ i ( u i ) | r d s ] d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ18_HTML.gif
(18)
From (18) and the Young inequality, we can conclude
D + V ( t ) Ω e 2 α t i = 1 m w i [ ( r n a i r 1 D i l | u i u i | r r n a i r | u i u i | r + ( r 1 ) j = 1 n a i r | u i u i | r + j = 1 n ( n r | b j i | r ( L j f ) r | v j v j | r ) + ( r 1 ) j = 1 n a i r | u i u i | r + j = 1 n | b ˜ j i | r n r | f ˜ j ( v j ( t θ j i ( t ) , x ) ) f ˜ j ( v j ) | r + ( r 1 ) j = 1 n ( a i r β j i r r 1 t k j i ( t s ) | u i u i | r d s ) + j = 1 n ( | b ¯ j i | r n r β j i r t k j i ( t s ) | f ¯ j ( v j ( s , x ) ) f ¯ j ( v j ) | r d s ) ) + 2 α n a i r 1 | u i u i | r + j = 1 n | b ˜ j i | r n r e θ 1 μ θ | f ˜ j ( v j ( t , x ) ) f ˜ j ( v j ) | r j = 1 n | b ˜ j i | r n r | f ˜ j ( v j ( t θ j i ( t ) , x ) ) f ˜ j ( v j ) | r + j = 1 n | b ¯ j i | r n r β j i r 0 + e 2 α s k j i ( s ) | f ¯ j ( v j ( t , x ) ) f ¯ j ( v j ) | r d s j = 1 n | b ¯ j i | r n r β j i r 0 + k j i ( s ) | f ¯ j ( v j ( t s , x ) ) f ¯ j ( v j ) | r d s ] d x + Ω e 2 α t j = 1 n w m + j [ ( r m c j r 1 D j l | v j v j | r r m c j r | v j v j | r + ( r 1 ) i = 1 m c j r | v j v j | r + i = 1 m ( | d i j | r m r ( L i g ) r | u i u i | r ) + ( r 1 ) i = 1 m ( c j r | v j v j | r ) + i = 1 m | d ˜ i j | r m r | g ˜ i ( u i ( t τ i j ( t ) , x ) ) g ˜ i ( u i ) | r + ( r 1 ) i = 1 m ( c j r γ i j r r 1 t k ¯ i j ( t s ) | v j v j | r d s ) + i = 1 m ( | d ¯ i j | r m r γ i j r t k ¯ i j ( t s ) | g ¯ i ( u i ( s , x ) ) g ¯ i ( u i ) | r d s ) ) + 2 α m c j r 1 | v j v j | r + i = 1 m | d ˜ i j | r m r e τ 1 μ τ | g ˜ i ( u i ( t , x ) ) g ˜ i ( u i ) | r i = 1 m | d ˜ i j | r m r | g ˜ i ( u i ( t τ i j ( t ) , x ) ) g ˜ i ( u i ) | r + i = 1 m | d ¯ i j | r m r γ i j r 0 + e 2 α s k ¯ i j ( s ) | g ¯ i ( u i ( t , x ) ) g ¯ i ( u i ) | r d s i = 1 m | d ¯ i j | r m r γ i j r 0 + k ¯ i j ( s ) | g ¯ i ( u i ( t s , x ) ) g ¯ i ( u i ) | r d s ] d x Ω e 2 α t i = 1 m [ w i ( r n a i r 1 D i l r n a i r + 2 ( r 1 ) j = 1 n a i r + 2 α n a i r 1 + ( r 1 ) j = 1 n ( a i r β j i r r 1 t k j i ( t s ) d s ) ) + j = 1 n w m + j m r ( | d i j | r ( L i g ) r + | d ˜ i j | r e τ 1 μ τ ( L i g ˜ ) r + | d ¯ i j | r γ i j r 0 + e 2 α s k ¯ i j ( s ) ( L i g ¯ i ) r d s ) ] | u i u i | r d x + Ω e 2 α t j = 1 n [ w m + j ( r m c j r 1 D j l r m c j r + 2 ( r 1 ) i = 1 m c j r + ( r 1 ) i = 1 m ( c j r γ i j r r 1 t k ¯ i j ( t s ) d s ) + 2 α m c j r 1 ) + i = 1 m w i n r ( | b j i | r ( L j f ) r + | b ˜ j i | r e θ 1 μ θ ( L j f ˜ ) r + | b ¯ j i | r β j i r ( L j f ¯ ) r 0 + e 2 α s k j i ( s ) d s ) ] | v j v j | r d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ19_HTML.gif
