Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 701087, 10 pages
http://dx.doi.org/10.1155/2013/701087
Research Article
Periodic Solution for Impulsive Cellar Neural Networks with
Time-Varying Delays in the Leakage Terms
Bingwen Liu1 and Shuhua Gong2
1
2
College of Mathematics and Computer Science, Hunan University of Arts and Science, Changde, Hunan 415000, China
College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, Zhejiang 314001, China
Correspondence should be addressed to Shuhua Gong; shuhuagong@yahoo.cn
Received 29 January 2013; Accepted 12 March 2013
Academic Editor: Chuangxia Huang
Copyright © 2013 B. Liu and S. Gong. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper is concerned with impulsive cellular neural networks with time-varying delays in leakage terms. Without assuming
bounded and monotone conditions on activation functions, we establish sufficient conditions on existence and exponential stability
of periodic solutions by using Lyapunov functional method and differential inequality techniques. Our results are complement to
some recent ones.
1. Introduction
It is well known that impulsive differential equations are
mathematical apparatus for simulation of process and phenomena observed in control theory, physics, chemistry,
population dynamics, biotechnologies, industrial robotics,
economics, and so forth [1–3]. Thus, many neural networks
with impulses have been studied extensively, and a great deal
of literature is focused on the existence and stability of an
equilibrium point [4–7]. In [8–10], the authors discussed the
existence and global exponential stability of periodic solution
of a class of cellular neural networks (CNNs) with impulse.
Recently, Wang et al. [11] considered the following CNNs with
impulses and leakage delays:
𝑛
𝑥𝑖󸀠 (𝑡) = −𝑎𝑖 𝑥𝑖 (𝑡 − 𝜏𝑖 ) + ∑ 𝛼𝑖𝑗 (𝑡) 𝑓𝑗 (𝑥𝑗 (𝑡))
𝑗=1
𝑛
+ ∑𝛽𝑖𝑗 (𝑡) 𝑓𝑗 (𝑥𝑗 (𝑡 − 𝜎𝑖𝑗 )) + 𝐼𝑖 (𝑡) ,
𝑗=1
𝑡 ≥ 0,
(1)
𝑡 ≠ 𝑡𝑘 ,
Δ𝑥𝑖 (𝑡𝑘 ) = 𝑥𝑖 (𝑡𝑘+ ) − 𝑥𝑖 (𝑡𝑘− ) = 𝑑𝑖𝑘 𝑥𝑖 (𝑡𝑘 ) ,
where Δ𝑥𝑖 (𝑡𝑘 ) are the impulses at moments 𝑡𝑘 and 𝑡1 < 𝑡2 <
⋅ ⋅ ⋅ is a strictly increasing sequence such that lim𝑘 → ∞ 𝑡𝑘 =
+∞; 𝑎𝑖 > 0 and 𝜏𝑖 > 0 are constants, and 𝛼𝑖𝑗 (𝑡), 𝐼𝑖 (𝑡), and 𝛽𝑖𝑗 (𝑡)
are continuous periodic functions with period 𝑇. Suppose
that the following conditions are satisfied.
𝑓
(𝐴 1 ) There exist constants 𝐿 𝑗 , 𝑗 = 1, 2, . . . 𝑛, such that, for
any 𝛼, 𝛽 ∈ 𝑅,
0<
𝑓𝑗 (𝛼) − 𝑓𝑗 (𝛽)
𝛼−𝛽
𝑓
< 𝐿𝑗 ,
𝛼 ≠ 𝛽, 𝑗 = 1, 2, . . . 𝑛.
(2)
(𝐴 2 ) 𝑓𝑖 (0) = 0 and for 𝑖 = 1, 2, . . . , 𝑛, there exists a constant
0 < 𝑀𝑖 < +∞, such that
󵄨
󵄨󵄨
󵄨󵄨𝑓𝑖 (𝛼)󵄨󵄨󵄨 ≤ 𝑀𝑖 ,
𝛼 ∈ 𝑅.
(3)
By using the continuation theorem of coincidence degree
theory and a suitable degenerate Lyapunov-Krasovskii functional together with model transformation technique, some
results were obtained in [11] to guarantee that all solutions
of system (1) converge exponentially to a periodic function.
However, to the best of our knowledge, few authors have
considered the existence and stability of periodic solutions
of system (1) without the assumptions (𝐴 1 ) and (𝐴 2 ). Thus,
it is worthwhile to continue to investigate the convergence
behavior of system (1) in this case. In view of the fact that
the coefficients and delays in neural networks are usually time
varying in the real world, motivated by the above discussions,
2
Abstract and Applied Analysis
in this paper, we will consider the problem on periodic
solution of the following impulsive CNNs with time-varying
delays in the leakage terms:
𝑗=1
𝑡 ≥ 0,
(4)
𝑡 ≠ 𝑡𝑘 ,
in which 𝑛 corresponds to the number of units in a neural
network, 𝑥𝑖 (𝑡) corresponds to the state vector of the 𝑖th unit
at the time 𝑡, and 𝑐𝑖 (𝑡) represents the rate with which the 𝑖th
unit will reset its potential to the resting state in isolation
when disconnected from the network and external inputs at
the time 𝑡. 𝑎𝑖𝑗 (𝑡) and 𝑏𝑖𝑗 (𝑡) are the connection weights at the
time 𝑡, 𝜂𝑖 (𝑡) and 𝜏𝑖𝑗 (𝑡) denote the transmission delays, 𝐼𝑖 (𝑡)
denotes the external bias on the 𝑖th unit at the time 𝑡, 𝑓𝑗 and
𝑔𝑗 are activation functions of signal transmission, Δ𝑥𝑖 (𝑡𝑘 ) are
the impulses at moments 𝑡𝑘 , and 0 ≤ 𝑡1 < 𝑡2 < ⋅ ⋅ ⋅ is a strictly
increasing sequence such that lim𝑘 → ∞ 𝑡𝑘 = +∞, and 𝑖, 𝑗 =
1, 2, . . . , 𝑛. It is obvious that when 𝑓 = 𝑔 and 𝜂𝑖 (𝑡) is a constant
function, (1) is a special case of (4).
The main purpose of this paper is to give the conditions for the existence and exponential stability of the periodic solutions for system (4). By applying Lyapunov functional method and differential inequality techniques, without
assuming (𝐴 1 ) and (𝐴 2 ), we derive some new sufficient conditions ensuring the existence, uniqueness, and exponential
stability of the periodic solution for system (4), which are
new and complement previously known results. Moreover, an
example is also provided to illustrate the effectiveness of our
results.
Throughout this paper, we assume that the following
conditions hold.
(𝐻1 ) For 𝑖, 𝑗 = 1, 2, . . . , 𝑛, 𝐼𝑖 , 𝑎𝑖𝑗 , 𝑏𝑖𝑗 : 𝑅 → 𝑅 and 𝑐𝑖 , 𝜂𝑖 ,
𝜏𝑖𝑗 : 𝑅 → 𝑅+ are continuous periodic functions with period
𝑇 > 0, and 𝑡 − 𝜂𝑖 (𝑡) ≥ 0 for all 𝑡 ≥ 0. In addition, there exist
constants 𝑐𝑖+ , 𝜂𝑖+ , 𝐼𝑖+ , 𝜏𝑖 , 𝑎𝑖𝑗+ , 𝑏𝑖𝑗+ , and 𝜏𝑖𝑗+ such that
𝑡∈𝑅
𝐼𝑖+
󵄨
󵄨
= sup 󵄨󵄨󵄨𝐼𝑖 (𝑡)󵄨󵄨󵄨 ,
𝑡∈𝑅
󵄨
󵄨
𝑏𝑖𝑗+ = sup 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 (𝑡)󵄨󵄨󵄨󵄨 ,
𝑡∈𝑅
𝜂𝑖+ = sup𝜂𝑖 (𝑡) ,
𝑡∈𝑅
𝑎𝑖𝑗+
󵄨
󵄨
= sup 󵄨󵄨󵄨󵄨𝑎𝑖𝑗 (𝑡)󵄨󵄨󵄨󵄨 ,
𝑡∈𝑅
𝜏𝑖𝑗+ = sup𝜏𝑖𝑗 (𝑡) ,
(𝑘 ∈ 𝑍+ ) .
(6)
󵄨
󵄨󵄨
󵄨󵄨𝑔𝑗 (𝑢) − 𝑔𝑗 (V)󵄨󵄨󵄨 ≤ 𝐿𝑔𝑗 |𝑢 − V| ,
󵄨
󵄨
󵄨󵄨
󵄨
󵄨󵄨𝑓𝑗 (𝑢) − 𝑓𝑗 (V)󵄨󵄨󵄨 ≤ 𝐿𝑓𝑗 |𝑢 − V| .
