Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 427827, 10 pages
http://dx.doi.org/10.1155/2013/427827
Research Article
Asymptotic Stability of Impulsive Cellular Neural Networks with
Infinite Delays via Fixed Point Theory
Yutian Zhang and Yuanhong Guan
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
Correspondence should be addressed to Yutian Zhang; ytzhang81@163.com
Received 1 November 2012; Accepted 8 February 2013
Academic Editor: Qi Luo
Copyright © 2013 Y. Zhang and Y. Guan. 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.
We employ the new method of fixed point theory to study the stability of a class of impulsive cellular neural networks with infinite
delays. Some novel and concise sufficient conditions are presented ensuring the existence and uniqueness of solution and the
asymptotic stability of trivial equilibrium at the same time. These conditions are easily checked and do not require the boundedness
and differentiability of delays.
1. Introduction
Cellular neural networks (CNNs), proposed by Chua and
Yang in 1988 [1, 2], have become a hot topic for their
numerous successful applications in various fields such as
optimization, linear and nonlinear programming, associative
memory, pattern recognition, and computer vision.
Due to the finite switching speed of neurons and amplifiers in the implementation of neural networks, it turns out
that the time delays should not be neglected, and therefore,
the model of delayed cellular neural networks (DCNNs) is put
forward, which is naturally of better realistic significances.
In fact, besides delay effects, stochastic and impulsive as
well as diffusing effects are also likely to exist in neural
networks. Accordingly many experts are showing a growing
interest in the research on the dynamic behaviors of complex
CNNs such as impulsive delayed reaction-diffusion CNNs
and stochastic delayed reaction-diffusion CNNs, with a result
of many achievements [3–9] obtained.
Synthesizing the reported results about complex CNNs,
we find that the existing research methods for dealing with
stability are mainly based on Lyapunov theory. However,
we also notice that there are still lots of difficulties in the
applications of corresponding results to specific problems;
correspondingly it is necessary to seek some new techniques
to overcome those difficulties.
Encouragingly, in recent few years, Burton and other
authors have applied the fixed point theory to investigate
the stability of deterministic systems and obtained some
more applicable results; for example, see the monograph
[10] and papers [11–22]. In addition, more recently, there
have been a few publications where the fixed point theory
is employed to deal with the stability of stochastic (delayed)
differential equations; see [23–29]. Particularly, in [24–26],
Luo used the fixed point theory to study the exponential
stability of mild solutions to stochastic partial differential
equations with bounded delays and with infinite delays. In
[27, 28], Sakthivel used the fixed point theory to investigate
the asymptotic stability in 𝑝th moment of mild solutions to
nonlinear impulsive stochastic partial differential equations
with bounded delays and with infinite delays. In [29], Luo
used the fixed point theory to study the exponential stability
of stochastic Volterra-Levin equations.
Naturally, for complex CNNs which have high application values, we wonder if we can utilize the fixed point
theory to investigate their stability, not just the existence
and uniqueness of solution. With this motivation, in the
present paper, we aim to discuss the stability of impulsive
CNNs with infinite delays via the fixed point theory. It
is worth noting that our research skill is the contraction
mapping theory which is different from the usual method
of Lyapunov theory. We employ the fixed point theorem
2
Abstract and Applied Analysis
to prove the existence and uniqueness of solution and the
asymptotic stability of trivial equilibrium all at once. Some
new and concise algebraic criteria are provided, and these
conditions are easy to verify and, moreover, do not require
the boundedness and differentiability of delays.
(𝑘 = 1, 2, . . .), where it is left continuous; that is, the following
relations are valid:
𝑥𝑖 (𝑡𝑘 − 0) = 𝑥𝑖 (𝑡𝑘 ) ,
𝑥𝑖 (𝑡𝑘 + 0) = 𝑥𝑖 (𝑡𝑘 ) + 𝐼𝑖𝑘 (𝑥𝑖 (𝑡𝑘 )) ,
(4)
𝑖 ∈ N, 𝑘 = 1, 2, . . . .
2. Preliminaries
Let 𝑅𝑛 denote the 𝑛-dimensional Euclidean space and let ‖ ⋅ ‖
represent the Euclidean norm. N ≜ {1, 2, . . . , 𝑛}. 𝑅+ = [0, ∞).
𝐶[𝑋, 𝑌] corresponds to the space of continuous mappings
from the topological space 𝑋 to the topological space 𝑌.
In this paper, we consider the following impulsive cellular
neural network with infinite delays:
𝑛
d𝑥𝑖 (𝑡)
= − 𝑎𝑖 𝑥𝑖 (𝑡) + ∑𝑏𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑡))
d𝑡
𝑗=1
𝑛
+ ∑ 𝑐𝑖𝑗 𝑔𝑗 (𝑥𝑗 (𝑡 − 𝜏𝑗 (𝑡))) ,
(1)
𝑗=1
= 𝐼𝑖𝑘 (𝑥𝑖 (𝑡𝑘 )) ,
𝑘 = 1, 2, . . . ,
𝜗 ≤ 𝑠 ≤ 0, 𝑖 ∈ N,
𝑇
𝑛
𝑡 ≥ 0.
(5)
Definition 2. The trivial equilibrium x = 0 is said to be
asymptotically stable if the trivial equilibrium x = 0 is
stable, and for any initial condition 𝜑(𝑠) ∈ 𝐶[[𝜗, 0], 𝑅𝑛 ],
lim𝑡 → ∞ ‖x(𝑡; 𝑠, 𝜑)‖ = 0 holds.
The consideration of this paper is based on the following
fixed point theorem.
(2)
3. Main Results
where 𝑖 ∈ N and 𝑛 is the number of neurons in the neural
network. 𝑥𝑖 (𝑡) corresponds to the state of the 𝑖th neuron at
time 𝑡. 𝑓𝑗 (⋅), 𝑔𝑗 (⋅) ∈ 𝐶[𝑅, 𝑅] denote the activation functions,
respectively. 𝜏𝑗 (𝑡) ∈ 𝐶[𝑅+ , 𝑅+ ] corresponds to the known
transmission delay satisfying 𝜏𝑗 (𝑡) → ∞ and 𝑡 − 𝜏𝑗 (𝑡) → ∞
as 𝑡 → ∞. Denote 𝜗 = inf{𝑡 − 𝜏𝑗 (𝑡), 𝑡 ≥ 0, 𝑗 ∈ N}.
The constant 𝑏𝑖𝑗 represents the connection weight of the 𝑗th
neuron on the 𝑖th neuron at time 𝑡. The constant 𝑐𝑖𝑗 denotes
the connection strength of the 𝑗th neuron on the 𝑖th neuron
at time 𝑡 − 𝜏𝑗 (𝑡). The constant 𝑎𝑖 > 0 represents the rate with
which the ith neuron will reset its potential to the resting state
when disconnected from the network and external inputs.
The fixed impulsive moments 𝑡𝑘 (𝑘 = 1, 2, . . .) satisfy 0 = 𝑡0 <
𝑡1 < 𝑡2 < ⋅ ⋅ ⋅ and lim𝑘 → ∞ 𝑡𝑘 = ∞. 𝑥𝑖 (𝑡𝑘 + 0) and 𝑥𝑖 (𝑡𝑘 − 0)
stand for the right-hand and left-hand limits of 𝑥𝑖 (𝑡) at time
𝑡𝑘 , respectively. 𝐼𝑖𝑘 (𝑥𝑖 (𝑡𝑘 )) shows the abrupt change of 𝑥𝑖 (𝑡) at
the impulsive moment 𝑡𝑘 and 𝐼𝑖𝑘 (⋅) ∈ 𝐶[𝑅, 𝑅].
Throughout this paper, we always assume that 𝑓𝑖 (0) =
𝑔𝑖 (0) = 𝐼𝑖𝑘 (0) = 0 for 𝑖 ∈ N and 𝑘 = 1, 2, . . .. Thereby,
problem (1) and (2) admits a trivial equilibrium x = 0.
Denote by x(𝑡) ≜ x(𝑡; 𝑠, 𝜑) = (𝑥1 (𝑡; 𝑠, 𝜑1 ), . . . ,
𝑥𝑛 (𝑡; 𝑠, 𝜑𝑛 ))𝑇 ∈ 𝑅𝑛 the solution to (1) and (2) with the initial
condition
𝑥𝑖 (𝑠) = 𝜑𝑖 (𝑠) ,
󵄩
󵄩󵄩
󵄩󵄩x (𝑡; 𝑠, 𝜑)󵄩󵄩󵄩 < 𝜀,
Theorem 3 (see [30]). Let Υ be a contraction operator on a
complete metric space Θ, then there exists a unique point 𝜁 ∈ Θ
for which Υ(𝜁) = 𝜁.
𝑡 ≥ 0, 𝑡 ≠ 𝑡𝑘 ,
Δ𝑥𝑖 (𝑡𝑘 ) = 𝑥𝑖 (𝑡𝑘 + 0) − 𝑥𝑖 (𝑡𝑘 )
Definition 1. The trivial equilibrium x = 0 is said to be stable,
if, for any 𝜀 > 0, there exists 𝛿 > 0 such that for any initial
condition 𝜑(𝑠) ∈ 𝐶[[𝜗, 0], 𝑅𝑛 ] satisfying |𝜑| < 𝛿:
(3)
where 𝜑(𝑠) = (𝜑1 (𝑠), . . . , 𝜑𝑛 (𝑠)) ∈ 𝑅 and 𝜑𝑖 (𝑠) ∈ 𝐶[[𝜗, 0],
𝑅]. Denote |𝜑| = sup𝑠∈[𝜗,0] ‖𝜑(𝑠)‖.