(19)
From (6), we can conclude
D + V ( t ) 0 and so V ( t ) V ( 0 ) , t 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ20_HTML.gif
(20)
Since
V ( 0 ) = Ω i = 1 m w i [ n a i r 1 | u i ( 0 , x ) u i | r + j = 1 n | b ˜ j i | r n r e θ 1 μ θ θ j i ( t ) 0 | f ˜ j ( v j ( ξ , x ) ) f ˜ j ( v j ) | r d ξ + j = 1 n | b ¯ j i | r n r β j i r 0 + k j i ( s ) s 0 e 2 α ( s + ξ ) | f ¯ j ( v j ( ξ , x ) ) f ¯ j ( v j ) | r d ξ d s ] d x + Ω j = 1 n w m + j [ m c j r 1 | v j ( 0 , x ) v j | r + i = 1 m | d ˜ i j | r m r e τ 1 μ τ τ i j ( t ) 0 | g ˜ i ( u i ( ξ , x ) ) g ˜ i ( u i ) | r d ξ + i = 1 m | d ¯ i j | r m r γ i j r 0 + k ¯ i j ( s ) s 0 e 2 α ( s + ξ ) | g ¯ i ( u i ( ξ , x ) ) g ¯ i ( u i ) | r d ξ d s ] d x Ω i = 1 m max 1 i m { w i } [ n a i r 1 | u i ( 0 , x ) u i | r + j = 1 n | b ˜ j i | r n r e θ 1 μ θ ( L j f ˜ ) r θ j i 0 | v j ( ξ , x ) v j | r d ξ + j = 1 n | b ¯ j i | r n r ( L j f ¯ ) r β j i r 0 + k j i ( s ) s 0 e 2 α ( s + ξ ) | v j ( ξ , x ) v j | r d ξ d s ] d x + Ω j = 1 n max 1 j n { w m + j } [ m c j r 1 | v j ( 0 , x ) v j | r + i = 1 m | d ˜ i j | r ( L i g ˜ ) r m r e τ 1 μ τ τ i j 0 | u i ( ξ , x ) u i | r d ξ + j = 1 n | d ¯ i j | r m r ( L i g ¯ ) r γ i j r 0 + k ¯ j i ( s ) s 0 e 2 α ( s + ξ ) | u i ( ξ , x ) u i | r d ξ d s ] d x { max 1 i m { w i } + max 1 j n { w m + j } max 1 j n [ i = 1 m | d ¯ i j | r m r ( L i g ¯ ) r γ i j r 0 + k ¯ i j ( s ) s e 2 α s d s ] + max 1 j n { w m + j } max 1 j n [ i = 1 m | d ˜ i j | r ( L i g ˜ ) r m r e τ τ 1 μ τ ] } φ u ( s , x ) u r + { max 1 j n { w m + j } + max 1 i m { w i } max 1 i m [ j = 1 n | b ¯ j i | r n r ( L j f ¯ ) r β j i r 0 + s e 2 α s k j i ( s ) d s ] + max 1 i m { w i } max 1 i m [ j = 1 n | b ˜ j i | r n r ( L j g ˜ ) r e θ θ 1 μ θ ] } φ v ( s , x ) v r . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ21_HTML.gif
(21)
Noting that
e 2 α t ( min 1 i m + n w i ) ( u ( t , x ) u + v ( t , x ) v ) V ( t ) , t 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ22_HTML.gif
(22)
Let
β = max { max 1 i m { w i } + max 1 j n { w m + j } max 1 j n [ i = 1 m | d ¯ i j | r m r ( L i g ¯ ) r γ i j r 0 + k ¯ i j ( s ) s e 2 α s d s ] β = + max 1 j n { w m + j } max 1 j n [ i = 1 m | d ˜ i j | r ( L i g ˜ ) r m r e τ τ 1 μ τ ] , max 1 j n { w m + j } + max 1 i m { w i } max 1 i m [ j = 1 n | b ¯ j i | r n r ( L j f ¯ ) r β j i r 0 + s e 2 α s k j i ( s ) d s ] + max 1 i m { w i } max 1 i m [ j = 1 n | b ˜ j i | r n r ( L j g ˜ ) r e θ θ 1 μ θ ] } / min 1 i m + n { w i } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equn_HTML.gif