󵄨
󵄨
(7)
(𝐻5 ) For all 𝑡 > 0 and 𝑖 ∈ {1, 2, . . . , 𝑛}, there exist constants 𝜉𝑖 > 0 and 𝜂 > 0 such that
Δ𝑥𝑖 (𝑡𝑘 ) = 𝑥𝑖 (𝑡𝑘+ ) − 𝑥𝑖 (𝑡𝑘− ) = 𝑑𝑖𝑘 𝑥𝑖 (𝑡𝑘 ) ,
𝑐𝑖+ = sup𝑐𝑖 (𝑡) ,
𝑡𝑘+𝑞 = 𝑡𝑘 + 𝑇,
𝑔𝑗 (0) = 𝑓𝑗 (0) = 0,
𝑛
𝑗=1
𝑑𝑖(𝑘+𝑞) = 𝑑𝑖𝑘 ,
(𝐻4 ) For each 𝑗 ∈ {1, 2, . . . , 𝑛}, there exist nonnegative
𝑓
𝑔
constants 𝐿 𝑗 and 𝐿 𝑗 such that, for all 𝑢, V ∈ 𝑅,
𝑛
𝑥𝑖󸀠 (𝑡) = −𝑐𝑖 (𝑡) 𝑥𝑖 (𝑡 − 𝜂𝑖 (𝑡)) + ∑𝑎𝑖𝑗 (𝑡) 𝑓𝑗 (𝑥𝑗 (𝑡))
+ ∑ 𝑏𝑖𝑗 (𝑡) 𝑔𝑗 (𝑥𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡))) + 𝐼𝑖 (𝑡) ,
(𝐻3 ) There exists a 𝑞 ∈ 𝑍+ such that
(5)
𝑡∈𝑅
𝜏𝑖 = max {𝜂𝑖+ , max 𝜏𝑗𝑖+ } .
𝑗=1,2,...,𝑛
(𝐻2 ) The sequence of times 𝑡𝑘 (𝑘 ∈ 𝑁) satisfies 𝑡𝑘 < 𝑡𝑘+1
and lim𝑘 → +∞ 𝑡𝑘 = +∞, and 𝑑𝑖𝑘 satisfies −2 ≤ 𝑑𝑖𝑘 ≤ 0 for
𝑖 ∈ {1, 2, . . . , 𝑛} and 𝑘 ∈ 𝑍+ , where 𝑍+ denotes the set of all
positive integers.
−𝜂 > − [𝑐𝑖 (𝑡) − 𝑐𝑖 (𝑡) 𝜂𝑖 (𝑡) 𝑐𝑖+ ] 𝜉𝑖
𝑛
󵄨
󵄨
𝑓
+ ∑ (󵄨󵄨󵄨󵄨𝑎𝑖𝑗 (𝑡)󵄨󵄨󵄨󵄨 + 𝑐𝑖 (𝑡) 𝜂𝑖 (𝑡) 𝑎𝑖𝑗+ ) 𝐿 𝑗 𝜉𝑗
𝑗=1
(8)
𝑛
󵄨
󵄨
𝑔
+ ∑ (󵄨󵄨󵄨󵄨𝑏𝑖𝑗 (𝑡)󵄨󵄨󵄨󵄨 + 𝑐𝑖 (𝑡) 𝜂𝑖 (𝑡) 𝑏𝑖𝑗+ ) 𝐿 𝑗 𝜉𝑗 .
𝑗=1
For convenience, let 𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 )𝑇 ∈ 𝑅𝑛 , in
which “𝑇” denotes the transposition. We define |𝑥| =
(|𝑥1 |, |𝑥2 |, . . . , |𝑥𝑛 |)𝑇 and ‖𝑥‖ = max1≤𝑖≤𝑛 {|𝑥𝑖 |}. As usual in
the theory of impulsive differential equations, at the points of
discontinuity 𝑡𝑘 of the solution 𝑡 󳨃→ (𝑥1 (𝑡), 𝑥2 (𝑡), . . . , 𝑥𝑛 (𝑡))𝑇 ,
we assume that (𝑥1 (𝑡), 𝑥2 (𝑡), . . . , 𝑥𝑛 (𝑡))𝑇 ≡ (𝑥1 (𝑡 − 0), 𝑥2 (𝑡 −
0), . . . , 𝑥𝑛 (𝑡 − 0))𝑇 . It is clearly that, in general, the derivative
𝑥𝑖󸀠 (𝑡𝑘 ) does not exist. On the other hand, according to system
(4), there exists the limit 𝑥𝑖󸀠 (𝑡𝑘 ∓ 0). In view of the above
convention, we assume that 𝑥𝑖󸀠 (𝑡𝑘 ) ≡ 𝑥𝑖󸀠 (𝑡𝑘 − 0).
The initial conditions associated with (4) are assumed to
be of the form
𝑥𝑖 (𝑠) = 𝜙𝑖 (𝑠) ,
𝑠 ∈ [−𝜏𝑖 , 0] , 𝑖 = 1, 2, . . . , 𝑛,
(9)
where 𝜙𝑖 (⋅) denotes a real-valued continuous function defined
on [−𝜏𝑖 , 0].
2. Preliminary Results
The following lemmas will be used to prove our main results
in Section 3.
Lemma 1. Let (H1 )–(H5 ) hold. Suppose that 𝑥(𝑡) = (𝑥1 (𝑡),
𝑥2 (𝑡), . . . , 𝑥𝑛 (𝑡))𝑇 is a solution of system (1) with the initial
conditions
𝛾
󵄨
󵄨
𝑥𝑖 (𝑠) = 𝜑𝑖 (𝑠) , 󵄨󵄨󵄨𝜑𝑖 (𝑠)󵄨󵄨󵄨 < 𝜉𝑖 , 𝑠 ∈ [−𝜏𝑖 , 0] ,
(10)
𝜂
where 𝛾 = 1 + max𝑖=1,2,...,𝑛 {[𝑐𝑖+ 𝜂𝑖+ + 1]𝐼𝑖+ }, 𝑖 = 1, 2, . . . , 𝑛. Then
𝛾
󵄨
󵄨󵄨
󵄨󵄨𝑥𝑖 (𝑡)󵄨󵄨󵄨 < 𝜉𝑖 ,
𝜂
∀𝑡 ≥ 0, 𝑖 = 1, 2, . . . , 𝑛.
(11)
Proof. Assume that (11) does not hold. From (𝐻2 ), we have
󵄨
󵄨󵄨
󵄨󵄨
󵄨 󵄨
󵄨
+ 󵄨
(12)
󵄨󵄨𝑥𝑖 (𝑡𝑘 )󵄨󵄨󵄨 = 󵄨󵄨󵄨(1 + 𝑑𝑖𝑘 )󵄨󵄨󵄨 󵄨󵄨󵄨𝑥𝑖 (𝑡𝑘 )󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨𝑥𝑖 (𝑡𝑘 )󵄨󵄨󵄨 .
Abstract and Applied Analysis
3
So, if |𝑥𝑖 (𝑡𝑘+ )| > 𝛾, then |𝑥𝑖 (𝑡𝑘 )| > 𝛾. Thus, we may assume that
there exist 𝑖 ∈ {1, 2, . . . , 𝑛} and 𝑡∗ ∈ (𝑡𝑘 , 𝑡𝑘+1 ) such that
𝛾
󵄨󵄨
󵄨
󵄨󵄨𝑥𝑖 (𝑡∗ )󵄨󵄨󵄨 = 𝜉𝑖 ,
𝜂
∀𝑡 ∈ [−𝜏𝑖 , 𝑡∗ ) ,
𝛾
󵄨
󵄨󵄨
󵄨󵄨𝑥𝑗 (𝑡)󵄨󵄨󵄨 < 𝜉𝑗
󵄨
󵄨
𝜂
󵄨
󵄨
= −𝑐𝑖 (𝑡∗ ) 󵄨󵄨󵄨𝑥𝑖 (𝑡∗ )󵄨󵄨󵄨 + 𝑐𝑖 (𝑡∗ )
󵄨󵄨
󵄨󵄨
󵄨󵄨 − 𝑐 (𝑠) 𝑥 (𝑠 − 𝜂 (𝑠))
𝑖
𝑖
󵄨󵄨 𝑖
𝑡∗ −𝜂𝑖 (𝑡∗ ) 󵄨󵄨
󵄨
𝑡∗
×∫
𝑛
(13)
+ ∑𝑎𝑖𝑗 (𝑠) 𝑓𝑗 (𝑥𝑗 (𝑠))
𝑗 = 1, 2, . . . , 𝑛.