The solution x(𝑡) ≜ x(𝑡; 𝑠, 𝜑) ∈ 𝑅𝑛 of (1)–(3) is, for
the time variable 𝑡, a piecewise continuous vector-valued
function with the first kind discontinuity at the points 𝑡𝑘
In this section, we will consider the existence and uniqueness
of solution and the asymptotic stability of trivial equilibrium
by means of the contraction mapping principle. Before
proceeding, we introduce some assumptions listed as follows.
(A1) There exist nonnegative constants 𝑙𝑗 such that, for any
𝜂, 𝜐 ∈ 𝑅,
󵄨
󵄨󵄨
󵄨󵄨𝑓𝑗 (𝜂) − 𝑓𝑗 (𝜐)󵄨󵄨󵄨 ≤ 𝑙𝑗 󵄨󵄨󵄨󵄨𝜂 − 𝜐󵄨󵄨󵄨󵄨 ,
󵄨
󵄨
𝑗 ∈ N.
(6)
(A2) There exist nonnegative constants 𝑘𝑗 such that, for any
𝜂, 𝜐 ∈ 𝑅,
󵄨
󵄨󵄨
󵄨󵄨𝑔𝑗 (𝜂) − 𝑔𝑗 (𝜐)󵄨󵄨󵄨 ≤ 𝑘𝑗 󵄨󵄨󵄨󵄨𝜂 − 𝜐󵄨󵄨󵄨󵄨 ,
󵄨
󵄨
𝑗 ∈ N.
(7)
(A3) There exist nonnegative constants 𝑝𝑗𝑘 such that, for
any 𝜂, 𝜐 ∈ 𝑅,
󵄨
󵄨󵄨
󵄨󵄨𝐼𝑗𝑘 (𝜂) − 𝐼𝑗𝑘 (𝜐)󵄨󵄨󵄨 ≤ 𝑝𝑗𝑘 󵄨󵄨󵄨󵄨𝜂 − 𝜐󵄨󵄨󵄨󵄨 ,
󵄨
󵄨
𝑗 ∈ N, 𝑘 = 1, 2, . . . . (8)
Let H = H1 × ⋅ ⋅ ⋅ × H𝑛 , and let H𝑖 (𝑖 ∈ N) be the
space consisting of functions 𝜙𝑖 (𝑡) : [𝜗, ∞) → 𝑅, where 𝜙𝑖 (𝑡)
satisfies the following:
(1) 𝜙𝑖 (𝑡) is continuous on 𝑡 ≠ 𝑡𝑘 (𝑘 = 1, 2, . . .);
(2) lim𝑡 → 𝑡𝑘− 𝜙𝑖 (𝑡) and lim𝑡 → 𝑡𝑘+ 𝜙𝑖 (𝑡) exist; furthermore,
lim𝑡 → 𝑡𝑘− 𝜙𝑖 (𝑡) = 𝜙𝑖 (𝑡𝑘 ) for 𝑘 = 1, 2, . . .;
(3) 𝜙𝑖 (𝑠) = 𝜑𝑖 (𝑠) on 𝑠 ∈ [𝜗, 0];
(4) 𝜙𝑖 (𝑡) → 0 as 𝑡 → ∞;
Abstract and Applied Analysis
3
here 𝑡𝑘 (𝑘 = 1, 2, . . .) and 𝜑𝑖 (𝑠) (𝑠 ∈ [𝜗, 0]) are defined as
shown in Section 2. Also H is a complete metric space when
it is equipped with the following metric:
𝑛
󵄨
󵄨
𝑑 (q (𝑡) , h (𝑡)) = ∑sup 󵄨󵄨󵄨𝑞𝑖 (𝑡) − ℎ𝑖 (𝑡)󵄨󵄨󵄨 ,
𝑖=1 𝑡≥𝜗
(9)
Letting 𝜀 → 0 in (11), we have
𝑥𝑖 (𝑡) 𝑒𝑎𝑖 𝑡 = 𝑥𝑖 (𝑡𝑘−1 + 0) 𝑒𝑎𝑖 𝑡𝑘−1
+∫
𝑡
𝑡𝑘−1
𝑒𝑎𝑖 𝑠
{𝑛
× {∑ 𝑏𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑠))
{𝑗=1
where q(𝑡) = (𝑞1 (𝑡), . . . , 𝑞𝑛 (𝑡)) ∈ H and h(𝑡) = (ℎ1 (𝑡), . . . ,
ℎ𝑛 (𝑡)) ∈ H.
In what follows, we will give the main result of this paper.
𝑛
}
+ ∑𝑐𝑖𝑗 𝑔𝑗 (𝑥𝑗 (𝑠 − 𝜏𝑗 (𝑠)))} d𝑠,
𝑗=1
}
Theorem 4. Assume that conditions (A1)–(A3) hold. Provided
that
(i) there exists a constant 𝜇 such that inf 𝑘=1,2,... {𝑡𝑘 −𝑡𝑘−1 } ≥
𝜇,
(ii) there exist constants 𝑝𝑖 such that 𝑝𝑖𝑘 ≤ 𝑝𝑖 𝜇 for 𝑖 ∈ N
and 𝑘 = 1, 2, . . .,
∗
∑𝑛𝑖=1 {(1/𝑎𝑖 )max𝑗∈N |𝑏𝑖𝑗 𝑙𝑗 |+(1/𝑎𝑖 )max𝑗∈N |𝑐𝑖𝑗 𝑘𝑗 |}+
(iii) 𝜆 ≜
max𝑖∈N {𝑝𝑖 (𝜇 + (1/𝑎𝑖 ))} < 1,
(iv) max𝑖∈N {𝜆 𝑖 } < 1/√𝑛, where 𝜆 𝑖 = (1/𝑎𝑖 ) ∑𝑛𝑗=1 |𝑏𝑖𝑗 𝑙𝑗 | +
(1/𝑎𝑖 ) ∑𝑛𝑗=1 |𝑐𝑖𝑗 𝑘𝑗 | + 𝑝𝑖 (𝜇 + (1/𝑎𝑖 )),
for 𝑡 ∈ (𝑡𝑘−1 , 𝑡𝑘 ) (𝑘 = 1, 2, . . .). Setting 𝑡 = 𝑡𝑘 − 𝜀 (𝜀 > 0) in
(12), we get
𝑥𝑖 (𝑡𝑘 − 𝜀) 𝑒𝑎𝑖 (𝑡𝑘 −𝜀)
= 𝑥𝑖 (𝑡𝑘−1 + 0) 𝑒𝑎𝑖 𝑡𝑘−1
+∫
𝑡𝑘 −𝜀
𝑡𝑘−1
{𝑛
𝑒𝑎𝑖 𝑠 {∑ 𝑏𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑠))
{𝑗=1
which generates by letting 𝜀 → 0
𝑥𝑖 (𝑡𝑘 − 0) 𝑒𝑎𝑖 𝑡𝑘 = 𝑥𝑖 (𝑡𝑘−1 + 0) 𝑒𝑎𝑖 𝑡𝑘−1
d𝑒𝑎𝑖 𝑡 𝑥𝑖 (𝑡) = 𝑒𝑎𝑖 𝑡 d𝑥𝑖 (𝑡) + 𝑎𝑖 𝑥𝑖 (𝑡) 𝑒𝑎𝑖 𝑡 d𝑡
{𝑛
= 𝑒𝑎𝑖 𝑡 {∑ 𝑏𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑡))
{𝑗=1
{𝑛
𝑒𝑎𝑖 𝑠 { ∑𝑏𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑠))
𝑡𝑘−1
{𝑗=1
𝑡𝑘
+∫
(10)
𝑛
}
+ ∑𝑐𝑖𝑗 𝑔𝑗 (𝑥𝑗 (𝑠 − 𝜏𝑗 (𝑠)))} d𝑠.
𝑗=1
}
(14)
𝑛
}
+ ∑𝑐𝑖𝑗 𝑔𝑗 (𝑥𝑗 (𝑡 − 𝜏𝑗 (𝑡)))} d𝑡,
𝑗=1
}
which yields after integrating from 𝑡𝑘−1 + 𝜀 (𝜀 > 0) to 𝑡 ∈
(𝑡𝑘−1 , 𝑡𝑘 ) (𝑘 = 1, 2, . . .)
𝑥𝑖 (𝑡) 𝑒𝑎𝑖 𝑡 = 𝑥𝑖 (𝑡𝑘−1 + 𝜀) 𝑒𝑎𝑖 (𝑡𝑘−1 +𝜀)
{𝑛
𝑒𝑎𝑖 𝑠 { ∑𝑏𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑠))
𝑡𝑘−1 +𝜀
{𝑗=1
+∫
(13)
𝑛
}
+ ∑𝑐𝑖𝑗 𝑔𝑗 (𝑥𝑗 (𝑠 − 𝜏𝑗 (𝑠)))} d𝑠,
𝑗=1
}
then the trivial equilibrium x = 0 is asymptotically stable.
