Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 942309, 13 pages
http://dx.doi.org/10.1155/2013/942309
Research Article
Almost Periodic Solutions for Neutral-Type BAM Neural
Networks with Delays on Time Scales
Yongkun Li and Li Yang
Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China
Correspondence should be addressed to Yongkun Li; yklie@ynu.edu.cn
Received 5 May 2013; Accepted 17 June 2013
Academic Editor: Sabri Arik
Copyright © 2013 Y. Li and L. Yang. 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.
Using the existence of the exponential dichotomy of linear dynamic equations on time scales, a fixed point theorem and the theory of
calculus on time scales, we obtain some sufficient conditions for the existence and exponential stability of almost periodic solutions
for a class of neutral-type BAM neural networks with delays on time scales. Finally, a numerical example illustrates the feasibility
of our results and also shows that the continuous-time neural network and its discrete-time analogue have the same dynamical
behaviors. The results of this paper are completely new even if the time scale T = R or Z and complementary to the previously
known results.
1. Introduction
The bidirectional associative memory (BAM) neural network, which was introduced by Kosko (see [1]), is a special
recurrent neural network that can store bipolar vector pairs
and is composed of neurons arranged in two layers. The
neurons in one layer are fully interconnected to the neurons
in the other layer, while there are no interconnections among
neurons in the same layer.
Recently, due to its wide range of applications, for
instance, pattern recognition, associative memory, and combinatorial optimization, BAM neural network has received
much attention. For example, in [2–4], some sufficient conditions were obtained for the stability of the equilibrium points
of BAM neural networks; in [5, 6], authors investigated the
periodic solutions of BAM neural networks by using the
continuation theorem of coincidence degree theory; in [7–9],
authors studied the almost periodic solution for BAM neural
networks by using the exponential dichotomy and fixed point
theorems; for other results about BAM neural networks, the
reader may see [10–13] and reference therein.
Since it is natural and important that systems will contain
some information about the derivative of the past state to
further describe and model the dynamics for such complex
neural reactions [14], many authors investigated the dynamical behaviors of neutral-type neural networks with delays
[15–26]. For example, in [21], under the assumptions that
the activation functions satisfy boundedness and Lipschitz
conditions, authors discussed global asymptotic stability of
neutral-type BAM neural networks with delays as follows:
𝑚
𝑚
𝑗=1
𝑗=1
𝑥𝑖󸀠 (𝑡)+ ∑𝑒𝑖𝑗 𝑥𝑗󸀠 (𝑡 − ℎ) = −𝑎𝑖 𝑥𝑖 (𝑡)− ∑𝑠𝑖𝑗 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝜏))+𝐼𝑖 ,
𝑖 = 1, 2, . . . , 𝑚,
𝑚
𝑚
𝑖=1
𝑖=1
𝑦𝑗󸀠 (𝑡)+ ∑]𝑗𝑖 𝑦𝑗󸀠 (𝑡 − 𝑑) = −𝑐𝑗 𝑦𝑗 (𝑡)− ∑𝑡𝑗𝑖 𝑔𝑖 (𝑥𝑖 (𝑡 − 𝛿))+𝐽𝑗 ,
𝑗 = 1, 2, . . . , 𝑚.
(1)
Also, it is well known that the study of dynamical systems
on time scales is now an active area of research. One of
the reasons for this is the fact that the study on time scales
unifies the study of both discrete and continuous processes,
besides many others. The pioneering works in this direction
are [27–30]. The theory of time scales was initiated by Stefan
Hilger in his Ph.D. thesis in 1988, providing a rich theory
that unifies and extends discrete and continuous analysis [31,
32]. The time scales calculus has a tremendous potential for
applications in some mathematical models of real processes
2
Journal of Applied Mathematics
and phenomena studied in physics, chemical technology,
population dynamics, biotechnology and economics, neural
networks, and social sciences.
In fact, both continuous and discrete systems are very
important in implementation and applications. But, it is troublesome to study the the dynamical properties for continuous
and discrete systems, respectively. Therefore, it is meaningful
to study that on time scale which can unify the continuous
and discrete situations (see [13, 31, 33–40]).
Motivated by the above, in this paper, we propose a
neutral-type BAM neural network with delays on time scales
as follows:
𝑥𝑖Δ
𝑚
(𝑡) = − 𝑎𝑖 (𝑡) 𝑥𝑖 (𝑡) + ∑ 𝑎𝑗𝑖 (𝑡) 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝜏𝑗𝑖 (𝑡)))
𝑗=1
𝑚
+ ∑𝑝𝑗𝑖 (𝑡) 𝑔𝑗 (𝑦𝑗Δ (𝑡 − 𝜎𝑗𝑖 (𝑡))) + 𝐼𝑖 (𝑡) ,
𝑗=1
𝑡 ∈ T, 𝑖 = 1, 2, . . . , 𝑛,
𝑦𝑗Δ
𝑛
and |𝜓Δ |0 = max1≤𝑗≤𝑚 (𝜓𝑗Δ (𝑡))+ , and 𝐶1 (T, R) is the set of
continuous functions with continuous derivatives on T, then
X is a Banach space.
The initial condition of (2) is
𝑥𝑖 (𝑠) = 𝜑𝑖 (𝑠) ,
𝑦𝑗 (𝑠) = 𝜓𝑗 (𝑠) ,
𝑠 ∈ [−𝜃, 0]T ,
(3)
where 𝜃 = max{max(𝑖,𝑗) {𝜏𝑗𝑖+ , 𝜎𝑗𝑖+ , 𝜁𝑖𝑗+ , 𝜍𝑖𝑗+ }}, 𝜑𝑖 , 𝜓𝑗 ∈ 𝐶1 ([−𝜃, 0]T ,
R), 𝑖 = 1, 2, . . . , 𝑛, 𝑗 = 1, 2, . . . , 𝑚.