𝑗=1
𝑛
+ ∑𝑏𝑖𝑗 (𝑠) 𝑔𝑗 (𝑥𝑗 (𝑠 − 𝜏𝑖𝑗 (𝑠)))
According to (4), we get
𝑗=1
󵄨󵄨
󵄨󵄨
+𝐼𝑖 (𝑠) 󵄨󵄨󵄨󵄨 𝑑𝑠
󵄨󵄨
󵄨
𝑛
󵄨󵄨
󵄨
+ ∑ 󵄨󵄨󵄨𝑎𝑖𝑗 (𝑡∗ ) 𝑓𝑗 (𝑥𝑗 (𝑡∗ ))󵄨󵄨󵄨󵄨
𝑛
𝑥𝑖󸀠 (𝑡) = −𝑐𝑖 (𝑡) 𝑥𝑖 (𝑡 − 𝜂𝑖 (𝑡)) + ∑𝑎𝑖𝑗 (𝑡) 𝑓𝑗 (𝑥𝑗 (𝑡))
𝑗=1
𝑛
𝑗=1
+ ∑ 𝑏𝑖𝑗 (𝑡) 𝑔𝑗 (𝑥𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡))) + 𝐼𝑖 (𝑡)
𝑛
󵄨
󵄨󵄨
󵄨 󵄨
󵄨
+ ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 (𝑡∗ )󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑔𝑗 (𝑥𝑗 (𝑡∗ − 𝜏𝑖𝑗 (𝑡∗ )))󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨𝐼𝑖 (𝑡∗ )󵄨󵄨󵄨
𝑗=1
𝑗=1
= −𝑐𝑖 (𝑡) 𝑥𝑖 (𝑡) + 𝑐𝑖 (𝑡) [𝑥𝑖 (𝑡) − 𝑥𝑖 (𝑡 − 𝜂𝑖 (𝑡))]
󵄨
󵄨
≤ − [𝑐𝑖 (𝑡∗ ) − 𝑐𝑖 (𝑡∗ ) 𝜂𝑖 (𝑡∗ ) 𝑐𝑖+ ] 󵄨󵄨󵄨𝑥𝑖 (𝑡∗ )󵄨󵄨󵄨
𝑛
+ ∑ 𝑎𝑖𝑗 (𝑡) 𝑓𝑗 (𝑥𝑗 (𝑡))
𝑗=1
𝑛
+ ∑ 𝑏𝑖𝑗 (𝑡) 𝑔𝑗 (𝑥𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡))) + 𝐼𝑖 (𝑡)
(14)
𝑗=1
= −𝑐𝑖 (𝑡) 𝑥𝑖 (𝑡) + 𝑐𝑖 (𝑡) ∫
𝑡
𝑡−𝜂𝑖 (𝑡)
𝑥𝑖󸀠 (𝑠) 𝑑𝑠
𝑛
{
= { − [𝑐𝑖 (𝑡∗ ) − 𝑐𝑖 (𝑡∗ ) 𝜂𝑖 (𝑡∗ ) 𝑐𝑖+ ] 𝜉𝑖
{
𝑛
󵄨
󵄨
𝑓
+ ∑ (󵄨󵄨󵄨󵄨𝑎𝑖𝑗 (𝑡∗ )󵄨󵄨󵄨󵄨 + 𝑐𝑖 (𝑡∗ ) 𝜂𝑖 (𝑡∗ ) 𝑎𝑖𝑗+ ) 𝐿 𝑗 𝜉𝑗
𝑗=1
𝑛
+ ∑ 𝑏𝑖𝑗 (𝑡) 𝑔𝑗 (𝑥𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡))) + 𝐼𝑖 (𝑡) ,
𝑗=1
𝑡 ≠ 𝑡𝑘 ,
𝑗=1
𝑖 = 1, 2, . . . , 𝑛.
Calculating the upper left derivative of |𝑥𝑖 (𝑡)|, together
with (13), (14), (𝐻5 ), and
𝛾>
[𝑐𝑖+ 𝜂𝑖+
+
𝑛
󵄨
󵄨
𝑔 𝛾
+ ∑ (󵄨󵄨󵄨󵄨𝑏𝑖𝑗 (𝑡∗ )󵄨󵄨󵄨󵄨 + 𝑐𝑖 (𝑡∗ ) 𝜂𝑖 (𝑡∗ ) 𝑏𝑖𝑗+ ) 𝐿 𝑗 𝜉𝑗
𝜂
𝑗=1
+ [𝑐𝑖+ 𝜂𝑖+ + 1] 𝐼𝑖+
+ ∑ 𝑎𝑖𝑗 (𝑡) 𝑓𝑗 (𝑥𝑗 (𝑡))
𝑡 > 0,
𝑛
󵄨
󵄨
𝑓 𝛾
+ ∑ (󵄨󵄨󵄨󵄨𝑎𝑖𝑗 (𝑡∗ )󵄨󵄨󵄨󵄨 + 𝑐𝑖 (𝑡∗ ) 𝜂𝑖 (𝑡∗ ) 𝑎𝑖𝑗+ ) 𝐿 𝑗 𝜉𝑗
𝜂
𝑗=1
1] 𝐼𝑖+ ,
(15)
𝑛
󵄨
󵄨
𝑔 } 𝛾
+ ∑ (󵄨󵄨󵄨󵄨𝑏𝑖𝑗 (𝑡∗ )󵄨󵄨󵄨󵄨 + 𝑐𝑖 (𝑡∗ ) 𝜂𝑖 (𝑡∗ ) 𝑏𝑖𝑗+ ) 𝐿 𝑗 𝜉𝑗 }
𝜂
𝑗=1
}
+ [𝑐𝑖+ 𝜂𝑖+ + 1] 𝐼𝑖+
𝛾
< −𝜂 + [𝑐𝑖+ 𝜂𝑖+ + 1] 𝐼𝑖+
𝜂
< 0.
we obtain
(16)
󵄨
󵄨
0 ≤ 𝐷− 󵄨󵄨󵄨𝑥𝑖 (𝑡∗ )󵄨󵄨󵄨
𝑡
󵄨󵄨 󸀠 󵄨󵄨
󵄨󵄨𝑥𝑖 (𝑠)󵄨󵄨 𝑑𝑠
󵄨
󵄨
𝑡∗ −𝜂𝑖 (𝑡∗ )
∗
󵄨
󵄨
≤ −𝑐𝑖 (𝑡∗ ) 󵄨󵄨󵄨𝑥𝑖 (𝑡∗ )󵄨󵄨󵄨 + 𝑐𝑖 (𝑡∗ ) ∫
𝑛
󵄨
󵄨
+ ∑ 󵄨󵄨󵄨󵄨𝑎𝑖𝑗 (𝑡∗ ) 𝑓𝑗 (𝑥𝑗 (𝑡∗ ))󵄨󵄨󵄨󵄨
𝑗=1
𝑛
󵄨
󵄨󵄨
󵄨 󵄨
󵄨
+ ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 (𝑡∗ )󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑔𝑗 (𝑥𝑗 (𝑡∗ − 𝜏𝑖𝑗 (𝑡∗ )))󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨𝐼𝑖 (𝑡∗ )󵄨󵄨󵄨
𝑗=1
It is a contradiction and shows that (11) holds. The proof is
now completed.
Remark 2. After the conditions (𝐻1 )–(𝐻5 ), the solution of
system (4) always exists (see [1, 2]). In view of the boundedness of this solution, from the theory of impulsive differential
equations in [1], it follows that the solution of system (4) with
initial conditions (10) can be defined on [0, +∞).
Lemma 3. Suppose that (H1 )–(H5 ) are true. Let 𝑥∗ (𝑡) =
(𝑥1∗ (𝑡), 𝑥2∗ (𝑡), . . . , 𝑥𝑛∗ (𝑡))𝑇 be the solution of system (4) with
4
Abstract and Applied Analysis
initial value 𝜑∗ = (𝜑1∗ (𝑡), 𝜑2∗ (𝑡), . . . , 𝜑𝑛∗ (𝑡))𝑇 , and let 𝑥(𝑡) =
(𝑥1 (𝑡), 𝑥2 (𝑡), . . . , 𝑥𝑛 (𝑡))𝑇 be the solution of system (4) with
initial value 𝜑 = (𝜑1 (𝑡), 𝜑2 (𝑡), . . . , 𝜑𝑛 (𝑡))𝑇 . Then, there exists
a positive constant 𝜆 such that
𝑥𝑖 (𝑡) − 𝑥𝑖∗ (𝑡) = 𝑂 (𝑒−𝜆𝑡 ) ,
𝑖 = 1, 2, . . . , 𝑛.