Proof. Multiplying both sides of (1) with 𝑒𝑎𝑖 𝑡 gives, for 𝑡 > 0
and 𝑡 ≠ 𝑡𝑘 ,
(12)
𝑡
𝑛
}
+ ∑𝑐𝑖𝑗 𝑔𝑗 (𝑥𝑗 (𝑠 − 𝜏𝑗 (𝑠)))} d𝑠.
𝑗=1
}
(11)
Noting 𝑥𝑖 (𝑡𝑘 − 0) = 𝑥𝑖 (𝑡𝑘 ), (14) can be rearranged as
𝑥𝑖 (𝑡𝑘 ) 𝑒𝑎𝑖 𝑡𝑘 = 𝑥𝑖 (𝑡𝑘−1 + 0) 𝑒𝑎𝑖 𝑡𝑘−1
{𝑛
𝑒𝑎𝑖 𝑠 { ∑𝑏𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑠))
𝑡𝑘−1
{𝑗=1
𝑡𝑘
+∫
𝑛
}
+ ∑𝑐𝑖𝑗 𝑔𝑗 (𝑥𝑗 (𝑠 − 𝜏𝑗 (𝑠)))} d𝑠.
𝑗=1
}
(15)
4
Abstract and Applied Analysis
Combining (12) and (15), we reach that
𝑥𝑖 (𝑡1 ) 𝑒𝑎𝑖 𝑡1 = 𝜑𝑖 (0)
𝑡1
{𝑛
+ ∫ 𝑒𝑎𝑖 𝑠 {∑ 𝑏𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑠))
0
{𝑗=1
𝑥𝑖 (𝑡) 𝑒𝑎𝑖 𝑡 = 𝑥𝑖 (𝑡𝑘−1 + 0) 𝑒𝑎𝑖 𝑡𝑘−1
{𝑛
𝑒𝑎𝑖 𝑠 {∑𝑏𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑠))
𝑡𝑘−1
{𝑗=1
𝑡
+∫
𝑛
}
+ ∑𝑐𝑖𝑗 𝑔𝑗 (𝑥𝑗 (𝑠 − 𝜏𝑗 (𝑠)))} d𝑠,
𝑗=1
}
(18)
(16)
𝑛
}
+ ∑𝑐𝑖𝑗 𝑔𝑗 (𝑥𝑗 (𝑠 − 𝜏𝑗 (𝑠)))} d𝑠
𝑗=1
}
which produces, for 𝑡 > 0,
𝑥𝑖 (𝑡) = 𝜑𝑖 (0) 𝑒−𝑎𝑖 𝑡
is true for 𝑡 ∈ (𝑡𝑘−1 , 𝑡𝑘 ] (𝑘 = 1, 2, . . .). Further,
𝑡
{𝑛
+ 𝑒−𝑎𝑖 𝑡 ∫ 𝑒𝑎𝑖 𝑠 { ∑𝑏𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑠))
0
{𝑗=1
𝑥𝑖 (𝑡) 𝑒𝑎𝑖 𝑡 = 𝑥𝑖 (𝑡𝑘−1 ) 𝑒𝑎𝑖 𝑡𝑘−1
{𝑛
+ ∫ 𝑒𝑎𝑖 𝑠 { ∑𝑏𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑠))
𝑡𝑘−1
{𝑗=1
𝑛
}
+ ∑𝑐𝑖𝑗 𝑔𝑗 (𝑥𝑗 (𝑠 − 𝜏𝑗 (𝑠)))} d𝑠
𝑗=1
}
𝑡
(17)
𝑛
}
+ ∑𝑐𝑖𝑗 𝑔𝑗 (𝑥𝑗 (𝑠 − 𝜏𝑗 (𝑠)))} d𝑠
𝑗=1
}
𝑎𝑖 𝑡𝑘−1
+ 𝐼𝑖(𝑘−1) (𝑥𝑖 (𝑡𝑘−1 )) 𝑒
+ 𝑒−𝑎𝑖 𝑡 ∑ {𝐼𝑖𝑘 (𝑥𝑖 (𝑡𝑘 )) 𝑒𝑎𝑖 𝑡𝑘 } .
0<𝑡𝑘 <𝑡
Note 𝑥𝑖 (0) = 𝜑𝑖 (0) in (19). We then define the following
operator 𝜋 acting on H, for y(𝑡) = (𝑦1 (𝑡), . . . , 𝑦𝑛 (𝑡)) ∈ H:
𝜋 (y) (𝑡) = (𝜋 (𝑦1 ) (𝑡) , . . . , 𝜋 (𝑦𝑛 ) (𝑡)) ,
holds for 𝑡 ∈ (𝑡𝑘−1 , 𝑡𝑘 ] (𝑘 = 1, 2, ⋅ ⋅ ⋅). Hence,
+∫
𝑡𝑘−2
(20)
where 𝜋(𝑦𝑖 )(𝑡) : [𝜗, ∞) → 𝑅 (𝑖 ∈ N) obeys the rules as
follows:
𝜋 (𝑦𝑖 ) (𝑡) = 𝜑𝑖 (0) 𝑒−𝑎𝑖 𝑡
𝑥𝑖 (𝑡𝑘−1 ) 𝑒𝑎𝑖 𝑡𝑘−1 = 𝑥𝑖 (𝑡𝑘−2 ) 𝑒𝑎𝑖 𝑡𝑘−2
𝑡𝑘−1
(19)
{𝑛
𝑒𝑎𝑖 𝑠 { ∑𝑏𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑠))
{𝑗=1
𝑡
{𝑛
+ 𝑒−𝑎𝑖 𝑡 ∫ 𝑒𝑎𝑖 𝑠 {∑ 𝑏𝑖𝑗 𝑓𝑗 (𝑦𝑗 (𝑠))
0
{𝑗=1
𝑛
}
+ ∑𝑐𝑖𝑗 𝑔𝑗 (𝑦𝑗 (𝑠 − 𝜏𝑗 (𝑠)))} d𝑠
𝑗=1
}
𝑛
}
+ ∑𝑐𝑖𝑗 𝑔𝑗 (𝑥𝑗 (𝑠 − 𝜏𝑗 (𝑠)))} d𝑠
𝑗=1
}
+ 𝐼𝑖(𝑘−2) (𝑥𝑖 (𝑡𝑘−2 )) 𝑒𝑎𝑖 𝑡𝑘−2 ,
+ 𝑒−𝑎𝑖 𝑡 ∑ {𝐼𝑖𝑘 (𝑦𝑖 (𝑡𝑘 )) 𝑒𝑎𝑖 𝑡𝑘 } ,
0<𝑡𝑘 <𝑡
(21)
..
.
𝑥𝑖 (𝑡2 ) 𝑒𝑎𝑖 𝑡2 = 𝑥𝑖 (𝑡1 ) 𝑒𝑎𝑖 𝑡1
𝑡2
{𝑛
+ ∫ 𝑒𝑎𝑖 𝑠 {∑ 𝑏𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑠))
𝑡1
{𝑗=1
𝑛
}
+ ∑𝑐𝑖𝑗 𝑔𝑗 (𝑥𝑗 (𝑠 − 𝜏𝑗 (𝑠)))} d𝑠
𝑗=1
}
+ 𝐼𝑖1 (𝑥𝑖 (𝑡1 )) 𝑒𝑎𝑖 𝑡1 ,
on 𝑡 ≥ 0 and 𝜋(𝑦𝑖 )(𝑠) = 𝜑𝑖 (𝑠) on 𝑠 ∈ [𝜗, 0].
The subsequent part is the application of the contraction
mapping principle, which can be divided into two steps.
Step 1. We need to prove 𝜋(H) ⊂ H. Choosing 𝑦𝑖 (𝑡) ∈ H𝑖
(𝑖 ∈ N), it is necessary to testify 𝜋(𝑦𝑖 )(𝑡) ⊂ H𝑖 .