Throughout this paper, we assume that the following
conditions hold:
(H1 ) 𝑎𝑗𝑖 , 𝑝𝑗𝑖 , 𝑏𝑖𝑗 , 𝑞𝑖𝑗 , 𝐼𝑖 , 𝐽𝑗 ∈ 𝐶(T, R), 𝑎𝑖 , 𝑏𝑗 ∈ 𝐶(T, R+ ) with
−𝑎𝑖 , −𝑏𝑗 ∈ R+ , 𝜏𝑗𝑖 , 𝜎𝑗𝑖 , 𝜁𝑖𝑗 , and 𝜍𝑖𝑗 ∈ 𝐶(T, T ∩ R+ ) are
all almost periodic functions, where R+ denotes the
set of positively regressive functions from T to R, 𝑖 =
1, 2, . . . , 𝑛, 𝑗 = 1, 2, . . . , 𝑚;
(H2 ) 𝑓𝑗 , 𝑔𝑗 , ℎ𝑖 , 𝑘𝑖 ∈ 𝐶(R, R), and there exist positive con𝑓
(2)
(𝑡) = − 𝑏𝑗 (𝑡) 𝑦𝑗 (𝑡) + ∑𝑏𝑖𝑗 (𝑡) ℎ𝑖 (𝑥𝑖 (𝑡 − 𝜁𝑖𝑗 (𝑡)))
𝑖=1
𝑛
+ ∑𝑞𝑖𝑗 (𝑡) 𝑘𝑖 (𝑥𝑖Δ (𝑡 − 𝜍𝑖𝑗 (𝑡))) + 𝐽𝑗 (𝑡) ,
𝑖=1
𝑡 ∈ T, 𝑗 = 1, 2, . . . , 𝑚,
where T is an almost periodic time scale; 𝑛, 𝑚 are the number
of neurons in layers; 𝑥𝑖 (𝑡) and 𝑦𝑗 (𝑡) denote the activations of
the 𝑖th neuron and the 𝑗th neuron at time 𝑡; 𝑎𝑖 and 𝑏𝑗 represent
the rate with which the 𝑖th neuron and 𝑗th neuron will reset
their potential to the resting state in isolation when they are
disconnected from the network and the external inputs at
time 𝑡; 𝑓𝑗 , 𝑔𝑗 , ℎ𝑖 , and 𝑘𝑖 are the input-output functions (the
activation functions); 𝜏𝑗𝑖 , 𝜎𝑗𝑖 , 𝜁𝑖𝑗 , and 𝜍𝑖𝑗 are transmission
delays at time 𝑡 and satisfy 𝑡 − 𝜏𝑗𝑖 (𝑡) ∈ T, 𝑡 − 𝜎𝑗𝑖 (𝑡) ∈ T,
𝑡 − 𝜁𝑖𝑗 (𝑡) ∈ T, and 𝑡 − 𝜍𝑖𝑗 (𝑡) ∈ T for 𝑡 ∈ T; 𝑎𝑗𝑖 , 𝑏𝑖𝑗 are
elements of feedback templates at time 𝑡; 𝑝𝑗𝑖 , 𝑞𝑖𝑗 are elements
of feed-forward templates at time 𝑡; and 𝐼𝑖 , 𝐽𝑗 denote biases of
the 𝑖th neuron and the 𝑗th neuron at time 𝑡, 𝑖 = 1, 2, . . . , 𝑛,
𝑗 = 1, 2, . . . , 𝑚.
Our main purpose of this paper is, using the exponential
dichotomy of linear dynamic equations on time scales, a fixed
point theorem and the theory of calculus on time scales,
to study the existence and exponential stability of almost
periodic solutions for (2). Our results of this paper are new
and complementary to the previously known results even if
the time scale T = R or Z.
For convenience, we denote [𝑎, 𝑏]T = {𝑡 | 𝑡 ∈ [𝑎, 𝑏] ∩
T}. For an almost periodic function 𝑓 : T → R, denote
𝑓+ = sup𝑡∈T |𝑓(𝑡)|, 𝑓− = inf 𝑡∈T |𝑓(𝑡)|. Set that X = {𝜙 =
(𝜑1 , 𝜑2 , . . . , 𝜑𝑛 , 𝜓1 , 𝜓2 , . . .,𝜓𝑚 )𝑇 |𝜑𝑖 , 𝜓𝑗 ∈ 𝐶1 (T, R), 𝜑𝑖 , 𝜓𝑗 are
almost periodic functions on T, 𝑖 = 1, 2, . . . , 𝑛, 𝑗 =
1, 2, . . . , 𝑚} with the norm ‖𝜙‖ = max{|𝜑|1 , |𝜓|1 }, where
|𝜑|1 = max{|𝜑|0 , |𝜑Δ |0 }, |𝜓|1 = max{|𝜓|0 , |𝜓Δ |0 }, |𝜑|0 =
max1≤𝑖≤𝑛 𝜑𝑖+ , |𝜑Δ |0 = max1≤𝑖≤𝑛 (𝜑𝑖Δ (𝑡))+ , |𝜓|0 = max1≤𝑗≤𝑚 𝜓𝑗+ ,
𝑔
stants 𝐻𝑗 , 𝐻𝑗 , 𝐻𝑖ℎ , and 𝐻𝑖𝑘 such that
󵄨󵄨
󵄨
󵄨󵄨𝑓𝑗 (𝑢) − 𝑓𝑗 (V)󵄨󵄨󵄨 ≤ 𝐻𝑗𝑓 |𝑢 − V| ,
󵄨
󵄨
󵄨󵄨󵄨𝑔 (𝑢) − 𝑔 (V)󵄨󵄨󵄨 ≤ 𝐻𝑔 |𝑢 − V| ,
󵄨󵄨 𝑗
󵄨󵄨
𝑗
𝑗
󵄨󵄨
󵄨
ℎ
󵄨󵄨ℎ𝑖 (𝑢) − ℎ𝑖 (V)󵄨󵄨󵄨 ≤ 𝐻𝑖 |𝑢 − V| ,
󵄨
󵄨󵄨
𝑘
󵄨󵄨𝑘𝑖 (𝑢) − 𝑘𝑖 (V)󵄨󵄨󵄨 ≤ 𝐻𝑖 |𝑢 − V| ,
(4)
for all 𝑢, V ∈ R, 𝑖 = 1, 2, . . . , 𝑛, 𝑗 = 1, 2, . . . , 𝑚.