which, together with the continuity of Γ𝑖 (𝜔), implies that we
can choose two positive constants 𝜆 and 𝜂 such that
−𝜂 > Γ𝑖 (𝜆)
= − [𝑐𝑖 (𝑡) 𝑒𝜆𝜂𝑖 (𝑡) − 𝜆
(17)
+
−𝑐𝑖 (𝑡) 𝑒𝜆𝜂𝑖 (𝑡) 𝜂𝑖 (𝑡) (𝜆 + 𝑐𝑖+ 𝑒𝜆𝜂𝑖 )] 𝜉𝑖
∗
Proof. Let 𝑦(𝑡) = 𝑥(𝑡) − 𝑥 (𝑡). Then, for 𝑖 ∈ {1, 2, . . . , 𝑛}, it is
followed by
𝑛
󵄨
󵄨
𝑓
+ ∑ (󵄨󵄨󵄨󵄨𝑎𝑖𝑗 (𝑡)󵄨󵄨󵄨󵄨 + 𝑎𝑖𝑗+ 𝑐𝑖 (𝑡) 𝑒𝜆𝜂𝑖 (𝑡) 𝜂𝑖 (𝑡)) 𝐿 𝑗 𝜉𝑗
𝑗=1
𝑦𝑖󸀠 (𝑡) = −𝑐𝑖 (𝑡) (𝑥𝑖 (𝑡 − 𝜂𝑖 (𝑡)) − 𝑥𝑖∗ (𝑡 − 𝜂𝑖 (𝑡)))
𝑛
+ ∑𝑎𝑖𝑗 (𝑡) [𝑓𝑗 (𝑥𝑗 (𝑡)) −
𝑗=1
𝑓𝑗 (𝑥𝑗∗
𝑛
󵄨
󵄨
+ ∑ ( 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 (𝑡)󵄨󵄨󵄨󵄨 𝑒𝜆𝜏𝑖𝑗 (𝑡) + 𝑏𝑖𝑗+ 𝑐𝑖 (𝑡) 𝑒𝜆𝜂𝑖 (𝑡)
(𝑡))]
𝑗=1
𝑛
+
+ ∑𝑏𝑖𝑗 (𝑡) [𝑔𝑗 (𝑥𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡)))
(18)
𝑗=1
−𝑔𝑗 (𝑥𝑗∗ (𝑡 − 𝜏𝑖𝑗 (𝑡)))] ,
𝑡 ≥ 0,
𝑦𝑖 (𝑡𝑘+ )
= (1 + 𝑑𝑖𝑘 ) 𝑦𝑖 (𝑡𝑘 ) ,
𝑔
× 𝜂𝑖 (𝑡) 𝑒𝜆𝜏𝑖𝑗 ) 𝐿 𝑗 𝜉𝑗 ,
𝑡 ≥ 0, 𝑖 = 1, 2, . . . , 𝑛.
(21)
Let
𝑡 ≠ 𝑡𝑘 ,
𝑌𝑖 (𝑡) = 𝑦𝑖 (𝑡) 𝑒𝜆𝑡 ,
𝑘 = 1, 2, . . . .
𝑖 = 1, 2, . . . , 𝑛.
(22)
Then
Define continuous functions Γ𝑖 (𝜔) by setting
𝑌𝑖󸀠 (𝑡) = 𝜆𝑌𝑖 (𝑡) − 𝑐𝑖 (𝑡) 𝑒𝜆𝑡 𝑦𝑖 (𝑡 − 𝜂𝑖 (𝑡))
Γ𝑖 (𝜔) = − [𝑐𝑖 (𝑡) 𝑒𝜔𝜂𝑖 (𝑡) − 𝜔
{𝑛
+ 𝑒𝜆𝑡 { ∑𝑎𝑖𝑗 (𝑡) [𝑓𝑗 (𝑥𝑗 (𝑡)) − 𝑓𝑗 (𝑥𝑗∗ (𝑡))]
{𝑗=1
+
−𝑐𝑖 (𝑡) 𝑒𝜔𝜂𝑖 (𝑡) 𝜂𝑖 (𝑡) (𝜔 + 𝑐𝑖+ 𝑒𝜔𝜂𝑖 )] 𝜉𝑖
𝑛
𝑛
󵄨
󵄨
𝑓
+ ∑ (󵄨󵄨󵄨󵄨𝑎𝑖𝑗 (𝑡)󵄨󵄨󵄨󵄨 + 𝑎𝑖𝑗+ 𝑐𝑖 (𝑡) 𝑒𝜔𝜂𝑖 (𝑡) 𝜂𝑖 (𝑡)) 𝐿 𝑗 𝜉𝑗
𝑗=1
+ ∑𝑏𝑖𝑗 (𝑡) [𝑔𝑗 (𝑥𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡)))
(19)
𝑛
󵄨
󵄨
+ ∑ ( 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 (𝑡)󵄨󵄨󵄨󵄨 𝑒𝜔𝜏𝑖𝑗 (𝑡)
𝑗=1
}
−𝑔𝑗 (𝑥𝑗∗ (𝑡 − 𝜏𝑖𝑗 (𝑡)))] }
}
𝑗=1
+
𝑔
+𝑏𝑖𝑗+ 𝑐𝑖 (𝑡) 𝑒𝜔𝜂𝑖 (𝑡) 𝜂𝑖 (𝑡) 𝑒𝜔𝜏𝑖𝑗 ) 𝐿 𝑗 𝜉𝑗 ,
𝜔 ≥ 0,
𝑡 ≥ 0,
= 𝜆𝑌𝑖 (𝑡) − 𝑐𝑖 (𝑡) 𝑒𝜆𝜂𝑖 (𝑡) 𝑌𝑖 (𝑡)
𝑖 = 1, 2, . . . , 𝑛.
+ 𝑐𝑖 (𝑡) 𝑒𝜆𝜂𝑖 (𝑡) [𝑌𝑖 (𝑡) − 𝑌𝑖 (𝑡 − 𝜂𝑖 (𝑡))]
Then
Γ𝑖 (0) = − [𝑐𝑖 (𝑡) −
{𝑛
+ 𝑒𝜆𝑡 { ∑𝑎𝑖𝑗 (𝑡) [𝑓𝑗 (𝑥𝑗 (𝑡)) − 𝑓𝑗 (𝑥𝑗∗ (𝑡))]
{𝑗=1
𝑐𝑖 (𝑡) 𝜂𝑖 (𝑡) 𝑐𝑖+ ] 𝜉𝑖
𝑛
󵄨
󵄨
𝑓
+ ∑ (󵄨󵄨󵄨󵄨𝑎𝑖𝑗 (𝑡)󵄨󵄨󵄨󵄨 + 𝑎𝑖𝑗+ 𝑐𝑖 (𝑡) 𝜂𝑖 (𝑡)) 𝐿 𝑗 𝜉𝑗
𝑗=1
𝑛
󵄨
󵄨
𝑔
+ ∑ (󵄨󵄨󵄨󵄨𝑏𝑖𝑗 (𝑡)󵄨󵄨󵄨󵄨 + 𝑏𝑖𝑗+ 𝑐𝑖 (𝑡) 𝜂𝑖 (𝑡)) 𝐿 𝑗 𝜉𝑗
𝑗=1
< −𝜂,
𝑡 ≥ 0, 𝑖 = 1, 2, . . . , 𝑛,
𝑛
(20)
+ ∑𝑏𝑖𝑗 (𝑡) [𝑔𝑗 (𝑥𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡)))
𝑗=1
}
−𝑔𝑗 (𝑥𝑗∗ (𝑡 − 𝜏𝑖𝑗 (𝑡)))] }
}
Abstract and Applied Analysis
5
= 𝜆𝑌𝑖 (𝑡) − 𝑐𝑖 (𝑡) 𝑒𝜆𝜂𝑖 (𝑡) 𝑌𝑖 (𝑡)
+ 𝑐𝑖 (𝑡) 𝑒𝜆𝜂𝑖 (𝑡) ∫
𝑡
𝑡−𝜂𝑖 (𝑡)
Let 𝐾 be a positive number such that
𝑌𝑖󸀠 (𝑠) 𝑑𝑠
󵄨
󵄨󵄨
󵄨󵄨𝑌𝑖 (𝑡)󵄨󵄨󵄨 ≤ 𝑀 < 𝐾𝜉𝑖
𝑛
{
+ 𝑒𝜆𝑡 { ∑𝑎𝑖𝑗 (𝑡) [𝑓𝑗 (𝑥𝑗 (𝑡)) − 𝑓𝑗 (𝑥𝑗∗ (𝑡))]
{𝑗=1
∀𝑡 ∈ [−𝜏𝑖 , 0] , 𝑖 = 1, 2, . . . , 𝑛.
(26)
We claim that
𝑛
+ ∑𝑏𝑖𝑗 (𝑡) [𝑔𝑗 (𝑥𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡)))
𝑗=1
−𝑔𝑗 (𝑥𝑗∗
󵄨󵄨
󵄨
󵄨󵄨𝑌𝑖 (𝑡)󵄨󵄨󵄨 < 𝐾𝜉𝑖 ,
}
(𝑡 − 𝜏𝑖𝑗 (𝑡)))] }
}
= 𝜆𝑌𝑖 (𝑡) − 𝑐𝑖 (𝑡) 𝑒𝜆𝜂𝑖 (𝑡) 𝑌𝑖 (𝑡) + 𝑐𝑖 (𝑡) 𝑒𝜆𝜂𝑖 (𝑡)
(27)
Obviously, (27) holds for 𝑡 = 0. We first prove that (27) is
true for 0 < 𝑡 ≤ 𝑡1 . Otherwise, there exist 𝑖 ∈ {1, 2, . . . , 𝑛}
and 𝜌 ∈ (0, 𝑡1 ] such that one of the following two cases must
occur;
{
𝜆𝑌𝑖 (𝑠) − 𝑐𝑖 (𝑠) 𝑒𝜆𝑠 𝑦𝑖 (𝑠 − 𝜂𝑖 (𝑠))
{
𝑡−𝜂𝑖 (𝑡)
{
×∫
∀𝑡 > 0, 𝑖 = 1, 2, . . . , 𝑛.