First, since 𝜋(𝑦𝑖 )(𝑠) = 𝜑𝑖 (𝑠) on 𝑠 ∈ [𝜗, 0] and 𝜑𝑖 (𝑠) ∈
𝐶[[𝜗, 0], 𝑅], we know 𝜋(𝑦𝑖 )(𝑠) is continuous on 𝑠 ∈ [𝜗, 0]. For
a fixed time 𝑡 > 0, it follows from (21) that
𝜋 (𝑦𝑖 ) (𝑡 + 𝑟) − 𝜋 (𝑦𝑖 ) (𝑡) = 𝑄1 + 𝑄2 + 𝑄3 + 𝑄4 ,
(22)
Abstract and Applied Analysis
5
where
= 𝑒−𝑎𝑖 (𝑡𝑘 +𝑟) { ∑ {𝐼𝑖𝑚 (𝑦𝑖 (𝑡𝑚 )) 𝑒𝑎𝑖 𝑡𝑚 }
0<𝑡𝑚 <𝑡𝑘
−𝑎𝑖 (𝑡+𝑟)
𝑄1 = 𝜑𝑖 (0) 𝑒
𝑄2 = 𝑒−𝑎𝑖 (𝑡+𝑟) ∫
𝑡+𝑟
0
−𝑎𝑖 𝑡
− 𝜑𝑖 (0) 𝑒
,
(23)
+𝐼𝑖𝑘 (𝑦𝑖 (𝑡𝑘 )) 𝑒𝑎𝑖 𝑡𝑘 }
𝑛
𝑒𝑎𝑖 𝑠 ∑ 𝑏𝑖𝑗 𝑓𝑗 (𝑦𝑗 (𝑠)) d𝑠
𝑗=1
𝑡
𝑛
0
𝑗=1
− 𝑒−𝑎𝑖 𝑡𝑘 ∑
0<𝑡𝑚 <𝑡𝑘
− 𝑒−𝑎𝑖 𝑡 ∫ 𝑒𝑎𝑖 𝑠 ∑ 𝑏𝑖𝑗 𝑓𝑗 (𝑦𝑗 (𝑠)) d𝑠,
𝑄3 = 𝑒−𝑎𝑖 (𝑡+𝑟) ∫
𝑡+𝑟
0
= {𝑒−𝑎𝑖 (𝑡𝑘 +𝑟) − 𝑒−𝑎𝑖 𝑡𝑘 }
𝑛
𝑒𝑎𝑖 𝑠 ∑ 𝑐𝑖𝑗 𝑔𝑗 (𝑦𝑗 (𝑠 − 𝜏𝑗 (𝑠))) d𝑠
𝑗=1
𝑡
𝑛
0
𝑗=1
× ∑
0<𝑡𝑚 <𝑡𝑘
(24)
∑
0<𝑡𝑘 <(𝑡+𝑟)
(26)
{𝐼𝑖𝑘 (𝑦𝑖 (𝑡𝑘 )) 𝑒𝑎𝑖 𝑡𝑘 }
− 𝑒−𝑎𝑖 𝑡 ∑ {𝐼𝑖𝑘 (𝑦𝑖 (𝑡𝑘 )) 𝑒𝑎𝑖 𝑡𝑘 } .
0<𝑡𝑘 <𝑡
Owing to 𝑦𝑖 (𝑡) ∈ H𝑖 , we see that 𝑦𝑖 (𝑡) is continuous on
𝑡 ≠ 𝑡𝑘 (𝑘 = 1, 2, . . .); moreover, lim𝑡 → 𝑡𝑘− 𝑦𝑖 (𝑡) and lim𝑡 → 𝑡𝑘+ 𝑦𝑖 (𝑡)
exist, and lim𝑡 → 𝑡𝑘− 𝑦𝑖 (𝑡) = 𝑦𝑖 (𝑡𝑘 ).
Consequently, when 𝑡 ≠ 𝑡𝑘 (𝑘 = 1, 2, . . .) in (22), it is easy
to find that 𝑄𝑖 → 0 as 𝑟 → 0 for 𝑖 = 1, . . . , 4, and so 𝜋(𝑦𝑖 )(𝑡)
is continuous on the fixed time 𝑡 ≠ 𝑡𝑘 (𝑘 = 1, 2, . . .).
On the other hand, as 𝑡 = 𝑡𝑘 (𝑘 = 1, 2, . . .) in (22), it is
not difficult to find that 𝑄𝑖 → 0 as 𝑟 → 0 for 𝑖 = 1, 2, 3.
Furthermore, if letting 𝑟 < 0 be small enough, we derive
𝑄4 = 𝑒−𝑎𝑖 (𝑡𝑘 +𝑟)
𝐼𝑖𝑚 (𝑦𝑖 (𝑡𝑚 )) 𝑒𝑎𝑖 𝑡𝑚
∑
0<𝑡𝑚 <(𝑡𝑘 +𝑟)
− 𝑒−𝑎𝑖 𝑡𝑘 ∑
0<𝑡𝑚 <𝑡𝑘
𝐼𝑖𝑚 (𝑦𝑖 (𝑡𝑚 )) 𝑒𝑎𝑖 𝑡𝑚
0<𝑡𝑚 <𝑡𝑘
which yields lim𝑟 → 0+ 𝑄4 = 𝑒−𝑎𝑖 𝑡𝑘 𝐼𝑖𝑘 (𝑦𝑖 (𝑡𝑘 ))𝑒𝑎𝑖 𝑡𝑘 as 𝑡 = 𝑡𝑘 .
According to the above discussion, we find that 𝜋(𝑦𝑖 )(𝑡) :
[𝜗, ∞) → 𝑅 is continuous on 𝑡 ≠ 𝑡𝑘 (𝑘 = 1, 2, . . .); moreover, lim𝑡 → 𝑡𝑘− 𝜋(𝑦𝑖 )(𝑡) and lim𝑡 → 𝑡𝑘+ 𝜋(𝑦𝑖 )(𝑡) exist; in addition,
lim𝑡 → 𝑡𝑘− 𝜋(𝑦𝑖 )(𝑡) = 𝜋(𝑦𝑖 )(𝑡𝑘 ) ≠ lim𝑡 → 𝑡𝑘+ 𝜋(𝑦𝑖 )(𝑡).
Next, we will prove 𝜋(𝑦𝑖 )(𝑡) → 0 as 𝑡 → ∞. For
convenience, denote
𝜋 (𝑦𝑖 ) (𝑡) = 𝐽1 + 𝐽2 + 𝐽3 + 𝐽4 ,
𝑡
𝑡
𝐽4 = 𝑒−𝑎𝑖 𝑡 ∑0<𝑡𝑘 <𝑡 {𝐼𝑖𝑘 (𝑦𝑖 (𝑡𝑘 ))𝑒𝑎𝑖 𝑡𝑘 }, and 𝐽3 = 𝑒−𝑎𝑖 𝑡 ∫0 𝑒𝑎𝑖 𝑠
∑𝑛𝑗=1 𝑐𝑖𝑗 𝑔𝑗 (𝑦𝑗 (𝑠 − 𝜏𝑗 (𝑠)))d𝑠.
Due to 𝑦𝑗 (𝑡) ∈ H𝑗 (𝑗 ∈ N), we know lim𝑡 → ∞ 𝑦𝑗 (𝑡) = 0.
Then for any 𝜀 > 0, there exists a 𝑇𝑗 > 0 such that 𝑡 ≥ 𝑇𝑗
implies |𝑦𝑗 (𝑡) | < 𝜀. Choose 𝑇∗ = max𝑗∈N {𝑇𝑗 }. It is derived
from (A1) that, for 𝑡 ≥ 𝑇∗ ,
𝑛
𝑡
󵄨
󵄨 󵄨󵄨
𝐽2 ≤ 𝑒−𝑎𝑖 𝑡 ∫ 𝑒𝑎𝑖 𝑠 ∑ {󵄨󵄨󵄨󵄨𝑏𝑖𝑗 𝑙𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑦𝑗 (𝑠)󵄨󵄨󵄨󵄨} d𝑠
0
{𝐼𝑖𝑚 (𝑦𝑖 (𝑡𝑚 )) 𝑒𝑎𝑖 𝑡𝑚 } ,
𝑗=1
∗
𝑛
𝑇
󵄨
󵄨 󵄨󵄨
= 𝑒−𝑎𝑖 𝑡 ∫ 𝑒𝑎𝑖 𝑠 ∑ {󵄨󵄨󵄨󵄨𝑏𝑖𝑗 𝑙𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑦𝑗 (𝑠)󵄨󵄨󵄨󵄨} d𝑠
which implies lim𝑟 → 0− 𝑄4 = 0 as 𝑡 = 𝑡𝑘 . While letting 𝑟 >
0 tend to zero gives
𝑄4 = 𝑒
∑
𝐼𝑖𝑚 (𝑦𝑖 (𝑡𝑚 )) 𝑒
0<𝑡𝑚 <(𝑡𝑘 +𝑟)
− 𝑒−𝑎𝑖 𝑡𝑘 ∑ 𝐼𝑖𝑚 (𝑦𝑖 (𝑡𝑚 )) 𝑒𝑎𝑖 𝑡𝑚
0<𝑡𝑚 <𝑡𝑘