This paper is organized as follows. In Section 2, we
introduce some notations and definitions and state some
preliminary results which are needed in later sections. In
Section 3, we establish some sufficient conditions for the
existence of almost periodic solutions of (2). In Section 4, we
prove that the almost periodic solution obtained in Section 3
is exponentially stable. In Section 5, we give an example to
illustrate the feasibility of our results obtained in previous
sections. We draw a conclusion in Section 6.
2. Preliminaries
In this section, we introduce some definitions and state some
preliminary results.
Definition 1 (see [32]). Let T be a nonempty closed subset
(time scale) of R. The forward and backward jump operators
𝜎, 𝜌 : T → T and the graininess 𝜇 : T → R+ are defined,
respectively, by
𝜎 (𝑡) = inf {𝑠 ∈ T : 𝑠 > 𝑡} ,
𝜌 (𝑡) = sup {𝑠 ∈ T : 𝑠 < 𝑡} ,
(5)
𝜇 (𝑡) = 𝜎 (𝑡) − 𝑡.
Lemma 2 (see [32]). Assume that 𝑝, 𝑞 : T → R are two
regressive functions, then
Journal of Applied Mathematics
3
(i) 𝑒0 (𝑡, 𝑠) ≡ 1 and 𝑒𝑝 (𝑡, 𝑡) ≡ 1;
Definition 10 (see [34]). Let T be an almost periodic time
scale. A function 𝑓(𝑡) : T → R𝑛 is said to be almost periodic
on T, if for any 𝜀 > 0, the set
󵄨
󵄨
𝐸 (𝜀, 𝑓) = {𝜏 ∈ Π : 󵄨󵄨󵄨𝑓 (𝑡 + 𝜏) − 𝑓 (𝑡)󵄨󵄨󵄨 < 𝜀, ∀𝑡 ∈ T}
(9)
(ii) 𝑒𝑝 (𝑡, 𝑠) = 1/𝑒𝑝 (𝑠, 𝑡) = 𝑒⊖𝑝 (𝑠, 𝑡);
(iii) 𝑒𝑝 (𝑡, 𝑠)𝑒𝑝 (𝑠, 𝑟) = 𝑒𝑝 (𝑡, 𝑟);
(iv) (𝑒𝑝 (𝑡, 𝑠))Δ = 𝑝(𝑡)𝑒𝑝 (𝑡, 𝑠).
Lemma 3 (see [32]). Let 𝑓, 𝑔 be Δ-differentiable functions on
𝑇. Then
(i) (]1 𝑓 + ]2 𝑔)Δ = ]1 𝑓Δ + ]2 𝑔Δ , for any constants ]1 , ]2 ;
is relatively dense in T; that is, for any 𝜀 > 0, there exists
a constant 𝑙(𝜀) > 0 such that each interval of length 𝑙(𝜀)
contains at least one 𝜏 ∈ 𝐸(𝜀, 𝑓) such that
󵄨
󵄨󵄨
(10)
󵄨󵄨𝑓 (𝑡 + 𝜏) − 𝑓 (𝑡)󵄨󵄨󵄨 < 𝜀, ∀𝑡 ∈ T.
(ii) (𝑓𝑔)Δ (𝑡) = 𝑓Δ (𝑡)𝑔(𝑡) + 𝑓(𝜎(𝑡))𝑔Δ (𝑡) = 𝑓(𝑡)𝑔Δ (𝑡) +
𝑓Δ (𝑡)𝑔(𝜎(𝑡)).
The set 𝐸(𝜀, 𝑓) is called the 𝜀-translation set of 𝑓(𝑡), 𝜏 is
called the 𝜀-translation number of 𝑓(𝑡), and 𝑙(𝜀) is called the
inclusion of 𝐸(𝜀, 𝑓).
Lemma 4 (see [32]). Assume that 𝑝(𝑡) ≥ 0 for 𝑡 ≥ 𝑠. Then
𝑒𝑝 (𝑡, 𝑠) ≥ 1.
Lemma 11 (see [34]). If 𝑓 ∈ 𝐶(T, R𝑛 ) is an almost periodic
function, then 𝑓 is bounded on T.
Definition 5 (see [32]). A function 𝑓 : T → R is positively
regressive if 1 + 𝜇(𝑡)𝑓(𝑡) > 0 for all 𝑡 ∈ T.
Lemma 12 (see [34]). If 𝑓, 𝑔 ∈ (T, R𝑛 ) are almost periodic
functions, then 𝑓 + 𝑔, 𝑓𝑔 are also almost periodic.