Clearly, β 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq67_HTML.gif.

It follows that
u ( t , x ) u + v ( t , x ) v β e 2 α t ( φ u ( s , x ) u + φ v ( s , x ) v ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equo_HTML.gif

for any t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq49_HTML.gif, where β 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq67_HTML.gif is a constant. This implies that the solution of (1) is globally exponentially stable. This completes the proof of Theorem 1. □

Remark 3 In this paper, the derived sufficient condition includes diffusion terms. Unfortunately, in the proof in the previous papers [2124], a negative integral term with gradient is left out in their deduction. This leads to the fact that those criteria are irrelevant to the diffusion term. Obviously, Lyapunov functional to construct is more general and our results expand the model in [22, 29].

When b ˜ j i = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq42_HTML.gif and d ˜ i j = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq43_HTML.gif ( i = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq31_HTML.gif, j = 1 , 2 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq97_HTML.gif), system (1) becomes the following BAM NNs with distributed delays and reaction-diffusion terms:
u i t = k = 1 l x k ( D i k u i x k ) p i ( u i ( t , x ) ) + j = 1 n ( b j i f j ( v j ( t , x ) ) ) u i t = + j = 1 n b ¯ j i t k j i ( t s ) f ¯ j ( v j ( s , x ) ) d s + I i ( t ) , v j t = k = 1 l x k ( D j k v j x k ) q j ( v j ( t , x ) ) + i = 1 m ( d i j g i ( u i ( t , x ) ) ) v j t = + i = 1 m d ¯ i j t k ¯ i j ( t s ) g ¯ i ( u i ( s , x ) ) d s + J j ( t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ23_HTML.gif
(23)

For (23), we get the following result.

Corollary 1 Let (A1)-(A4) be in force. If there exist w i > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq73_HTML.gif ( i = 1 , 2 , , n + m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq74_HTML.gif), r 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq65_HTML.gif, γ i j > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq98_HTML.gif, β j i > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq76_HTML.gif such that
w i ( r n a i r 1 D i l r n a i r + 2 ( r 1 ) j = 1 n a i r + ( r 1 ) j = 1 n a i r β j i r r 1 ) + j = 1 n w m + j m r ( | d i j | r ( L i g ) r + | d ¯ i j | r γ i j r ( L i g ¯ ) r ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equp_HTML.gif
and
w m + j ( r m c j r 1 D j l r m c j r + 2 ( r 1 ) i = 1 m c j r + ( r 1 ) i = 1 m c j r γ i j r r 1 ) + i = 1 m w i n r ( | b j i | r ( L j f ) r + | b ¯ j i | r β j i r ( L j f ¯ ) r ) < 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ24_HTML.gif
(24)

where i = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq31_HTML.gif, j = 1 , 2 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq99_HTML.gif, L j f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq52_HTML.gif, L j f ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq53_HTML.gif, L j f ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq54_HTML.gif, L i g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq55_HTML.gif, L i g ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq56_HTML.gif and L i g ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq57_HTML.gif are Lipschitz constants. Then the equilibrium point ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq80_HTML.gif of system (1) is unique and globally exponentially stable.

4 Periodic solutions

In this section, we consider the stability criterion for periodic oscillatory solutions of system (1), in which external input I i : R + R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq100_HTML.gif, i = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq31_HTML.gif, and J j : R + R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq101_HTML.gif, j = 1 , 2 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq77_HTML.gif, are continuously periodic functions with period ω, that is,
I i ( t + ω ) = I i ( t ) , J j ( t + ω ) = J j ( t ) , i = 1 , 2 , , m , j = 1 , 2 , , n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equq_HTML.gif

By constructing a Poincaré mapping, the existence of a unique ω-periodic solution and its stability are readily established.