𝑎𝑖 𝑡𝑚
(27)
where 𝐽1 = 𝜑𝑖 (0)𝑒−𝑎𝑖 𝑡 , 𝐽2 = 𝑒−𝑎𝑖 𝑡 ∫0 𝑒𝑎𝑖 𝑠 ∑𝑛𝑗=1 𝑏𝑖𝑗 𝑓𝑗 (𝑦𝑗 (𝑠))d𝑠,
0
−𝑎𝑖 (𝑡𝑘 +𝑟)
𝑡 > 0,
(25)
= {𝑒−𝑎𝑖 (𝑡𝑘 +𝑟) − 𝑒−𝑎𝑖 𝑡𝑘 }
× ∑
{𝐼𝑖𝑚 (𝑦𝑖 (𝑡𝑚 )) 𝑒𝑎𝑖 𝑡𝑚 }
+ 𝑒−𝑎𝑖 (𝑡𝑘 +𝑟) 𝐼𝑖𝑘 (𝑦𝑖 (𝑡𝑘 )) 𝑒𝑎𝑖 𝑡𝑘 ,
− 𝑒−𝑎𝑖 𝑡 ∫ 𝑒𝑎𝑖 𝑠 ∑ 𝑐𝑖𝑗 𝑔𝑗 (𝑦𝑗 (𝑠 − 𝜏𝑗 (𝑠))) d𝑠,
𝑄4 = 𝑒−𝑎𝑖 (𝑡+𝑟)
{𝐼𝑖𝑚 (𝑦𝑖 (𝑡𝑚 )) 𝑒𝑎𝑖 𝑡𝑚 }
𝑗=1
𝑛
𝑡
󵄨
󵄨 󵄨󵄨
+ 𝑒−𝑎𝑖 𝑡 ∫ 𝑒𝑎𝑖 𝑠 ∑ {󵄨󵄨󵄨󵄨𝑏𝑖𝑗 𝑙𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑦𝑗 (𝑠)󵄨󵄨󵄨󵄨} d𝑠
𝑇∗
−𝑎𝑖 𝑡
≤𝑒
𝑗=1
∗
𝑛
𝑇
󵄨
󵄨
󵄨 󵄨
∑ {󵄨󵄨󵄨󵄨𝑏𝑖𝑗 𝑙𝑗 󵄨󵄨󵄨󵄨 sup 󵄨󵄨󵄨󵄨𝑦𝑗 (𝑠)󵄨󵄨󵄨󵄨} {∫ 𝑒𝑎𝑖 𝑠 d𝑠}
0
𝑠∈[0,𝑇∗ ]
𝑗=1
𝑛
𝑡
󵄨 󵄨
+ 𝜀 ∑ {󵄨󵄨󵄨󵄨𝑏𝑖𝑗 𝑙𝑗 󵄨󵄨󵄨󵄨} 𝑒−𝑎𝑖 𝑡 ∫ 𝑒𝑎𝑖 𝑠 d𝑠
𝑇∗
𝑗=1
6
Abstract and Applied Analysis
𝑛
󵄨
󵄨
󵄨 󵄨
≤ 𝑒−𝑎𝑖 𝑡 ∑ {󵄨󵄨󵄨󵄨𝑏𝑖𝑗 𝑙𝑗 󵄨󵄨󵄨󵄨 sup 󵄨󵄨󵄨󵄨𝑦𝑗 (𝑠)󵄨󵄨󵄨󵄨}
𝑠∈[0,𝑇∗ ]
Furthermore, from (A3), we know that |𝐼𝑖𝑘 (𝑦𝑖 (𝑡𝑘 ))| ≤
𝑝𝑖𝑘 |𝑦𝑖 (𝑡𝑘 )|. So
𝑗=1
𝑇∗
× {∫ 𝑒𝑎𝑖 𝑠 d𝑠} +
0
󵄨
󵄨
𝐽4 ≤ 𝑒−𝑎𝑖 𝑡 ∑ {𝑝𝑖𝑘 󵄨󵄨󵄨𝑦𝑖 (𝑡𝑘 )󵄨󵄨󵄨 𝑒𝑎𝑖 𝑡𝑘 } .
𝜀 𝑛 󵄨󵄨 󵄨󵄨
∑ {󵄨󵄨𝑏 𝑙 󵄨󵄨} .
𝑎𝑖 𝑗=1 󵄨 𝑖𝑗 𝑗 󵄨
0<𝑡𝑘 <𝑡
(28)
Moreover, as lim𝑡 → ∞ 𝑒−𝑎𝑖 𝑡 = 0, we can find a 𝑇 > 0 for the
given 𝜀 such that 𝑡 ≥ 𝑇 implies 𝑒−𝑎𝑖 𝑡 < 𝜀, which leads to
× {∫ 𝑒𝑎𝑖 𝑠 𝑑𝑠} +
0
𝑛
1
󵄨 󵄨 }
∑ {󵄨󵄨󵄨󵄨𝑏𝑖𝑗 𝑙𝑗 󵄨󵄨󵄨󵄨}} ,
𝑎𝑖 𝑗=1
}
As 𝑦𝑖 (𝑡) ∈ H𝑖 , we have lim𝑡 → ∞ 𝑦𝑖 (𝑡) = 0. Then for any
𝜀 > 0, there exists a nonimpulsive point 𝑇𝑖 > 0 such that
𝑠 ≥ 𝑇𝑖 implies |𝑦𝑖 (𝑠)| < 𝜀. It then follows from conditions (i)
and (ii) that
{
󵄨
󵄨
𝐽4 ≤ 𝑒−𝑎𝑖 𝑡 { ∑ {𝑝𝑖𝑘 󵄨󵄨󵄨𝑦𝑖 (𝑡𝑘 )󵄨󵄨󵄨 𝑒𝑎𝑖 𝑡𝑘 }
{0<𝑡𝑘 <𝑇𝑖
{𝑛
󵄨
󵄨
󵄨 󵄨
𝐽2 ≤ 𝜀 { ∑ {󵄨󵄨󵄨󵄨𝑏𝑖𝑗 𝑙𝑗 󵄨󵄨󵄨󵄨 sup 󵄨󵄨󵄨󵄨𝑦𝑗 (𝑠)󵄨󵄨󵄨󵄨}
∗
𝑠∈[0,𝑇
]
{𝑗=1
𝑇∗
(33)
(29)
𝑇𝑖 <𝑡𝑘 <𝑡
≤ 𝑒−𝑎𝑖 𝑡 ∑
𝑡 ≥ max {𝑇∗ , 𝑇} ;
0<𝑡𝑘 <𝑇𝑖
󵄨
󵄨
{𝑝𝑖𝑘 󵄨󵄨󵄨𝑦𝑖 (𝑡𝑘 )󵄨󵄨󵄨 𝑒𝑎𝑖 𝑡𝑘 }
+ 𝑒−𝑎𝑖 𝑡 𝑝𝑖 𝜀 ∑
namely,
}
󵄨
󵄨
{𝑝𝑖𝑘 󵄨󵄨󵄨𝑦𝑖 (𝑡𝑘 )󵄨󵄨󵄨 𝑒𝑎𝑖 𝑡𝑘 }}
}
+ ∑
{𝜇𝑒𝑎𝑖 𝑡𝑘 }
𝑇𝑖 <𝑡𝑘 <𝑡
𝐽2 󳨀→ 0 as 𝑡 󳨀→ ∞.
(30)
On the other hand, since 𝑡 − 𝜏𝑗 (𝑡) → ∞ as 𝑡 → ∞, we
get lim𝑡 → ∞ 𝑦𝑗 (𝑡 − 𝜏𝑗 (𝑡)) = 0. Then for any 𝜀 > 0, there also
exists a 𝑇𝑗󸀠 > 0 such that 𝑠 ≥ 𝑇𝑗󸀠 implies |𝑦𝑗 (𝑠 − 𝜏𝑗 (𝑠))| < 𝜀.
Select 𝑇 = max𝑗∈N {𝑇𝑗󸀠 }. It follows from (A2) that
≤ 𝑒−𝑎𝑖 𝑡 ∑
0<𝑡𝑘 <𝑇𝑖
󵄨
󵄨
{𝑝𝑖𝑘 󵄨󵄨󵄨𝑦𝑖 (𝑡𝑘 )󵄨󵄨󵄨 𝑒𝑎𝑖 𝑡𝑘 }
{
+ 𝑒−𝑎𝑖 𝑡 𝑝𝑖 𝜀 { ∑ {𝑒𝑎𝑖 𝑡𝑟 (𝑡𝑟+1 − 𝑡𝑟 )}
{𝑇𝑖 <𝑡𝑟 <𝑡𝑘
}
+𝜇𝑒𝑎𝑖 𝑡𝑘 }
}
𝑛
𝑡
󵄨󵄨
󵄨
󵄨
𝐽3 ≤ 𝑒−𝑎𝑖 𝑡 ∫ 𝑒𝑎𝑖 𝑠 ∑ {󵄨󵄨󵄨󵄨𝑐𝑖𝑗 𝑘𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑦𝑗 (𝑠 − 𝜏𝑗 (𝑠))󵄨󵄨󵄨󵄨} d𝑠
0
𝑗=1
𝑛
𝑇
󵄨
󵄨󵄨
󵄨
= 𝑒−𝑎𝑖 𝑡 ∫ 𝑒𝑎𝑖 𝑠 ∑ {󵄨󵄨󵄨󵄨𝑐𝑖𝑗 𝑘𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑦𝑗 (𝑠 − 𝜏𝑗 (𝑠))󵄨󵄨󵄨󵄨} d𝑠
0
−𝑎𝑖 𝑡
+𝑒
≤ 𝑒−𝑎𝑖 𝑡 ∑
0<𝑡𝑘 <𝑇𝑖
𝑗=1
𝑡
𝑛
𝑗=1
𝑛
{󵄨
󵄨}
󵄨
󵄨
≤ ∑ {󵄨󵄨󵄨󵄨𝑐𝑖𝑗 𝑘𝑗 󵄨󵄨󵄨󵄨 sup 󵄨󵄨󵄨󵄨𝑦𝑗 (𝑠)󵄨󵄨󵄨󵄨} 𝑒−𝑎𝑖 𝑡 ∫ 𝑒𝑎𝑖 𝑠 d𝑠
0
𝑗=1
𝑠∈[𝜗,𝑇]
}
{
≤ 𝑒−𝑎𝑖 𝑡 ∑
𝑇
−𝑎𝑖 𝑡
≤𝑒
+
+
𝑡
𝑇𝑖
𝑒𝑎𝑖 𝑠 d𝑠 + 𝜇𝑒𝑎𝑖 𝑡 )
󵄨
󵄨
{𝑝𝑖𝑘 󵄨󵄨󵄨𝑦𝑖 (𝑡𝑘 )󵄨󵄨󵄨 𝑒𝑎𝑖 𝑡𝑘 }
𝜀𝑝𝑖
+ 𝑝𝑖 𝜀𝜇,
𝑎𝑖
which produces
𝑇
𝐽4 󳨀→ 0
𝑛
{󵄨
󵄨} 𝑇
󵄨
󵄨
∑ {󵄨󵄨󵄨󵄨𝑐𝑖𝑗 𝑘𝑗 󵄨󵄨󵄨󵄨 sup 󵄨󵄨󵄨󵄨𝑦𝑗 (𝑠)󵄨󵄨󵄨󵄨} ∫ 𝑒𝑎𝑖 𝑠 d𝑠
0
𝑗=1
𝑠∈[𝜗,𝑇]
}
{
as 𝑡 󳨀→ ∞.