Lemma 6 (see [32]). Suppose that 𝑝 ∈ R+ . Then
Definition 13 (see [35]). Letting 𝑋 ∈ R𝑛 and 𝐴(𝑡) be a 𝑛 × 𝑛
matrix-valued function on T, the linear system
(i) 𝑒𝑝 (𝑡, 𝑠) > 0, for all 𝑡, 𝑠 ∈ T;
(ii) if 𝑝(𝑡) ≤ 𝑞(𝑡) for all 𝑡 ≥ 𝑠, 𝑡, 𝑠 ∈ T, then 𝑒𝑝 (𝑡, 𝑠) ≤
𝑒𝑞 (𝑡, 𝑠) for all 𝑡 ≥ 𝑠.
Lemma 7 (see [32]). If 𝑝 ∈ R and 𝑎, 𝑏, 𝑐 ∈ T, then
Δ
𝜎
[𝑒𝑝 (𝑐, ⋅)] = −𝑝[𝑒𝑝 (𝑐, ⋅)] ,
(6)
𝑏
∫ 𝑝 (𝑡) 𝑒𝑝 (𝑐, 𝜎 (𝑡)) Δ𝑡 = 𝑒𝑝 (𝑐, 𝑎) − 𝑒𝑝 (𝑐, 𝑏) .
𝑎
Lemma 8 (see [32]). Let 𝑎 ∈ T 𝑘 , 𝑏 ∈ T, and assume that
𝑓 : T × T 𝑘 → R is continuous at (𝑡, 𝑡), where 𝑡 ∈ T 𝑘 with
𝑡 > 𝑎. Also assume that 𝑓Δ (𝑡, ⋅) is 𝑟𝑑-continuous on [𝑎, 𝜎(𝑡)].
Suppose that for each 𝜀 > 0, there exists a neighborhood 𝑈 of
𝜏 ∈ [𝑎, 𝜎(𝑡)] such that
󵄨
󵄨󵄨
󵄨󵄨𝑓 (𝜎 (𝑡) , 𝜏) − 𝑓 (𝑠, 𝜏) − 𝑓Δ (𝑡, 𝜏) (𝜎 (𝑡) − 𝑠)󵄨󵄨󵄨 ≤ 𝜀 |𝜎 (𝑡) − 𝑠| ,
󵄨
󵄨
∀𝑠 ∈ 𝑈,
(7)
𝑋Δ (𝑡) = 𝐴 (𝑡) 𝑋 (𝑡) ,
where 𝑓 denotes the derivative of 𝑓 with respect to the first
variable. Then,
𝑡
𝑡
𝑏
𝑏
(i) 𝑔(𝑡) := ∫𝑎 𝑓(𝑡, 𝜏)Δ𝜏 implies 𝑔Δ (𝑡) := ∫𝑎 𝑓Δ (𝑡, 𝜏)Δ𝜏 +
𝑓(𝜎(𝑡), 𝑡);
(ii) ℎ(𝑡) := ∫𝑡 𝑓(𝑡, 𝜏)Δ𝜏 implies ℎΔ (𝑡) := ∫𝑡 𝑓Δ (𝑡, 𝜏)Δ𝜏 −
𝑓(𝜎(𝑡), 𝑡).
Definition 9 (see [34]). A time scale T is called an almost
periodic time scale if
Π := {𝜏 ∈ R : 𝑡 + 𝜏 ∈ T, ∀𝑡 ∈ T} ≠ {0} .
(8)
(11)
is said to admit an exponential dichotomy on T if there exist
positive constants 𝑘𝑖 , 𝛼𝑖 , 𝑖 = 1, 2, projection 𝑃, and the
fundamental solution matrix 𝑋(𝑡) of (11) satisfying
󵄨
󵄨󵄨
󵄨󵄨𝑋 (𝑡) 𝑃𝑋−1 (𝑠)󵄨󵄨󵄨 ≤ 𝑘1 𝑒⊖𝛼1 (𝑡, 𝑠) , 𝑠, 𝑡 ∈ T, 𝑡 ≥ 𝑠,
󵄨
󵄨
(12)
󵄨
󵄨󵄨
−1
󵄨󵄨𝑋 (𝑡) (𝐼 − 𝑃) 𝑋 (𝑠)󵄨󵄨󵄨 ≤ 𝑘2 𝑒⊖𝛼2 (𝑠, 𝑡) , 𝑠, 𝑡 ∈ T, 𝑡 ≤ 𝑠,
󵄨
󵄨
where | ⋅ | is a matrix norm on T; that is, if 𝐴 = (𝑎𝑖𝑗 )𝑛×𝑚 , then
2 1/2
we can take |𝐴| = (∑𝑛𝑖=1 ∑𝑚
𝑗=1 |𝑎𝑖𝑗 | ) .
Lemma 14 (see [34]). If (11) admits an exponential dichotomy,
then the following almost periodic system
𝑋Δ (𝑡) = 𝐴 (𝑡) 𝑋 (𝑡) + 𝑔 (𝑡) ,
𝑡∈T
(13)
has an almost periodic solution as follows:
𝑋 (𝑡) = ∫
𝑡
−∞
−∫
𝑋 (𝑡) 𝑃𝑋−1 (𝜎 (𝑠)) 𝑔 (𝑠) Δ𝑠
+∞
𝑡
Δ
𝑡∈T
(14)
−1
𝑋 (𝑡) (𝐼 − 𝑃) 𝑋 (𝜎 (𝑠)) 𝑔 (𝑠) Δ𝑠,
where 𝑋(𝑡) is the fundamental solution matrix of (11).