𝑡
𝑛
+ 𝑒𝜆𝑠 ∑ 𝑎𝑖𝑗 (𝑠) [𝑓𝑗 (𝑥𝑗 (𝑠)) − 𝑓𝑗 (𝑥𝑗∗ (𝑠))]
(1) 𝑌𝑖 (𝜌) = 𝐾𝜉𝑖 ,
𝑗=1
∀𝑡 ∈ [0, 𝜌) ,
𝑛
+ 𝑒𝜆𝑠 ∑ 𝑏𝑖𝑗 (𝑠) [𝑔𝑗 (𝑥𝑗 (𝑠 − 𝜏𝑖𝑗 (𝑠)))
𝑗=1
(2) 𝑌𝑖 (𝜌) = −𝐾𝜉𝑖 ,
}
−𝑔𝑗 (𝑥𝑗∗ (𝑠 − 𝜏𝑖𝑗 (𝑠)))]}𝑑𝑠
}
{𝑛
+ 𝑒𝜆𝑡 { ∑𝑎𝑖𝑗 (𝑡) [𝑓𝑗 (𝑥𝑗 (𝑡)) − 𝑓𝑗 (𝑥𝑗∗ (𝑡))]
{𝑗=1
∀𝑡 ∈ [0, 𝜌) ,
󵄨
󵄨󵄨
󵄨󵄨𝑌𝑗 (𝑡)󵄨󵄨󵄨 < 𝐾𝜉𝑗
󵄨
󵄨
𝑗 = 1, 2, . . . , 𝑛,
󵄨
󵄨󵄨
󵄨󵄨𝑌𝑗 (𝑡)󵄨󵄨󵄨 < 𝐾𝜉𝑗
󵄨
󵄨
(28)
(29)
𝑗 = 1, 2, . . . , 𝑛.
Now, we distinguish two cases to finish the proof.
Case (i). If (28) holds. Then, from (21), (23), and (𝐻1 )–
(𝐻5 ), we have
𝑛
+ ∑𝑏𝑖𝑗 (𝑡) [𝑔𝑗 (𝑥𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡)))
𝑗=1
}
−𝑔𝑗 (𝑥𝑗∗ (𝑡 − 𝜏𝑖𝑗 (𝑡)))] } ,
}
𝑡 ≠ 𝑡𝑘 ,
󵄨
󵄨󵄨
󵄨
󵄨󵄨
+ 󵄨
󵄨󵄨𝑌𝑖 (𝑡𝑘 )󵄨󵄨󵄨 = 󵄨󵄨󵄨1 + 𝑑𝑖𝑘 󵄨󵄨󵄨 󵄨󵄨󵄨𝑌𝑖 (𝑡𝑘 )󵄨󵄨󵄨 ,
0 ≤ 𝑌𝑖󸀠 (𝜌)
= 𝜆𝑌𝑖 (𝜌) − 𝑐𝑖 (𝜌) 𝑒𝜆𝜂𝑖 (𝜌) 𝑌𝑖 (𝜌) + 𝑐𝑖 (𝜌) 𝑒𝜆𝜂𝑖 (𝜌)
𝑖 = 1, 2, . . . , 𝑛,
(23)
𝑖 = 1, 2, . . . , 𝑛.
(24)
{
𝜆𝑠
{𝜆𝑌𝑖 (𝑠) − 𝑐𝑖 (𝑠) 𝑒 𝑦𝑖 (𝑠 − 𝜂𝑖 (𝑠))
𝜌−𝜂𝑖 (𝜌)
{
×∫
𝜌
𝑛
+ 𝑒𝜆𝑠 ∑𝑎𝑖𝑗 (𝑠) [𝑓𝑗 (𝑥𝑗 (𝑠)) − 𝑓𝑗 (𝑥𝑗∗ (𝑠))]
𝑗=1
We define a positive constant 𝑀 as follows:
𝑛
+ 𝑒𝜆𝑠 ∑𝑏𝑖𝑗 (𝑠) [𝑔𝑗 (𝑥𝑗 (𝑠 − 𝜏𝑖𝑗 (𝑠)))
𝑗=1
{
󵄨
󵄨}
𝑀 = max { sup 󵄨󵄨󵄨𝑌𝑖 (𝑠)󵄨󵄨󵄨} .
1≤𝑖≤𝑛
}
{𝑠∈[−𝜏𝑖 , 0]
(25)
}
−𝑔𝑗 (𝑥𝑗∗ (𝑠 − 𝜏𝑖𝑗 (𝑠)))] } 𝑑𝑠
}
6
Abstract and Applied Analysis
Case (ii). If (29) holds. Then, from (21), (23), and (𝐻1 )–
(𝐻5 ), we get
{𝑛
+ 𝑒𝜆𝜌 { ∑𝑎𝑖𝑗 (𝜌) [𝑓𝑗 (𝑥𝑗 (𝜌)) − 𝑓𝑗 (𝑥𝑗∗ (𝜌))]
{𝑗=1
𝑛
0 ≥ 𝑌𝑖󸀠 (𝜌)
+ ∑ 𝑏𝑖𝑗 (𝜌) [𝑔𝑗 (𝑥𝑗 (𝜌 − 𝜏𝑖𝑗 (𝜌)))
𝑗=1
= 𝜆𝑌𝑖 (𝜌) − 𝑐𝑖 (𝜌) 𝑒𝜆𝜂𝑖 (𝜌) 𝑌𝑖 (𝜌) + 𝑐𝑖 (𝜌) 𝑒𝜆𝜂𝑖 (𝜌)
}
−𝑔𝑗 (𝑥𝑗∗ (𝜌 − 𝜏𝑖𝑗 (𝜌)))] }
}
{
𝜆𝑌𝑖 (𝑠) − 𝑐𝑖 (𝑠) 𝑒𝜆𝑠 𝑦𝑖 (𝑠 − 𝜂𝑖 (𝑠))
{
𝜌−𝜂𝑖 (𝜌)
{
×∫
≤ 𝜆𝑌𝑖 (𝜌) − 𝑐𝑖 (𝜌) 𝑒𝜆𝜂𝑖 (𝜌) 𝑌𝑖 (𝜌) + 𝑐𝑖 (𝜌) 𝑒𝜆𝜂𝑖 (𝜌)
𝜌
𝑛
+ 𝑒𝜆𝑠 ∑𝑎𝑖𝑗 (𝑠) [𝑓𝑗 (𝑥𝑗 (𝑠)) − 𝑓𝑗 (𝑥𝑗∗ (𝑠))]
{
󵄨
+ 𝜆𝜂𝑖 (𝑠) 󵄨󵄨
󵄨󵄨𝑌𝑖 (𝑠 − 𝜂𝑖 (𝑠))󵄨󵄨󵄨
{𝜆𝑌𝑖 (𝑠) + 𝑐𝑖 𝑒
𝜌−𝜂𝑖 (𝜌)
{
𝑛
󵄨
𝑓󵄨
+ ∑ 𝑎𝑖𝑗+ 𝐿 𝑗 󵄨󵄨󵄨󵄨𝑌𝑗 (𝑠)󵄨󵄨󵄨󵄨
×∫
𝜌
𝑗=1
𝑛
+ 𝑒𝜆𝑠 ∑𝑏𝑖𝑗 (𝑠) [𝑔𝑗 (𝑥𝑗 (𝑠 − 𝜏𝑖𝑗 (𝑠)))
𝑗=1
}
−𝑔𝑗 (𝑥𝑗∗ (𝑠 − 𝜏𝑖𝑗 (𝑠)))] } 𝑑𝑠
}
𝑗=1
𝑛
󵄨
󵄨}
𝑔
+ ∑𝑏𝑖𝑗+ 𝐿 𝑗 𝑒𝜆𝜏𝑖𝑗 (𝑠) 󵄨󵄨󵄨󵄨𝑌𝑗 (𝑠 − 𝜏𝑖𝑗 (𝑠))󵄨󵄨󵄨󵄨} 𝑑𝑠
𝑗=1
}
𝑛
󵄨󵄨
󵄨󵄨 𝑓 󵄨󵄨
󵄨󵄨
+ ∑ 󵄨󵄨󵄨𝑎𝑖𝑗 (𝜌)󵄨󵄨󵄨 𝐿 𝑗 󵄨󵄨󵄨𝑌𝑗 (𝜌)󵄨󵄨󵄨
{𝑛
+ 𝑒𝜆𝜌 { ∑𝑎𝑖𝑗 (𝜌) [𝑓𝑗 (𝑥𝑗 (𝜌)) − 𝑓𝑗 (𝑥𝑗∗ (𝜌))]
{𝑗=1
𝑛
𝑗=1
+ ∑ 𝑏𝑖𝑗 (𝜌) [𝑔𝑗 (𝑥𝑗 (𝜌 − 𝜏𝑖𝑗 (𝜌)))
𝑛
󵄨
󵄨
󵄨 𝑔
󵄨
+ ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 (𝜌)󵄨󵄨󵄨󵄨 𝐿 𝑗 𝑒𝜆𝜏𝑖𝑗 (𝜌) 󵄨󵄨󵄨󵄨𝑌𝑗 (𝜌 − 𝜏𝑖𝑗 (𝜌))󵄨󵄨󵄨󵄨