(35)
From (30), (32), and (35), we deduce 𝜋(𝑦𝑖 )(𝑡) → 0 as
𝑡 → ∞ for 𝑖 ∈ N. We therefore conclude that 𝜋(𝑦𝑖 )(𝑡) ⊂ H𝑖
(𝑖 ∈ N) which means 𝜋(H) ⊂ H.
𝜀 𝑛 󵄨󵄨
󵄨
∑ {󵄨󵄨󵄨𝑐𝑖𝑗 𝑘𝑗 󵄨󵄨󵄨󵄨} ,
𝑎𝑖 𝑗=1
which results in
𝐽3 󳨀→ 0 as 𝑡 󳨀→ ∞.
0<𝑡𝑘 <𝑇𝑖
(31)
𝑛
𝑡
󵄨
󵄨
+ 𝜀∑ {󵄨󵄨󵄨󵄨𝑐𝑖𝑗 𝑘𝑗 󵄨󵄨󵄨󵄨} 𝑒−𝑎𝑖 𝑡 ∫ 𝑒𝑎𝑖 𝑠 d𝑠
𝑗=1
󵄨
󵄨
{𝑝𝑖𝑘 󵄨󵄨󵄨𝑦𝑖 (𝑡𝑘 )󵄨󵄨󵄨 𝑒𝑎𝑖 𝑡𝑘 }
+ 𝑒−𝑎𝑖 𝑡 𝑝𝑖 𝜀 (∫
󵄨
󵄨󵄨
󵄨
∫ 𝑒 ∑ {󵄨󵄨󵄨󵄨𝑐𝑖𝑗 𝑘𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑦𝑗 (𝑠 − 𝜏𝑗 (𝑠))󵄨󵄨󵄨󵄨} d𝑠
𝑇
𝑎𝑖 𝑠
(34)
(32)
Step 2. We need to prove 𝜋 is contractive. For y =
(𝑦1 (𝑡), . . . , 𝑦𝑛 (𝑡)) ∈ H and 𝑧 = (𝑧1 (𝑡), . . . , 𝑧𝑛 (𝑡)) ∈ H, we
estimate
󵄨
󵄨󵄨
(36)
󵄨󵄨𝜋 (𝑦𝑖 ) (𝑡) − 𝜋 (𝑧𝑖 ) (𝑡)󵄨󵄨󵄨 ≤ 𝐼1 + 𝐼2 + 𝐼3 ,
Abstract and Applied Analysis
7
𝑡
𝑛
󵄨
󵄨
× ∑ { sup 󵄨󵄨󵄨󵄨𝑦𝑗 (𝑠) − 𝑧𝑗 (𝑠)󵄨󵄨󵄨󵄨}
𝑠∈[0,𝑡]
where 𝐼1 = 𝑒−𝑎𝑖 𝑡 ∫0 𝑒𝑎𝑖 𝑠 ∑𝑛𝑗=1 [|𝑏𝑖𝑗 ||𝑓𝑗 (𝑦𝑗 (𝑠)) − 𝑓𝑗 (𝑧𝑗 (𝑠))|]
d𝑠, 𝐼3 = 𝑒−𝑎𝑖 𝑡 ∑0<𝑡𝑘 <𝑡 {𝑒𝑎𝑖 𝑡𝑘 |𝐼𝑖𝑘 (𝑦𝑖 (𝑡𝑘 )) − 𝐼𝑖𝑘 (𝑧𝑖 (𝑡𝑘 ))|}, and 𝐼2 =
𝑗=1
𝑡
𝑒−𝑎𝑖 𝑡 ∫0 𝑒𝑎𝑖 𝑠 ∑𝑛𝑗=1 [|𝑐𝑖𝑗 | |𝑔𝑗 (𝑦𝑗 (𝑠 − 𝜏𝑗 (𝑠))) − 𝑔𝑗 (𝑧𝑗 (𝑠 − 𝜏𝑗 (𝑠)))|]d𝑠.
Note
+
𝑛
𝑡
󵄨
󵄨 󵄨󵄨
𝐼1 ≤ 𝑒−𝑎𝑖 𝑡 ∫ 𝑒𝑎𝑖 𝑠 ∑ [󵄨󵄨󵄨󵄨𝑏𝑖𝑗 𝑙𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑦𝑗 (𝑠) − 𝑧𝑗 (𝑠)󵄨󵄨󵄨󵄨] d𝑠
0
+ 𝑝𝑖 (𝜇 +
𝑗=1
𝑡
󵄨
󵄨 󵄨 𝑛
󵄨
≤ max 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 𝑙𝑗 󵄨󵄨󵄨󵄨 ∑ { sup 󵄨󵄨󵄨󵄨𝑦𝑗 (𝑠) − 𝑧𝑗 (𝑠)󵄨󵄨󵄨󵄨} 𝑒−𝑎𝑖 𝑡 ∫ 𝑒𝑎𝑖 𝑠 d𝑠
𝑗∈N
0
𝑠∈[0,𝑡]
𝑗=1
≤
𝑛
1
󵄨
󵄨 󵄨
󵄨
max 󵄨󵄨󵄨𝑏 𝑙 󵄨󵄨󵄨 ∑ { sup 󵄨󵄨󵄨𝑦 (𝑠) − 𝑧𝑗 (𝑠)󵄨󵄨󵄨󵄨} ,
𝑎𝑖 𝑗∈N 󵄨 𝑖𝑗 𝑗 󵄨 𝑗=1 𝑠∈[0,𝑡] 󵄨 𝑗
(38)
which implies
󵄨
󵄨
sup 󵄨󵄨󵄨𝜋 (𝑦𝑖 ) (𝑡) − 𝜋 (𝑧𝑖 ) (𝑡)󵄨󵄨󵄨
𝑡
≤
0
1
󵄨
󵄨 󵄨 𝑛
󵄨
max 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 𝑙𝑗 󵄨󵄨󵄨󵄨 ∑ { sup 󵄨󵄨󵄨󵄨𝑦𝑗 (𝑠) − 𝑧𝑗 (𝑠)󵄨󵄨󵄨󵄨}
𝑗∈N
𝑎𝑖
𝑠∈[𝜗,𝑇]
𝑗=1
𝑛
󵄨󵄨
󵄨
× ∑ [󵄨󵄨󵄨󵄨𝑐𝑖𝑗 𝑘𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑦𝑗 (𝑠 − 𝜏𝑗 (𝑠))
+
𝑗=1
󵄨
−𝑧𝑗 (𝑠 − 𝜏𝑗 (𝑠))󵄨󵄨󵄨󵄨] d𝑠
𝑛
𝑡
󵄨
󵄨
󵄨
󵄨
≤ max 󵄨󵄨󵄨󵄨𝑐𝑖𝑗 𝑘𝑗 󵄨󵄨󵄨󵄨 ∑ { sup 󵄨󵄨󵄨󵄨𝑦𝑗 (𝑠) − 𝑧𝑗 (𝑠)󵄨󵄨󵄨󵄨} 𝑒−𝑎𝑖 𝑡 ∫ 𝑒𝑎𝑖 𝑠 d𝑠
𝑗∈N
0
𝑠∈[𝜗,𝑡]
𝑗=1
1
󵄨
󵄨
󵄨 𝑛
󵄨
max 󵄨󵄨󵄨󵄨𝑐𝑖𝑗 𝑘𝑗 󵄨󵄨󵄨󵄨 ∑ { sup 󵄨󵄨󵄨󵄨𝑦𝑗 (𝑠) − 𝑧𝑗 (𝑠)󵄨󵄨󵄨󵄨} ,
𝑎𝑖 𝑗∈N
𝑗=1 𝑠∈[𝜗,𝑡]
𝑛
𝑖=1 𝑡∈[−𝜏,𝑇]
In view of condition (iii), we see 𝜋 is a contraction
mapping, and, thus there exists a unique fixed point y∗ (⋅) of
𝜋 in H which means the transposition of y∗ (⋅) is the vectorvalued solution to (1)–(3) and its norm tends to zero as 𝑡 →
∞.