Lemma 15 (see [35]). If 𝐴(𝑡) is a uniformly bounded 𝑟𝑑continuous 𝑛 × 𝑛 matrix-valued function on T, and there is a
𝛿 > 0 such that
2
𝑛
󵄨 1
󵄨
󵄨
󵄨
󵄨
󵄨󵄨
2
󵄨󵄨𝑎𝑖𝑖 (𝑡)󵄨󵄨󵄨 − ∑ 󵄨󵄨󵄨󵄨𝑎𝑖𝑗 (𝑡)󵄨󵄨󵄨󵄨 − 𝜇 (𝑡) ( ∑ 󵄨󵄨󵄨󵄨𝑎𝑖𝑗 (𝑡)󵄨󵄨󵄨󵄨) − 𝛿 𝜇 (𝑡) ≥ 2𝛿,
2
𝑗 ≠ 𝑖
𝑗=1
𝑡 ∈ T, 𝑖 = 1, 2, . . . , 𝑛,
(15)
then (11) admits an exponential dichotomy on T.
4
Journal of Applied Mathematics
Definition 16. Let 𝑧∗ (𝑡) = (𝑥1∗ (𝑡), 𝑥2∗ (𝑡), . . . , 𝑥𝑛∗ (𝑡), 𝑦1∗ (𝑡),
∗
(𝑡))𝑇 be an almost periodic solution of (2) with
𝑦2∗ (𝑡), . . . , 𝑦𝑚
initial value 𝜙∗ (𝑠) = (𝜑1∗ (𝑠), 𝜑2∗ (𝑠), . . . , 𝜑𝑛∗ (𝑠), 𝜓1∗ (𝑠), 𝜓2∗ (𝑠), . . . ,
∗
(𝑠))𝑇 . If there exists a positive constant 𝜆 with −𝜆 ∈
𝜓𝑚
+
R such that for 𝑡0 ∈ [−𝜃, 0]T , there exists 𝑀 > 1 such that
for an arbitrary solution 𝑧(𝑡) = (𝑥1 (𝑡), 𝑥2 (𝑡), . . . , 𝑥𝑛 (𝑡), 𝑦1 (𝑡),
𝑦2 (𝑡), . . . , 𝑦𝑚 (𝑡))𝑇 of (2) with initial value 𝜙(𝑠) = (𝜑1 (𝑠),
𝜑2 (𝑠), . . . , 𝜑𝑛 (𝑠), 𝜓1 (𝑠), 𝜓2 (𝑠), . . . , 𝜓𝑚 (𝑠))𝑇 satisfies
󵄨󵄨
󵄨
󵄩
∗
∗󵄩
󵄨󵄨𝑧 (𝑡) − 𝑧 (𝑡)󵄨󵄨󵄨1 ≤ 𝑀 󵄩󵄩󵄩𝜙 − 𝜙 󵄩󵄩󵄩 𝑒−𝜆 (𝑡, 𝑡0 ) ,
𝑡 ∈ [−𝜃, ∞)T , 𝑡 ≥ 𝑡0 ,
Proof. For any given 𝜙 ∈ X, we consider the following almost
periodic system:
𝑥𝑖Δ (𝑡) = −𝑎𝑖 (𝑡) 𝑥𝑖 (𝑡) + 𝐹𝑖 (𝑡, 𝜓) + 𝐼𝑖 (𝑡) ,
𝑦𝑗Δ (𝑡) = −𝑏𝑗 (𝑡) 𝑦𝑗 (𝑡) + 𝐺𝑗 (𝑡, 𝜑) + 𝐽𝑗 (𝑡) ,
𝑖 = 1, 2, . . . , 𝑛,
𝑗 = 1, 2, . . . , 𝑚,
(19)
where
(16)
𝑚
𝐹𝑖 (𝑡, 𝜓) = ∑ 𝑎𝑗𝑖 (𝑡) 𝑓𝑗 (𝜓𝑗 (𝑡 − 𝜏𝑗𝑖 (𝑡)))
where |𝑧(𝑡) − 𝑧∗ (𝑡)|1 = max{max1≤𝑖≤𝑛 {|𝑥𝑖 (𝑡) − 𝑥𝑖∗ (𝑡)|, |(𝑥𝑖 (𝑡) −
𝑥𝑖∗ (𝑡))Δ |}, max1≤𝑗≤𝑚 {|𝑦𝑗 (𝑡) − 𝑦𝑗∗ (𝑡)|, |(𝑦𝑗 (𝑡) − 𝑦𝑗∗ (𝑡))Δ |}}, ‖𝜙 −
𝜙∗ ‖ = max{max1≤𝑖≤𝑛 sup𝑠∈[−𝜃,0]T {|𝜑𝑖 (𝑠) − 𝜑𝑖∗ (𝑠)|, |(𝜑𝑖 (𝑠) −
𝜑𝑖∗ (𝑠))Δ |}, max1≤𝑗≤𝑚 sup𝑠∈[−𝜃,0]T {|𝜓𝑗 (𝑠) − 𝜓𝑗∗ (𝑠)|, |(𝜓𝑗 (𝑠) −
𝜓𝑗∗ (𝑠))Δ |}}. Then, the solution 𝑧∗ (𝑡) is said to be exponentially
stable.
𝑗=1
𝑚
+ ∑𝑝𝑗𝑖 (𝑡) 𝑔𝑗 (𝜓𝑗Δ (𝑡 − 𝜎𝑗𝑖 (𝑡))) ,
𝑖 = 1, 2, . . . , 𝑛,
𝑗=1
𝑛
𝐺𝑗 (𝑡, 𝜑) = ∑𝑏𝑖𝑗 (𝑡) ℎ𝑖 (𝜑𝑖 (𝑡 − 𝜁𝑖𝑗 (𝑡)))
𝑖=1
𝑛
+ ∑𝑞𝑖𝑗 (𝑡) 𝑘𝑖 (𝜑𝑖Δ (𝑡 − 𝜍𝑖𝑗 (𝑡))) ,
3. Existence of Almost Periodic Solutions
𝑗 = 1, 2, . . . , 𝑚.