Theorem 2 Let (A1)-(A4) be in force. There exists only one ω-periodic solution of system (1), and all other solutions converge exponentially to it as t + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq102_HTML.gif if there exist constants w i > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq73_HTML.gif ( i = 1 , 2 , , n + m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq103_HTML.gif), r 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq65_HTML.gif, γ i j > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq104_HTML.gif, β j i > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq76_HTML.gif ( i = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq31_HTML.gif, j = 1 , 2 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq97_HTML.gif) such that
w i ( r n a i r 1 D i l r n a i r + 2 ( r 1 ) j = 1 n a i r + ( r 1 ) j = 1 n a i r β j i r r 1 ) + j = 1 n w m + j m r ( | d i j | r ( L i g ) r + | d ˜ i j | r e τ 1 μ τ ( L i g ˜ ) r + | d ¯ i j | r γ i j r ( L i g ¯ ) r ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equr_HTML.gif
and
w m + j ( r m c j r 1 D j l r m c j r + 2 ( r 1 ) i = 1 m c j r + ( r 1 ) i = 1 m c j r γ i j r r 1 ) + i = 1 m w i n r ( | b j i | r ( L j f ) r + | b ˜ j i | r e θ 1 μ θ ( L j f ˜ ) r + | b ¯ j i | r β j i r ( L j f ¯ ) r ) < 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ25_HTML.gif
(25)

where i = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq31_HTML.gif and j = 1 , 2 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq33_HTML.gif, L j f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq52_HTML.gif, L j f ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq53_HTML.gif, L j f ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq54_HTML.gif, L i g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq55_HTML.gif, L i g ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq56_HTML.gif and L i g ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq57_HTML.gif are Lipschitz constants in (A3).

Proof For any ( φ u φ v ) , ( ψ u ψ v ) C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq105_HTML.gif, we denote the solutions of system (1) through ( ( 0 0 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq106_HTML.gif, ( ( φ u φ v ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq107_HTML.gif and ( ( 0 0 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq108_HTML.gif, ( ( ψ u ψ v ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq109_HTML.gif as
u ( t , φ u , x ) = ( u 1 ( t , φ u , x ) , , u m ( t , φ u , x ) ) T , v ( t , φ v , x ) = ( v 1 ( t , φ v , x ) , , v n ( t , φ v , x ) ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equs_HTML.gif
and
u ( t , ψ u , x ) = ( u 1 ( t , ψ u , x ) , , u m ( t , ψ u , x ) ) T , v ( t , ψ v , x ) = ( v 1 ( t , ψ v , x ) , , v n ( t , ψ v , x ) ) T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equt_HTML.gif
respectively. Define
u t ( φ u , x ) = u ( t + θ , φ u , x ) , θ ( , 0 ] , t 0 , v t ( φ v , x ) = v ( t + θ , φ v , x ) , θ ( , 0 ] , t 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equu_HTML.gif
Clearly, for any t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq49_HTML.gif, ( u t ( φ u ) v t ( φ v ) ) C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq110_HTML.gif. Now, we define
y i = u i ( t , φ u , x ) u i ( t , ψ u , x ) , z j = v j ( t , φ v , x ) v j ( t , ψ v , x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equv_HTML.gif
Thus, we can obtain from system (1) that
y i t = k = 1 l x k ( D i k y i x k ) ( p i ( u i ( t , φ u , x ) ) p i ( u i ( t , ψ u , x ) ) ) y i t = + j = 1 n ( b j i ( f j ( v j ( t , φ v , x ) ) f j ( v j ( t , ψ v , x ) ) ) ) y i t = + j = 1 n ( b ˜ j i ( f ˜ j ( v j ( t θ j i ( t ) , φ v , x ) ) f ˜ j ( v j ( t θ j i ( t ) , ψ v , x ) ) ) ) y i t = + j = 1 n ( b ¯ j i t k j i ( t s ) ( f ¯ j ( v j ( s , φ v , x ) ) f ¯ j ( v j ( s , ψ v , x ) ) ) ) d s , z j t = k = 1 l x k ( D j k z j x k ) ( q j ( v j ( t , φ v , x ) ) q j ( v j ( t , ψ v , x ) ) ) z j t = + i = 1 m ( d i j ( g i ( u i ( t , φ u , x ) ) g i ( u i ( t , ψ u , x ) ) ) ) z j t = + i = 1 m ( d ˜ i j ( g ˜ i ( u i ( t τ i j ( t ) , φ u , x ) ) g ˜ i ( u i ( t τ i j ( t ) , ψ u , x ) ) ) ) z j t = + i = 1 m ( d ¯ i j t k ¯ i j ( t s ) ( g ¯ i ( u i ( s , φ u , x ) ) g ¯ i ( u i ( s , ψ u , x ) ) ) d s ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equw_HTML.gif
We consider the following Lyapunov functional:
V ( t ) = Ω i = 1 m w i [ n a i r 1 | y i | r e 2 α t + j = 1 n | b ˜ j i | r n r ( 1 μ θ ) t θ j i ( t ) t | f ˜ j ( v j ( ξ , φ v , x ) ) f ˜ j ( v j ( ξ , ψ v , x ) ) | r d ξ + j = 1 n | b ¯ j i | r n r β j i r 0 + k j i ( s ) t s t e 2 α ( s + ξ ) × | f ¯ j ( v j ( ξ , φ v , x ) ) f ¯ j ( v j ( ξ , ψ v , x ) ) | r d ξ d s ] d x + Ω j = 1 n w m + j [ m c j r 1 | z j | r e 2 α t + i = 1 m | d ˜ i j | r m r ( 1 μ τ ) t τ i j ( t ) t | g ˜ i ( u i ( ξ , φ u , x ) ) g ˜ i ( u i ( ξ , ψ u , x ) ) | r d ξ d s + i = 1 m | d ¯ i j | r m r γ i j r 0 + k ¯ i j ( s ) t s t e 2 α ( s + ξ ) × | g ¯ i ( u i ( ξ , φ u , x ) ) g ¯ i ( u i ( ξ , ψ u , x ) ) | r d ξ d s ] d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equx_HTML.gif
By a minor modification of the proof of Theorem 1, we can easily get
u ( t , φ u , x ) u ( t , ψ u , x ) + v ( t , φ v , x ) v ( t , ψ v , x ) β e 2 α t ( φ u ψ u + φ v ψ v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ26_HTML.gif
(26)
for t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq49_HTML.gif,in which β 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq67_HTML.gif is a constant. Now, we can choose a positive integer N such that
β e α N ω 1 4 , β e α N ω 1 4 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ27_HTML.gif
(27)
Defining a Poincaré mapping P : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq111_HTML.gif by
P ( φ u φ v ) = ( u ω ( φ u ) v ω ( φ v ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ28_HTML.gif
(28)
due to the periodicity of system, we have
P N ( φ u φ v ) = ( u N ω ( φ u ) v N ω ( φ v ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equ29_HTML.gif
(29)
Let t = N ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq112_HTML.gif, then from (26)-(29) we can derive that
| P N ( φ u φ v ) P N ( ψ u ψ v ) | 1 2 | ( φ u φ v ) ( ψ u ψ v ) | , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equy_HTML.gif