To obtain the asymptotic stability, we still need to prove
that the trivial equilibrium x = 0 is stable. For any 𝜀 > 0,
from condition (iv), we can find 𝛿 satisfying 0 < 𝛿 < 𝜀 such
that 𝛿 + max𝑖∈N {𝜆 𝑖 }𝜀 ≤ 𝜀/√𝑛. Let |𝜑| < 𝛿. According to what
has been discussed above, we know that there exists a unique
solution x(𝑡; 𝑠, 𝜑) = (𝑥1 (𝑡; 𝑠, 𝜑1 ), . . . , 𝑥𝑛 (𝑡; 𝑠, 𝜑𝑛 ))𝑇 to (1)–(3);
moreover,
0<𝑡𝑘 <𝑡
{𝑒𝑎𝑖 𝑡𝑟 (𝑡𝑟+1 − 𝑡𝑟 )} + 𝑒𝑎𝑖 𝑡𝑘 𝜇}
󵄨
󵄨
≤ 𝑝𝑖 sup 󵄨󵄨󵄨𝑦𝑖 (𝑠) − 𝑧𝑖 (𝑠)󵄨󵄨󵄨 𝑒−𝑎𝑖 𝑡
𝑠∈[0,𝑡]
𝑡
× {∫ 𝑒𝑎𝑖 𝑠 d𝑠 + 𝑒𝑎𝑖 𝑡 𝜇}
0
≤ 𝑝𝑖 (𝜇 +
(40)
𝑗=1
𝑠∈[0,𝑡]
0<𝑡𝑟 <𝑡𝑘
𝑛
󵄨
󵄨
≤ 𝜆 ∑ { sup 󵄨󵄨󵄨󵄨𝑦𝑗 (𝑠) − 𝑧𝑗 (𝑠)󵄨󵄨󵄨󵄨} .
𝑠∈[𝜗,𝑇]
∗
󵄨
󵄨
≤ 𝑝𝑖 𝑒−𝑎𝑖 𝑡 sup 󵄨󵄨󵄨𝑦𝑖 (𝑠) − 𝑧𝑖 (𝑠)󵄨󵄨󵄨
×{ ∑
1
󵄨
󵄨
) sup 󵄨󵄨𝑦 (𝑠) − 𝑧𝑖 (𝑠)󵄨󵄨󵄨 .
𝑎𝑖 𝑠∈[𝜗,𝑇] 󵄨 𝑖
󵄨
󵄨
∑ sup 󵄨󵄨󵄨𝜋 (𝑦𝑖 ) (𝑡) − 𝜋 (𝑧𝑖 ) (𝑡)󵄨󵄨󵄨
󵄨
󵄨
sup 󵄨󵄨󵄨𝑦𝑖 (𝑠) − 𝑧𝑖 (𝑠)󵄨󵄨󵄨 ∑ {𝑒𝑎𝑖 𝑡𝑘 𝜇}
𝑠∈[0,𝑡]
(39)
Therefore,
0<𝑡𝑘 <𝑡
≤ 𝑝𝑖 𝑒
1
󵄨
󵄨
󵄨 𝑛
󵄨
max 󵄨󵄨󵄨󵄨𝑐𝑖𝑗 𝑘𝑗 󵄨󵄨󵄨󵄨 ∑ { sup 󵄨󵄨󵄨󵄨𝑦𝑗 (𝑠) − 𝑧𝑗 (𝑠)󵄨󵄨󵄨󵄨}
𝑎𝑖 𝑗∈N
𝑠∈[𝜗,𝑇]
𝑗=1
+ 𝑝𝑖 (𝜇 +
󵄨
󵄨
𝐼3 ≤ 𝑒−𝑎𝑖 𝑡 ∑ {𝑒𝑎𝑖 𝑡𝑘 𝑝𝑖𝑘 󵄨󵄨󵄨𝑦𝑖 (𝑡𝑘 ) − 𝑧𝑖 (𝑡𝑘 )󵄨󵄨󵄨}
−𝑎𝑖 𝑡
1
󵄨
󵄨
) sup 󵄨󵄨𝑦 (𝑠) − 𝑧𝑖 (𝑠)󵄨󵄨󵄨 ,
𝑎𝑖 𝑠∈[0,𝑡] 󵄨 𝑖
𝑡∈[𝜗,𝑇]
𝐼2 ≤ 𝑒−𝑎𝑖 𝑡 ∫ 𝑒𝑎𝑖 𝑠
≤
1
󵄨
󵄨
󵄨 𝑛
󵄨
max 󵄨󵄨󵄨󵄨𝑐𝑖𝑗 𝑘𝑗 󵄨󵄨󵄨󵄨 ∑ { sup 󵄨󵄨󵄨󵄨𝑦𝑗 (𝑠) − 𝑧𝑗 (𝑠)󵄨󵄨󵄨󵄨}
𝑎𝑖 𝑗∈N
𝑗=1 𝑠∈[𝜗,𝑡]
1
󵄨
󵄨
) sup 󵄨󵄨𝑦 (𝑠) − 𝑧𝑖 (𝑠)󵄨󵄨󵄨 .
𝑎𝑖 𝑠∈[0,𝑡] 󵄨 𝑖
𝑥𝑖 (𝑡) = 𝜋 (𝑥𝑖 ) (𝑡) = 𝐽1 + 𝐽2 + 𝐽3 + 𝐽4 ,
(37)
𝑡 ≥ 0;
(41)
𝑡
here 𝐽1 = 𝜑𝑖 (0)𝑒−𝑎𝑖 𝑡 , 𝐽2 = 𝑒−𝑎𝑖 𝑡 ∫0 𝑒𝑎𝑖 𝑠 ∑𝑛𝑗=1 𝑏𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑠))d𝑠,
𝑡
It hence follows from (37) that
󵄨󵄨
󵄨
󵄨󵄨𝜋 (𝑦𝑖 ) (𝑡) − 𝜋 (𝑧𝑖 ) (𝑡)󵄨󵄨󵄨
≤
1
󵄨 󵄨
max 󵄨󵄨󵄨𝑏 𝑙 󵄨󵄨󵄨
𝑎𝑖 𝑗∈N 󵄨 𝑖𝑗 𝑗 󵄨
𝐽3 = 𝑒−𝑎𝑖 𝑡 ∫0 𝑒𝑎𝑖 𝑠 ∑𝑛𝑗=1 𝑐𝑖𝑗 𝑔𝑗 (𝑥𝑗 (𝑠 − 𝜏𝑗 (𝑠)))d𝑠, and 𝐽4 = 𝑒−𝑎𝑖 𝑡
∑0<𝑡𝑘 <𝑡 {𝐼𝑖𝑘 (𝑥𝑖 (𝑡𝑘 ))𝑒𝑎𝑖 𝑡𝑘 }.
Suppose there exists 𝑡∗ > 0 such that ‖x(𝑡∗ ; 𝑠, 𝜑)‖ = 𝜀 and
‖x(𝑡; 𝑠, 𝜑)‖ < 𝜀 as 0 ≤ 𝑡 < 𝑡∗ . It follows from (41) that
󵄨
󵄨󵄨
∗ 󵄨
∗ 󵄨 󵄨
∗ 󵄨 󵄨
∗ 󵄨 󵄨
∗ 󵄨
󵄨󵄨𝑥𝑖 (𝑡 )󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨𝐽1 (𝑡 )󵄨󵄨󵄨 + 󵄨󵄨󵄨𝐽2 (𝑡 )󵄨󵄨󵄨 + 󵄨󵄨󵄨𝐽3 (𝑡 )󵄨󵄨󵄨 + 󵄨󵄨󵄨𝐽4 (𝑡 )󵄨󵄨󵄨 . (42)
8
Abstract and Applied Analysis
As
differentiability of delays, let alone the monotone decreasing
behavior of delays which is necessary in some relevant works.