𝑖=1
In this section, we will state and prove the sufficient conditions for the existence of almost periodic solutions of (2).
Let 𝜙0 (𝑡) = (𝜑10 (𝑡), 𝜑20 (𝑡), . . . , 𝜑𝑛0 (𝑡), 𝜓10 (𝑡), 𝜓20 (𝑡), . . . ,
𝑡
0
𝜓𝑚 (𝑡))𝑇 , where 𝜑𝑖0 (𝑡) = ∫−∞ 𝑒−𝑎𝑖 (𝑡, 𝜎(𝑠)) × 𝐼𝑖 (𝑠)Δ𝑠, 𝜓𝑗0 (𝑡) =
(20)
Since (H3 ) holds, it follows from Lemma 15 that the linear
system
𝑡
∫−∞ 𝑒−𝑏𝑗 (𝑡, 𝜎(𝑠))𝐽𝑗 (𝑠)Δ𝑠, 𝑖 = 1, 2, . . . , 𝑛, 𝑗 = 1, 2, . . . , 𝑚,
and let 𝐿 be a constant satisfying 𝐿 ≥ max{‖𝜙0 ‖,
max1≤𝑗≤𝑚 {|𝑓𝑗 (0)|, |𝑔𝑗 (0)|, max1≤𝑖≤𝑛 {|ℎ𝑖 (0)|, 𝑘𝑖 (0)|}. We have
the following theorem.
(H3 ) there exists a positive constant 𝛿 such that
1
𝜇 (𝑡) 𝑐𝑘2 (𝑡) − 𝛿2 𝜇 (𝑡) ≥ 2𝛿,
2
(17)
=
𝑡
(𝑡) = ∫
−∞
(18)
(𝑎𝑖+ /𝑎𝑖− ))𝜃𝑖 },
≤ (1/2),
𝛾𝑗 =
+
. . . , 𝑛, 𝑗 = 1, 2, . . . , 𝑚.
𝑞𝑖𝑗+ 𝐻𝑖𝑘
𝑗 = 1, 2, . . . , 𝑚.
For 𝜙 ∈ X0 , then ‖𝜙‖ ≤ ‖𝜙 − 𝜙0 ‖ + ‖𝜙0 ‖ ≤ 2𝐿. Define the
following operator:
𝑔
+
𝑏𝑖𝑗+
+
𝑞𝑖𝑗+ ),
𝑖 = 1, 2,
Then, (2) has a unique almost periodic solution in X0 =
{𝜙 ∈ X | ‖𝜙 − 𝜙0 ‖ ≤ 𝐿}, 𝜙(𝑡) = (𝜑1 (𝑡), 𝜑2 (𝑡),. . . , 𝜑𝑛 (𝑡),
𝜓1 (𝑡), 𝜓2 (𝑡), . . . , 𝜓𝑚 (𝑡))𝑇 .
(22)
𝑒−𝑏𝑗 (𝑡, 𝜎 (𝑠)) (𝐺𝑗 (𝑠, 𝜑) + 𝐽𝑖 (𝑠)) Δ𝑠,
+
+
+
+
∑𝑚
𝑗=1 (𝑎𝑗𝑖 𝐻𝑗 + 𝑝𝑗𝑖 𝐻𝑗 + 𝑎𝑗𝑖 + 𝑝𝑗𝑖 ),
∑𝑛𝑖=1 (𝑏𝑖𝑗+ 𝐻𝑖ℎ
(21)
𝑒−𝑎𝑖 (𝑡, 𝜎 (𝑠)) (𝐹𝑖 (𝑠, 𝜓) + 𝐼𝑖 (𝑠)) Δ𝑠,
𝑖 = 1, 2, . . . , 𝑛,
𝑘 = 𝑖, 𝑖 = 1, 2, . . . , 𝑛,
𝑘 = 𝑛 + 𝑗, 𝑗 = 1, 2, . . . , 𝑚;
𝑓
𝑡
−∞
𝜓
𝑦𝑗
where 𝜃𝑖
𝑗 = 1, 2, . . . , 𝑚
𝜑
where
(H4 ) max{max1≤𝑖≤𝑛 {(𝜃𝑖 /𝑎𝑖− ),
(1
+
max1≤𝑗≤𝑚 {(𝛾𝑗 /𝑏𝑗− ), (1 + (𝑏𝑗+ /𝑏𝑗− ))𝛾𝑗 }}
𝑦𝑗Δ (𝑡) = −𝑏𝑗 (𝑡) 𝑦𝑗 (𝑡) ,
𝑥𝑖 (𝑡) = ∫
𝑡 ∈ T, 𝑘 = 1, 2, . . . , 𝑛 + 𝑚,
𝑎 (𝑡) ,
𝑐𝑘 (𝑡) = { 𝑖
𝑏𝑗 (𝑡) ,
𝑖 = 1, 2, . . . , 𝑛,
admits an exponential dichotomy on T. Thus, by Lemma 14,
we obtain that (19) has an almost periodic solution, which is
expressed as follows:
Theorem 17. Let (H1 ) and (H2 ) hold. Suppose that
𝑐𝑘 (𝑡) −
𝑥𝑖Δ (𝑡) = −𝑎𝑖 (𝑡) 𝑥𝑖 (𝑡) ,
Φ : X0 󳨀→ X0 ,
𝑇
(𝜑1 , 𝜑2 , . . . , 𝜑𝑛 , 𝜓1 , 𝜓2 , . . . , 𝜓𝑚 )
𝜑
𝜑
𝜓
𝜓
(23)
𝑇
𝜓
󳨀→ (𝑥1 , 𝑥2 , . . . , 𝑥𝑛𝜑 , 𝑦1 , 𝑦2 , . . . , 𝑦𝑚
) .