𝑗=1
}
−𝑔𝑗 (𝑥𝑗∗ (𝜌 − 𝜏𝑖𝑗 (𝜌)))] }
}
𝑗=1
≤ − [𝑐𝑖 (𝜌) 𝑒𝜆𝜂𝑖 (𝜌) − 𝜆 − 𝑐𝑖 (𝜌) 𝑒𝜆𝜂𝑖 (𝜌) 𝜂𝑖 (𝜌)
× (𝜆 +
≥ 𝜆𝑌𝑖 (𝜌) − 𝑐𝑖 (𝜌) 𝑒𝜆𝜂𝑖 (𝜌) 𝑌𝑖 (𝜌) + 𝑐𝑖 (𝜌) 𝑒𝜆𝜂𝑖 (𝜌)
+
𝑐𝑖+ 𝑒𝜆𝜂𝑖 )] 𝐾𝜉𝑖
{
󵄨
+ 𝜆𝜂𝑖 (𝑠) 󵄨󵄨
󵄨󵄨𝑌𝑖 (𝑠 − 𝜂𝑖 (𝑠))󵄨󵄨󵄨
{𝜆𝑌𝑖 (𝑠) − 𝑐𝑖 𝑒
𝜌−𝜂𝑖 (𝜌)
{
𝑛
󵄨
𝑓󵄨
− ∑ 𝑎𝑖𝑗+ 𝐿 𝑗 󵄨󵄨󵄨󵄨𝑌𝑗 (𝑠)󵄨󵄨󵄨󵄨
×∫
𝑛
󵄨
󵄨
𝑓
+ ∑ (󵄨󵄨󵄨󵄨𝑎𝑖𝑗 (𝜌)󵄨󵄨󵄨󵄨 + 𝑎𝑖𝑗+ 𝑐𝑖 (𝜌) 𝑒𝜆𝜂𝑖 (𝜌) 𝜂𝑖 (𝜌)) 𝐿 𝑗 𝐾𝜉𝑗
𝑗=1
𝑛
󵄨
󵄨
+ ∑ ( 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 (𝜌)󵄨󵄨󵄨󵄨 𝑒𝜆𝜏𝑖𝑗 (𝜌) + 𝑏𝑖𝑗+ 𝑐𝑖 (𝜌) 𝑒𝜆𝜂𝑖 (𝜌)
𝑗=1
𝑗=1
𝜆𝜏𝑖𝑗+
𝑛
󵄨}
󵄨
𝑔
− ∑𝑏𝑖𝑗+ 𝐿 𝑗 𝑒𝜆𝜏𝑖𝑗 (𝑠) 󵄨󵄨󵄨󵄨𝑌𝑗 (𝑠 − 𝜏𝑖𝑗 (𝑠))󵄨󵄨󵄨󵄨} 𝑑𝑠
𝑗=1
}
𝑛
󵄨󵄨
󵄨󵄨 𝑓 󵄨󵄨
󵄨󵄨
− ∑ 󵄨󵄨󵄨𝑎𝑖𝑗 (𝜌)󵄨󵄨󵄨 𝐿 𝑗 󵄨󵄨󵄨𝑌𝑗 (𝜌)󵄨󵄨󵄨
𝑔
) 𝐿 𝑗 𝐾𝜉𝑗
× 𝜂𝑖 (𝜌) 𝑒
𝜌
{
= { − [𝑐𝑖 (𝜌) 𝑒𝜆𝜂𝑖 (𝜌) − 𝜆 − 𝑐𝑖 (𝜌) 𝑒𝜆𝜂𝑖 (𝜌) 𝜂𝑖 (𝜌)
{
𝑗=1
𝑛
󵄨
󵄨
󵄨 𝑔
󵄨
− ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 (𝜌)󵄨󵄨󵄨󵄨 𝐿 𝑗 𝑒𝜆𝜏𝑖𝑗 (𝜌) 󵄨󵄨󵄨󵄨𝑌𝑗 (𝜌 − 𝜏𝑖𝑗 (𝜌))󵄨󵄨󵄨󵄨
+
× (𝜆 + 𝑐𝑖+ 𝑒𝜆𝜂𝑖 )] 𝜉𝑖
𝑗=1
≥ − [𝑐𝑖 (𝜌) 𝑒𝜆𝜂𝑖 (𝜌) − 𝜆 − 𝑐𝑖 (𝜌) 𝑒𝜆𝜂𝑖 (𝜌) 𝜂𝑖 (𝜌)
𝑛
󵄨
󵄨
𝑓
+ ∑ (󵄨󵄨󵄨󵄨𝑎𝑖𝑗 (𝜌)󵄨󵄨󵄨󵄨 + 𝑎𝑖𝑗+ 𝑐𝑖 (𝜌) 𝑒𝜆𝜂𝑖 (𝜌) 𝜂𝑖 (𝜌)) 𝐿 𝑗 𝜉𝑗
𝑗=1
+
× (𝜆 + 𝑐𝑖+ 𝑒𝜆𝜂𝑖 )] (−𝐾𝜉𝑖 )
𝑛
󵄨
󵄨
+ ∑ ( 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 (𝜌)󵄨󵄨󵄨󵄨 𝑒𝜆𝜏𝑖𝑗 (𝜌) + 𝑏𝑖𝑗+ 𝑐𝑖 (𝜌) 𝑒𝜆𝜂𝑖 (𝜌)
𝑛
󵄨
󵄨
𝑓
+ ∑ (󵄨󵄨󵄨󵄨𝑎𝑖𝑗 (𝜌)󵄨󵄨󵄨󵄨 + 𝑎𝑖𝑗+ 𝑐𝑖 (𝜌) 𝑒𝜆𝜂𝑖 (𝜌) 𝜂𝑖 (𝜌)) 𝐿 𝑗 (−𝐾𝜉𝑗 )
𝑗=1
𝜆𝜏𝑖𝑗+
× 𝜂𝑖 (𝜌) 𝑒
< −𝜂𝐾 < 0.
𝑗=1
𝑔 }
) 𝐿 𝑗 𝜉𝑗 } 𝐾
𝑛
󵄨
󵄨
+ ∑ ( 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 (𝜌)󵄨󵄨󵄨󵄨 𝑒𝜆𝜏𝑖𝑗 (𝜌) + 𝑏𝑖𝑗+ 𝑐𝑖 (𝜌) 𝑒𝜆𝜂𝑖 (𝜌)
}
𝑗=1
+
(30)
𝑔
× 𝜂𝑖 (𝜌) 𝑒𝜆𝜏𝑖𝑗 ) 𝐿 𝑗 (−𝐾𝜉𝑗 )
Abstract and Applied Analysis
7
solution 𝑥(𝑡) can be defined for all 𝑡 ∈ [0, +∞). By hypothesis
(𝐻1 ), we have, for any natural number ℎ,
{
= { − [𝑐𝑖 (𝜌) 𝑒𝜆𝜂𝑖 (𝜌) − 𝜆 − 𝑐𝑖 (𝜌) 𝑒𝜆𝜂𝑖 (𝜌)
{
+
× 𝜂𝑖 (𝜌) (𝜆 + 𝑐𝑖+ 𝑒𝜆𝜂𝑖 ) ] 𝜉𝑖
[𝑥𝑖 (𝑡 + (ℎ + 1) 𝑇)]
𝑛
󵄨
󵄨
𝑓
+ ∑ (󵄨󵄨󵄨󵄨𝑎𝑖𝑗 (𝜌)󵄨󵄨󵄨󵄨 + 𝑎𝑖𝑗+ 𝑐𝑖 (𝜌) 𝑒𝜆𝜂𝑖 (𝜌) 𝜂𝑖 (𝜌)) 𝐿 𝑗 𝜉𝑗
= −𝑐𝑖 (𝑡) 𝑥𝑖 (𝑡 + (ℎ + 1) 𝑇 − 𝜂𝑖 (𝑡))
𝑛
𝑗=1
+ ∑ 𝑎𝑖𝑗 (𝑡) 𝑓𝑗 (𝑥𝑗 (𝑡 + (ℎ + 1) 𝑇))
𝑛
󵄨
󵄨
+ ∑ ( 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 (𝜌)󵄨󵄨󵄨󵄨 𝑒𝜆𝜏𝑖𝑗 (𝜌) + 𝑏𝑖𝑗+ 𝑐𝑖 (𝜌) 𝑒𝜆𝜂𝑖 (𝜌)
𝑗=1
+ ∑ 𝑏𝑖𝑗 (𝑡) 𝑔𝑗 (𝑥𝑗 (𝑡 + (ℎ + 1) 𝑇 − 𝜏𝑖𝑗 (𝑡)))
𝑗=1
+
𝑔 }
× 𝜂𝑖 (𝜌) 𝑒𝜆𝜏𝑖𝑗 ) 𝐿 𝑗 𝜉𝑗 } (−𝐾)
}
+ 𝐼𝑖 (𝑡) ,
> 𝜂𝐾 > 0.