󵄨󵄨
󵄨󵄨
∗ 󵄨
−𝑎 𝑡∗ 󵄨󵄨
󵄨󵄨𝐽1 (𝑡 )󵄨󵄨󵄨 = 󵄨󵄨󵄨𝜑𝑖 (0) 𝑒 𝑖 󵄨󵄨󵄨 ≤ 𝛿,
󵄨
󵄨
󵄨󵄨
∗ 󵄨
−𝑎 𝑡
󵄨󵄨𝐽2 (𝑡 )󵄨󵄨󵄨 ≤ 𝑒 𝑖
∗
𝑡∗
Provided that 𝐼𝑖𝑘 (⋅) ≡ 0, (1) and (2) will become the
following cellular neural network with infinite delays and
without impulsive effects:
𝑛
󵄨
󵄨
∫ 𝑒𝑎𝑖 𝑠 ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 𝑙𝑗 𝑥𝑗 (𝑠)󵄨󵄨󵄨󵄨 d𝑠
0
𝑗=1
𝑛
d𝑥𝑖 (𝑡)
= − 𝑎𝑖 𝑥𝑖 (𝑡) + ∑ 𝑏𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑡))
d𝑡
𝑗=1
𝜀 𝑛 󵄨 󵄨
< ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 𝑙𝑗 󵄨󵄨󵄨󵄨 ,
𝑎𝑖 𝑗=1
𝑛
𝑡∗
+ ∑𝑐𝑖𝑗 𝑔𝑗 (𝑥𝑗 (𝑡 − 𝜏𝑗 (𝑡))) ,
󵄨󵄨
∗ 󵄨
−𝑎 𝑡∗
𝑎𝑠
󵄨󵄨𝐽3 (𝑡 )󵄨󵄨󵄨 ≤ 𝑒 𝑖 ∫ 𝑒 𝑖
0
𝑗=1
𝑛
󵄨
󵄨
× ∑ 󵄨󵄨󵄨󵄨𝑐𝑖𝑗 𝑘𝑗 𝑥𝑗 (𝑠 − 𝜏𝑗 (𝑠))󵄨󵄨󵄨󵄨 d𝑠
𝑗=1
<
𝑖 ∈ N, 𝑡 ≥ 0,
(43)
𝑛
𝜀
󵄨
󵄨
∑ 󵄨󵄨󵄨𝑐 𝑘 󵄨󵄨󵄨 ,
𝑎𝑖 𝑗=1 󵄨 𝑖𝑗 𝑗 󵄨
󵄨󵄨
∗ 󵄨
−𝑎 𝑡∗
󵄨󵄨𝐽4 (𝑡 )󵄨󵄨󵄨 ≤ 𝑝𝑖 𝑒 𝑖 ∑
0<𝑡𝑘 <𝑡∗
𝑡
∗
∗
󵄨
󵄨
{𝜇 󵄨󵄨󵄨𝑥𝑖 (𝑡𝑘 )󵄨󵄨󵄨 𝑒𝑎𝑖 𝑡𝑘 }
∗
< 𝜀𝑝𝑖 𝑒−𝑎𝑖 𝑡 {∫ 𝑒𝑎𝑖 𝑠 𝑑𝑠 + 𝜇𝑒𝑎𝑖 𝑡 }
0
≤ 𝜀𝑝𝑖 (𝜇 +
(44)
1
),
𝑎𝑖
we obtain |𝑥𝑖 (𝑡∗ )| < 𝛿 + 𝜆 𝑖 𝜀.
2
2
So ‖x(𝑡∗ ; 𝑠, 𝜑)‖ = ∑𝑛𝑖=1 {|𝑥𝑖 (𝑡∗ )| } < ∑𝑛𝑖=1 {|𝛿 + 𝜆 𝑖 𝜀|2 } ≤
2
2
𝑛|𝛿 + max𝑖∈N {𝜆 𝑖 }𝜀| ≤ 𝜀 . This contradicts the assumption of
‖x(𝑡∗ ; 𝑠, 𝜑)‖ = 𝜀. Therefore, ‖x(𝑡; 𝑠, 𝜑)‖ < 𝜀 holds for all 𝑡 ≥ 0.
This completes the proof.
Corollary 5. Assume that conditions (A1)–(A3) hold. Provided that
(i) inf 𝑘=1,2,... {𝑡𝑘 − 𝑡𝑘−1 } ≥ 1,
(ii) there exist constants 𝑝𝑖 such that 𝑝𝑖𝑘 ≤ 𝑝𝑖 for 𝑖 ∈ N
and 𝑘 = 1, 2, . . .,
(iii) ∑𝑛𝑖=1 {(1/𝑎𝑖 )max𝑗∈N |𝑏𝑖𝑗 𝑙𝑗 | + (1/𝑎𝑖 )max𝑗∈N |𝑐𝑖𝑗 𝑘𝑗 |} +
max𝑖∈N {𝑝𝑖 (1 + (1/𝑎𝑖 ))} < 1,
(iv) max𝑖∈N {𝜆󸀠𝑖 } < 1/√𝑛, where 𝜆󸀠𝑖 = (1/𝑎𝑖 ) ∑𝑛𝑗=1 |𝑏𝑖𝑗 𝑙𝑗 | +
(1/𝑎𝑖 ) ∑𝑛𝑗=1 |𝑐𝑖𝑗 𝑘𝑗 | + 𝑝𝑖 (1 + (1/𝑎𝑖 )),
then the trivial equilibrium x = 0 is asymptotically stable.
Proof. Corollary 5 is a direct conclusion by letting 𝜇 = 1 in
Theorem 4.
Remark 6. In Theorem 4, we can see it is the fixed point
theory that deals with the existence and uniqueness of
solution and the asymptotic analysis of trivial equilibrium at
the same time, while Lyapunov method fails to do this.
Remark 7. The presented sufficient conditions in Theorems
4 and Corollary 5 do not require even the boundedness and
where 𝑎𝑖 , 𝑏𝑖𝑗 , 𝑐𝑖𝑗 , 𝑓𝑗 (⋅), 𝑔𝑗 (⋅), 𝜏𝑗 (𝑡), and 𝑥𝑖 (𝑡) are the same as
defined in Section 2. Obviously, (44) also admits a trivial
equilibrium x = 0. From Theorem 4, we reach the following.
Theorem 8. Assume that conditions (A1)-(A2) hold. Provided
that
(i) ∑𝑛𝑖=1 {(1/𝑎𝑖 )max𝑗∈N |𝑏𝑖𝑗 𝑙𝑗 | + (1/𝑎𝑖 )max𝑗∈N |𝑐𝑖𝑗 𝑘𝑗 |} < 1,
(ii) max𝑖∈N {𝜆󸀠󸀠𝑖 } < 1/√𝑛, where 𝜆󸀠󸀠𝑖 = (1/𝑎𝑖 ) ∑𝑛𝑗=1 |𝑏𝑖𝑗 𝑙𝑗 | +
(1/𝑎𝑖 ) ∑𝑛𝑗=1 |𝑐𝑖𝑗 𝑘𝑗 |,
then the trivial equilibrium x = 0 is asymptotically stable.
4. Example
Consider the following two-dimensional impulsive cellular
neural network with infinite delays:
2
d𝑥𝑖 (𝑡)
= − 𝑎𝑖 𝑥𝑖 (𝑡) + ∑ 𝑏𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑡))
d𝑡
𝑗=1
2
+ ∑ 𝑐𝑖𝑗 𝑔𝑗 (𝑥𝑗 (𝑡 − 𝜏𝑗 (𝑡))) ,
𝑗=1
(45)
𝑡 ≥ 0, 𝑡 ≠ 𝑡𝑘 ,
Δ𝑥𝑖 (𝑡𝑘 ) = 𝑥𝑖 (𝑡𝑘 + 0) − 𝑥𝑖 (𝑡𝑘 )
= arctan (0.4𝑥𝑖 (𝑡𝑘 )) ,
𝑘 = 1, 2, . . . ,
with the initial conditions 𝑥1 (𝑠) = cos(𝑠), 𝑥2 (𝑠) = sin(𝑠) on
−1 ≤ 𝑠 ≤ 0, where 𝜏𝑗 (𝑡) = 0.4𝑡 + 1, 𝑎1 = 𝑎2 = 7, 𝑏𝑖𝑗 = 0,
𝑐11 = 3/7, 𝑐12 = 2/7, 𝑐21 = 0, 𝑐22 = 1/7, 𝑓𝑗 (𝑠) = 𝑔𝑗 (𝑠) =
(|𝑠 + 1| − |𝑠 − 1|)/2, and 𝑡𝑘 = 𝑡𝑘−1 + 0.5𝑘.
It is easy to see that 𝜇 = 0.5, 𝑙𝑗 = 𝑘𝑗 = 1, and 𝑝𝑖𝑘 = 0.4. Let
𝑝𝑖 = 0.8 and compute
2
1
1
󵄨
󵄨
∑ { max 󵄨󵄨󵄨󵄨𝑐𝑖𝑗 𝑘𝑗 󵄨󵄨󵄨󵄨} + max {𝑝𝑖 (𝜇 + )} < 1,
𝑖=1,2
𝑎𝑖 𝑗=1,2
𝑎𝑖
𝑖=1
1
,
max {𝜆 𝑖 } <
𝑖∈N
√2
(46)
Abstract and Applied Analysis
where 𝜆 𝑖 = (1/𝑎𝑖 ) ∑𝑛𝑗=1 |𝑐𝑖𝑗 𝑘𝑗 |+𝑝𝑖 (𝜇+(1/𝑎𝑖 )). From Theorem 4,
we conclude that the trivial equilibrium x = 0 of this twodimensional impulsive cellular neural network with infinite
delays is asymptotically stable.
5. Conclusions
This work is devoted to seeking new methods to investigate
the stability of complex neural networks. From what has
been discussed above, we find that the fixed point theory is
feasible. With regard to a class of impulsive cellular neural
networks with infinite delays, we utilize the contraction
mapping principle to deal with the existence and uniqueness
of solution and the asymptotic analysis of trivial equilibrium
at the same time, for which Lyapunov method feels helpless.
Now that there are different kinds of fixed point theorems and
complex neural networks, our future work is to continue the
study on the application of fixed point theory to the stability
analysis of complex neural networks.
Acknowledgment
This work is supported by the National Natural Science
Foundation of China under Grants 60904028, 61174077, and
41105057.
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