We will show that Φ is a contraction.
Journal of Applied Mathematics
5
First, we show that for any 𝜙 ∈ X0 , we have Φ𝜙 ∈ X0 .
Note that
󵄨󵄨
󵄨 󵄨󵄨 𝑡
󵄨󵄨
󵄨󵄨(Φ (𝜙 − 𝜙0 )) (𝑡)󵄨󵄨󵄨 = 󵄨󵄨󵄨∫ 𝑒−𝑏 (𝑡, 𝜎 (𝑠)) 𝐺𝑗 (𝑠, 𝜑) Δ𝑠󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨 󵄨󵄨 −∞ 𝑗
󵄨󵄨
𝑛+𝑗
󵄨
󵄨
󵄨󵄨 𝑚
󵄨󵄨 󵄨󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨𝐹𝑖 (𝑠, 𝜓)󵄨󵄨 = 󵄨󵄨 ∑ 𝑎𝑗𝑖 (𝑠) 𝑓𝑗 (𝜓𝑗 (𝑠 − 𝜏𝑗𝑖 (𝑠)))
󵄨󵄨𝑗=1
󵄨
≤∫
−∞
≤∫
󵄨󵄨
𝑚
󵄨󵄨
Δ
+ ∑𝑝𝑗𝑖 (𝑠) 𝑔𝑗 (𝜓𝑗 (𝑠 − 𝜎𝑗𝑖 (𝑠)))󵄨󵄨󵄨󵄨
󵄨󵄨
𝑗=1
󵄨
≤
󵄨 󵄨
󵄨
󵄨
≤ ∑ 𝑎𝑗𝑖+ (󵄨󵄨󵄨󵄨𝑓𝑗 (𝜓𝑗 (𝑠 − 𝜏𝑗𝑖 (𝑠))) − 𝑓𝑗 (0)󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑓𝑗 (0)󵄨󵄨󵄨󵄨)
𝑗=1
󵄨
󵄨
+ ∑ 𝑝𝑗𝑖+ (󵄨󵄨󵄨󵄨𝑔𝑗 (𝜓𝑗Δ (𝑠 − 𝜎𝑗𝑖 (𝑠))) − 𝑔𝑗 (0)󵄨󵄨󵄨󵄨
𝑗=1
󵄨
󵄨
+ 󵄨󵄨󵄨󵄨𝑔𝑗 (0)󵄨󵄨󵄨󵄨 )
𝑚
𝑓
𝑔 󵄨 󵄨
≤ ∑ (𝑎𝑗𝑖+ 𝐻𝑗 + 𝑝𝑗𝑖+ 𝐻𝑗 ) 󵄨󵄨󵄨𝜓󵄨󵄨󵄨1
𝑗=1
+
󵄨
󵄨
󵄨󵄨
󵄨
󵄨󵄨𝑓𝑗 (0)󵄨󵄨󵄨 + 𝑝𝑗𝑖+ 󵄨󵄨󵄨𝑔𝑗 (0)󵄨󵄨󵄨)
󵄨
󵄨
󵄨
󵄨
×∫
𝑓
𝑔 󵄩 󵄩
≤ ∑ (𝑎𝑗𝑖+ 𝐻𝑗 + 𝑝𝑗𝑖+ 𝐻𝑗 ) 󵄩󵄩󵄩𝜙󵄩󵄩󵄩
≤ (1 +
𝑚
󵄨
󵄨
󵄨
󵄨
+ ∑ (𝑎𝑗𝑖+ 󵄨󵄨󵄨󵄨𝑓𝑗 (0)󵄨󵄨󵄨󵄨 + 𝑝𝑗𝑖+ 󵄨󵄨󵄨󵄨𝑔𝑗 (0)󵄨󵄨󵄨󵄨)
≤
+
2𝛾𝑗 𝐿
𝑏𝑗−
(24)
󵄨
󵄨
𝑒−𝑎𝑖 (𝑡, 𝜎 (𝑠)) 󵄨󵄨󵄨𝐹𝑖 (𝑠, 𝜓)󵄨󵄨󵄨 Δ𝑠
𝑎𝑖+
) 2𝜃𝑖 𝐿
𝑎𝑖−
max {
1≤𝑗≤𝑚
𝑛
(25)
𝑗 = 1, 2, . . . , 𝑚.
Therefore, we have
󵄨󵄨
󵄨󵄨
󵄨 󵄨󵄨 𝑡
󵄨󵄨(Φ (𝜙 − 𝜙0 )) (𝑡)󵄨󵄨󵄨 = 󵄨󵄨󵄨󵄨∫ 𝑒−𝑎𝑖 (𝑡, 𝜎 (𝑠)) 𝐹𝑖 (𝑠, 𝜓) Δ𝑠󵄨󵄨󵄨󵄨
󵄨
󵄨 󵄨󵄨 −∞
𝑖
󵄨󵄨
𝑡
−∞
≤∫
𝑡
−∞
≤
󵄨
󵄨
𝑒−𝑎𝑖 (𝑡, 𝜎 (𝑠)) 󵄨󵄨󵄨𝐹𝑖 (𝑠, 𝜓)󵄨󵄨󵄨 Δ𝑠
𝑒−𝑎𝑖− (𝑡, 𝜎 (𝑠)) 2𝜃𝑖 𝐿Δ𝑠
2𝜃𝑖 𝐿
,
𝑎𝑖−
𝑖 = 1, 2, . . . , 𝑛,
(28)
𝑎+
2𝜃 𝐿
󵄩󵄩
󵄩
󵄩󵄩Φ𝜙 − 𝜙0 󵄩󵄩󵄩 ≤ max {max { −𝑖 , (1 + 𝑖− ) 2𝜃𝑖 𝐿} ,
󵄩
󵄩
1≤𝑖≤𝑛
𝑎𝑖
𝑎𝑖
and similarly,
≤∫
(27)
In view of (H4 ), we have
𝑖 = 1, 2, . . . , 𝑛
𝑖=1
𝑗 = 1, 2, . . . , 𝑚.