which shows that P N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq113_HTML.gif is a contraction mapping. Therefore, there exists a unique fixed point ( φ u φ v ) C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq114_HTML.gif, namely, P N ( φ u φ v ) = ( φ u φ v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq115_HTML.gif.

Since P N ( P ( φ u φ v ) ) = P ( P N ( φ u φ v ) ) = P ( φ u φ v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq116_HTML.gif, then P ( φ u φ v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq117_HTML.gif is also a fixed point of P N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq113_HTML.gif. Because of the uniqueness of a fixed point of P N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq113_HTML.gif, then P ( φ u φ v ) = ( φ u φ v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq118_HTML.gif.

Let ( u ( t , φ u , x ) , v ( t , φ v , x ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq119_HTML.gif be the solution of system (1) through ( ( 0 0 ) , ( φ u φ v ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq120_HTML.gif, then ( u ( t + ω , φ u , x ) , v ( t + ω , φ v , x ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq121_HTML.gif is also a solution of system (1). Clearly,
( u t + ω ( φ u , x ) v t + ω ( φ v , x ) ) = ( u t ( u ω ( φ u ) ) v t ( v ω ( φ v ) ) ) = ( u t ( φ u , x ) v t ( φ v , x ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_Equz_HTML.gif

for t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq49_HTML.gif. Hence ( u ( t + ω , φ u , x ) , v ( t + ω , φ v , x ) ) T = ( u ( t , φ u , x ) , v ( t , φ v , x ) ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq122_HTML.gif for t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq49_HTML.gif.

This shows that ( u ( t , φ u , x ) , v ( t , φ v , x ) ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq123_HTML.gif is exactly one ω-periodic solution of system (1), and it is easy to see that all other solutions of system (1) converge exponentially to it as t + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-105/MediaObjects/13661_2012_Article_354_IEq102_HTML.gif. The proof is completed. □

5 Illustration example

In this section, a numerical example is given to illustrate the effectiveness of the obtained results.

Example 1 Consider the following system on Ω = { ( x 1 , x 2 ) T | 0 <