(31)
Therefore, (27) holds for 𝑡 ∈ [0, 𝑡1 ]. From (24) and (27), we
know that
(32)
∀𝑡 ∈ [𝑡1 , 𝑡2 ] , 𝑖 = 1, 2, . . . , 𝑛.
(33)
+
𝑥𝑖 ((𝑡𝑘 + (ℎ + 1) 𝑇) )
(37)
= (1 + 𝑑𝑖(𝑘+(ℎ+1)𝑞) ) 𝑥𝑖 (𝑡𝑘+(ℎ+1)𝑞 )
∀𝑡 > 0, 𝑖 = 1, 2, . . . , 𝑛.
(34)
∀𝑡 > 0, 𝑖 = 1, 2, . . . , 𝑛.
(35)
That is,
Remark 4. If 𝑥∗ (𝑡) = (𝑥1∗ (𝑡), 𝑥2∗ (𝑡), . . . , 𝑥𝑛∗ (𝑡))𝑇 is the 𝑇-periodic solution of system (4), it follows from Lemma 3 that
𝑥∗ (𝑡) is globally exponentially stable.
3. Main Results
In this section, we will study existence and exponential stability for periodic solutions of system (4).
Theorem 5. Suppose that all conditions in Lemma 3 are
satisfied. Then system (4) has exactly one 𝑇-periodic solution
𝑥∗ (𝑡). Moreover, 𝑥∗ (𝑡) is globally exponentially stable.
Proof. Let 𝑥(𝑡) = (𝑥1 (𝑡), 𝑥2 (𝑡), . . . , 𝑥𝑛 (𝑡))𝑇 be a solution
of system (4) with initial conditions (10). By Remark 2, the
𝑘 = 1, 2, . . . .
Thus, for any natural number ℎ, we obtain that 𝑥(𝑡 + (ℎ + 1)𝑇)
is a solution of system (4) for all 𝑡 + (ℎ + 1)𝑇 ≥ 0. Hence,
𝑥(𝑡 + 𝑇) is also a solution of (4) with initial values
𝑠 ∈ [−𝜏𝑖 , 0] , 𝑖 = 1, 2, . . . , 𝑛.
𝑥𝑖 (𝑠 + 𝑇) ,
Further, we have
󵄨
󵄨󵄨
∗
−𝜆𝑡
󵄨󵄨𝑥𝑖 (𝑡) − 𝑥𝑖 (𝑡)󵄨󵄨󵄨 ≤ 𝐾𝜉𝑖 𝑒 ,
Further, by hypothesis of (𝐻3 ), we obtain
= (1 + 𝑑𝑖𝑘 ) 𝑥𝑖 (𝑡𝑘 + (ℎ + 1) 𝑇) ,
Thus, for 𝑡 ∈ [𝑡1 , 𝑡2 ], we may repeat the above procedure and
obtain
󵄨 󵄨
󵄨 𝜆𝑡
󵄨󵄨
󵄨󵄨𝑌𝑖 (𝑡)󵄨󵄨󵄨 = 󵄨󵄨󵄨𝑦𝑖 (𝑡)󵄨󵄨󵄨 𝑒 < 𝐾𝜉𝑖 ,
𝑡 ≠ 𝑡𝑘 , 𝑖 = 1, 2, . . . , 𝑛.
+
)
= 𝑥𝑖 (𝑡𝑘+(ℎ+1)𝑞
𝑖 = 1, 2, . . . , 𝑛.
󵄨 󵄨
󵄨 𝜆𝑡
󵄨󵄨
󵄨󵄨𝑌𝑖 (𝑡)󵄨󵄨󵄨 = 󵄨󵄨󵄨𝑦𝑖 (𝑡)󵄨󵄨󵄨 𝑒 < 𝐾𝜉𝑖 ,
(36)
𝑛
𝑗=1
󵄨󵄨
󵄨 󵄨
󵄨 𝜆𝑡
󵄨󵄨𝑌𝑖 (𝑡1 )󵄨󵄨󵄨 = 󵄨󵄨󵄨𝑦𝑖 (𝑡1 )󵄨󵄨󵄨 𝑒 1 < 𝐾𝜉𝑖 , 𝑖 = 1, 2, . . . , 𝑛,
󵄨
󵄨󵄨
󵄨 󵄨
󵄨
󵄨󵄨
+ 󵄨
󵄨󵄨𝑌𝑖 (𝑡1 )󵄨󵄨󵄨 = 󵄨󵄨󵄨1 + 𝑑𝑖1 󵄨󵄨󵄨 󵄨󵄨󵄨𝑌𝑖 (𝑡1 )󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨𝑌𝑖 (𝑡1 )󵄨󵄨󵄨 < 𝐾𝜉𝑖 ,
󸀠
(38)
Then, by the proof of Lemma 3, there exists a constant 𝐾 > 0
such that for any natural number ℎ,
󵄨
󵄨󵄨
󵄨󵄨𝑥𝑖 (𝑡 + (ℎ + 1) 𝑇) − 𝑥𝑖 (𝑡 + ℎ𝑇)󵄨󵄨󵄨
󵄨
󵄨
= 󵄨󵄨󵄨𝑥𝑖 (𝑡 + ℎ𝑇 + 𝑇) − 𝑥𝑖 (𝑡 + ℎ𝑇)󵄨󵄨󵄨
≤ 𝐾𝜉𝑖 𝑒−𝜆(𝑡+ℎ𝑇)
= 𝐾𝜉𝑖 𝑒−𝜆𝑡 (
1
𝑒𝜆𝑇
𝑡 ≠ 𝑡𝑘 ,
ℎ
) ,
𝑡 + ℎ𝑇 ≥ 0,
𝑖 = 1, 2, . . . , 𝑛,
󵄨󵄨
󵄨
󵄨󵄨𝑥𝑖 ((𝑡𝑘 + (ℎ + 1) 𝑇)+ ) − 𝑥𝑖 ((𝑡𝑘 + ℎ𝑇)+ )󵄨󵄨󵄨
󵄨
󵄨
󵄨󵄨
󵄨
= (1 + 𝑑𝑖𝑘 ) 󵄨󵄨𝑥𝑖 (𝑡𝑘 + (ℎ + 1) 𝑇) − 𝑥𝑖 (𝑡𝑘 + ℎ𝑇)󵄨󵄨󵄨
≤ 𝐾𝜉𝑖 𝑒−𝜆(𝑡𝑘 +ℎ𝑇)
= 𝐾𝜉𝑖 𝑒−𝜆𝑡𝑘 (
1
ℎ
) ,
𝑒𝜆𝑇
∀𝑘 ∈ 𝑍+ , 𝑖 = 1, 2, . . . , 𝑛.
(39)
8
Abstract and Applied Analysis
Moreover, for any natural number 𝑚, we can obtain
4. An Example
𝑥𝑖 (𝑡 + (𝑚 + 1) 𝑇)
In this section, we give an example to demonstrate the results
obtained in the previous sections.
𝑚
= 𝑥𝑖 (𝑡) + ∑ [𝑥𝑖 (𝑡 + (ℎ + 1) 𝑇) − 𝑥𝑖 (𝑡 + ℎ𝑇)] ,
Example 6. Consider the following impulsive cellar neural
network consisting of two neurons with time-varying delays
in the leakage terms, which is described by
ℎ=0
𝑡 + ℎ𝑇 ≥ 0,
𝑡 ≠ 𝑡𝑘 ,
𝑖 = 1, 2, . . . , 𝑛,
+
(𝑥𝑖 ((𝑡𝑘 + (𝑚 + 1) 𝑇) ))
(40)
𝑚
+
= 𝑥𝑖 (𝑡) + ∑ [𝑥𝑖 ((𝑡𝑘 + (ℎ + 1) 𝑇) )
𝑥1󸀠 (𝑡) = −3 (|sin 𝜋𝑡| + 1) 𝑥1 (𝑡 −
ℎ=0
+
− (𝑥𝑖 ((𝑡𝑘 + ℎ𝑇) ))] ,
∀𝑘 ∈ 𝑍+ ,
𝑖 = 1, 2, . . . , 𝑛.
Combining (39) with (40), we know that 𝑥(𝑡 + 𝑚𝑇)
will converge uniformly to a piecewise continuous function
𝑥∗ (𝑡) = (𝑥1∗ (𝑡), 𝑥2∗ (𝑡), . . . , 𝑥𝑛∗ (𝑡))𝑇 on any compact set of 𝑅.
Now we are in the position of proving that 𝑥∗ (𝑡) is a
𝑇-periodic solution of system (4). It is easily known that 𝑥∗ (𝑡)
is 𝑇-periodic since
𝑥𝑖∗ (𝑡 + 𝑇) = lim 𝑥𝑖 (𝑡 + 𝑇 + 𝑚𝑇)
𝑚+1 → +∞
= 𝑥𝑖∗ (𝑡) ,
𝑡 ≠ 𝑡𝑘 ,
+
+
1
sin2 𝜋𝑡𝑓2 (𝑥2 (𝑡))
16
+
1
sin2 𝜋𝑡𝑔1 (𝑥1 (𝑡 − cos2 𝜋𝑡))
16
+
1
sin2 𝜋𝑡𝑔2 (𝑥2 (𝑡 − 2sin2 𝜋𝑡))
16
𝑡 ≠ 2𝑘 − 1,
(41)
+
𝑚 → +∞
𝑘 = 1, 2, . . . ,
where 𝑖 = 1, 2, . . . , 𝑛. Noting that the right side of (4) is
piecewise continuous, together with (36) and (37), we know
that {𝑥𝑖󸀠 (𝑡 + (𝑚 + 1)𝑇)} converges uniformly to a piecewise
continuous function on any compact set of 𝑅 \ {𝑡1 , 𝑡2 , . . .}.
Therefore, letting 𝑚 → +∞ on both sides of (36) and (37),
we get
+
1
sin3 𝜋𝑡𝑓2 (𝑥2 (𝑡))
16
+
1
sin3 𝜋𝑡𝑔1 (𝑥1 (𝑡 − cos2 𝜋𝑡))
16
+
1
sin3 𝜋𝑡𝑔2 (𝑥2 (𝑡 − 2sin2 𝜋𝑡))
16
+ 100 sin 𝜋𝑡
󸀠
𝑑𝑖(2𝑠) = −2,
𝑛
(43)
𝑡 ≠ 2𝑘 − 1,
𝑥𝑖 (𝑡𝑘+ ) = (1 + 𝑑𝑖𝑘 ) 𝑥𝑖 (𝑡𝑘 ) ,
𝑥𝑖∗ (𝑡) = −𝑐𝑖 (𝑡) 𝑥𝑖∗ (𝑡 − 𝜂𝑖 (𝑡))
𝑡𝑘 = 𝑘,
+ ∑𝑎𝑖𝑗 (𝑡) 𝑓𝑗 (𝑥𝑗∗ (𝑡))
sin4 𝜋𝑡
)
1000
1
+ cos3 𝜋𝑡𝑓1 (𝑥1 (𝑡))
16
𝑥𝑖∗ ((𝑡𝑘 + 𝑇) ) = lim 𝑥𝑖 ((𝑡𝑘 + 𝑇 + 𝑚𝑇) )
= 𝑥𝑖∗ (𝑡𝑘+ ) ,
1
cos2 𝜋𝑡𝑓1 (𝑥1 (𝑡))
16
+ 100 cos 𝜋𝑡
𝑥𝑖 (𝑡 + (𝑚 + 1) 𝑇)
lim
+
𝑥2󸀠 (𝑡) = −3 (|cos 𝜋𝑡| + 1) 𝑥1 (𝑡 −
𝑚 → +∞
=
sin2 𝜋𝑡
)
1000
𝑖 = 1, 2,
𝑑𝑖(2𝑠−1) = −1,
𝑘, 𝑠 = 1, 2, . . . .
𝑗=1
+
𝑛
Here, it is assumed that the activation functions are
∑𝑏𝑖𝑗 (𝑡) 𝑔𝑗 (𝑥𝑗∗
𝑗=1
+ 𝐼𝑖 (𝑡) ,
(𝑡 − 𝜏𝑖𝑗 (𝑡)))
(42)
𝑔1 (𝑥) = 𝑔2 (𝑥) = 𝑥 + 2 sin 𝑥,
𝑡 ≠ 𝑡𝑘 , 𝑖 = 1, 2, . . . , 𝑛,
𝑓1 (𝑥) = 𝑓2 (𝑥) = 𝑥 + 3 sin 𝑥.
𝑥𝑖∗ (𝑡𝑘+ ) = (1 + 𝑑𝑖𝑘 ) 𝑥𝑖∗ (𝑡𝑘 ) ,
𝑘 = 1, 2, . . . ,
𝑖 = 1, 2, . . . , 𝑛.
Thus, 𝑥∗ (𝑡) = (𝑥1∗ (𝑡), 𝑥2∗ (𝑡), . . . , 𝑥𝑛∗ (𝑡))𝑇 is a 𝑇-periodic
solution of system (4).
Finally, by Lemma 3, we can prove that 𝑥∗ (𝑡) is globally
exponentially stable. This completes the proof.
Noting that
𝜂1 (𝑡) =
sin2 𝜋𝑡
,
1000
𝑐1 (𝑡) = 3 (|sin 𝜋𝑡| + 1) ,
𝜂2 (𝑡) =
sin4 𝜋𝑡
,
1000
𝑐2 (𝑡) = 3 (|cos 𝜋𝑡| + 1) ,
(44)
Abstract and Applied Analysis
9
𝑎11 (𝑡) =
1
cos2 𝜋𝑡,
16
𝑎12 (𝑡) =
1
sin2 𝜋𝑡,
16
𝑏11 (𝑡) =
1
sin2 𝜋𝑡,
16
𝑏12 (𝑡) =
1
sin2 𝜋𝑡,
16
𝑎21 (𝑡) =
1
cos3 𝜋𝑡,
16
𝑎22 (𝑡) =
1
sin3 𝜋𝑡,
16
1
𝑏21 (𝑡) = sin3 𝜋𝑡,
16
of China (Grants nos. 11C0916 and 11C0915), the Natural
Scientific Research Fund of Zhejiang Provincial of China
(Grant no. LY12A01018), and the Natural Scientific Research
Fund of Zhejiang Provincial Education Department of China
(Grant no. Z201122436).
References
1
𝑏22 (𝑡) = sin3 𝜋𝑡,
16
𝜏11 (𝑡) = 𝜏21 (𝑡) = cos2 𝜋𝑡,
𝜏12 (𝑡) = 𝜏22 (𝑡) = 2sin2 𝜋𝑡,
(45)
then we obtain
− [𝑐𝑖 (𝑡) − 𝑐𝑖 (𝑡) 𝜂𝑖 (𝑡) 𝑐𝑖+ ] 𝜉𝑖
2
󵄨
󵄨
𝑓
+ ∑ (󵄨󵄨󵄨󵄨𝑎𝑖𝑗 (𝑡)󵄨󵄨󵄨󵄨 + 𝑐𝑖 (𝑡) 𝜂𝑖 (𝑡) 𝑎𝑖𝑗+ ) 𝐿 𝑗 𝜉𝑗
𝑗=1
2
󵄨
󵄨
𝑔
+ ∑ (󵄨󵄨󵄨󵄨𝑏𝑖𝑗 (𝑡)󵄨󵄨󵄨󵄨 + 𝑐𝑖 (𝑡) 𝜂𝑖 (𝑡) 𝑏𝑖𝑗+ ) 𝐿 𝑗 𝜉𝑗
𝑗=1
< − (3 − 6 ×
1
× 6)
1000
+ 2(
1
1
1
+6×
× )×3
16
1000 16
+ 2(
1
1
1
+6×
× )×4
16
1000 16
< −1,
(46)
𝜉𝑖 = 1, 𝑖 = 1, 2.
This yields that system (43) satisfies (𝐻1 )–(𝐻5 ). Hence, from
Theorem 5, system (43) has exactly one 2-periodic solution.
Moreover, the 2-periodic solution is globally exponentially
stable.
Remark 7. Since 𝑔1 (𝑥) = 𝑔2 (𝑥) = 𝑥 + 2 sin 𝑥, 𝑓1 (𝑥) =
𝑓2 (𝑥) = 𝑥 + 3 sin 𝑥 and CNNs (43) is a very simple form
of CNNs with time-varying delays in the leakage terms, it
is clear that the conditions (𝐴 1 ) and (𝐴 2 ) are not satisfied.
Therefore, all the results in [11–19] and the references therein
cannot be applicable to system (43) to obtain the existence
and exponential stability of the 2-periodic solutions.
Acknowledgments
The authors would like to express their sincere appreciation
to the reviewers for their helpful comments in improving
the presentation and quality of the paper. This work was
supported by the National Natural Science Foundation of
China (Grant no. 11201184), the Natural Scientific Research
Fund of Hunan Provincial of China (Grant no. 11JJ6006), the
Construct Program of the Key Discipline in Hunan Province
(Mechanical Design and Theory), the Natural Scientific
Research Fund of Hunan Provincial Education Department
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