𝑏+
󵄨
󵄨󵄨
󵄨󵄨(Φ (𝜙 − 𝜙0 ))Δ (𝑡)󵄨󵄨󵄨 ≤ (1 + 𝑗 ) 2𝛾𝑗 𝐿.
󵄨󵄨
󵄨󵄨
𝑛+𝑗
𝑏𝑗−
+ 𝑎𝑗𝑖+ + 𝑝𝑗𝑖+ ) 2𝐿 := 2𝜃𝑖 𝐿,
󵄨󵄨
󵄨
󵄨󵄨𝐺𝑗 (𝑠, 𝜑)󵄨󵄨󵄨 ≤ ∑ (𝑏𝑖𝑗+ 𝐻𝑖ℎ + 𝑞𝑖𝑗+ 𝐻𝑖𝑘 + 𝑏𝑖𝑗+ + 𝑞𝑖𝑗+ ) 2𝐿 := 2𝛾𝑗 𝐿,
󵄨
󵄨
,
and for 𝑗 = 1, 2, . . . , 𝑚,
𝑗=1
𝑔
𝑝𝑗𝑖+ 𝐻𝑗
𝑡
−∞
𝑗=1
𝑓
∑ (𝑎𝑗𝑖+ 𝐻𝑗
𝑗=1
𝑒−𝑏𝑗− (𝑡, 𝜎 (𝑠)) 2𝛾𝑗 𝐿Δ𝑠
(26)
𝑚
𝑚
󵄨
󵄨
𝑒−𝑏𝑗 (𝑡, 𝜎 (𝑠)) 󵄨󵄨󵄨󵄨𝐺𝑗 (𝑠, 𝜑)󵄨󵄨󵄨󵄨 Δ𝑠
On the other hand, for 𝑖 = 1, 2, . . . , 𝑛, we have
󵄨󵄨
󵄨
󵄨󵄨(Φ (𝜙 − 𝜙0 ))Δ (𝑡)󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨
𝑖
󵄨󵄨 𝑡
Δ 󵄨󵄨
󵄨󵄨
󵄨󵄨
= 󵄨󵄨󵄨(∫ 𝑒−𝑎𝑖 (𝑡, 𝜎 (𝑠)) 𝐹𝑖 (𝑠, 𝜓) Δ𝑠) 󵄨󵄨󵄨
󵄨󵄨󵄨 −∞
𝑡 󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨
𝑡
󵄨
󵄨
= 󵄨󵄨󵄨𝐹𝑖 (𝑡, 𝜓)−𝑎𝑖 (𝑡) ∫ 𝑒−𝑎𝑖 (𝑡, 𝜎 (𝑠)) 𝐹𝑖 (𝑠, 𝜓) Δ𝑠󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨
−∞
󵄨
󵄨
󵄨 󵄨
≤ 󵄨󵄨󵄨𝐹𝑖 (𝑡, 𝜓)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑎𝑖 (𝑡)󵄨󵄨󵄨
𝑚
∑ (𝑎𝑗𝑖+
𝑗=1
𝑡
−∞
𝑚
𝑚
𝑡
2𝛾𝑗 𝐿
𝑏𝑗−
, (1 +
𝑏𝑗+
𝑏𝑗−
) 2𝛾𝑗 𝐿}} ≤ 𝐿,
(29)
that is, Φ𝜙 ∈ X0 . Next, we show that Φ is a contraction. For 𝜙 = (𝜑1 , 𝜑2 , . . . , 𝜑𝑛 , 𝜓1 , 𝜓2 , . . . , 𝜓𝑚 )𝑇 , 𝜗 =
(𝜉1 , 𝜉2 , . . . , 𝜉𝑛 , 𝜂1 , 𝜂2 , . . . , 𝜂𝑚 )𝑇 ∈ X0 , for 𝑖 = 1, 2, . . . , 𝑛, denote
by
𝐹𝑖(1) (𝑠, 𝜓, 𝜂)
𝑚
= ∑𝑎𝑗𝑖 (𝑠) (𝑓𝑗 (𝜓𝑗 (𝑠 − 𝜏𝑗𝑖 (𝑠))) − 𝑓𝑗 (𝜂𝑗 (𝑠 − 𝜏𝑗𝑖 (𝑠)))) ,
𝑗=1
𝐹𝑖(2) (𝑠, 𝜓, 𝜂)
𝑚
= ∑𝑝𝑗𝑖 (𝑠) (𝑔𝑗 (𝜓𝑗Δ (𝑠 − 𝜎𝑗𝑖 (𝑠))) − 𝑔𝑗 (𝜂𝑗Δ (𝑠 − 𝜎𝑗𝑖 (𝑠)))) .
𝑗=1
(30)
6
Journal of Applied Mathematics
Then, we have
In a similar way, we have
𝑛
󵄨
󵄨󵄨
󵄨󵄨(Φ𝜙 − Φ𝜁) (𝑡)󵄨󵄨󵄨 ≤ 1 ∑ (𝑏+ 𝐻ℎ + 𝑞+ 𝐻𝑘 ) 󵄨󵄨󵄨𝜑 − 𝜉󵄨󵄨󵄨 ,
𝑖𝑗 𝑖
𝑖𝑗 𝑖 󵄨
󵄨1
󵄨󵄨 𝑏−
󵄨󵄨
𝑛+𝑗
𝑗 𝑖=1
󵄨
󵄨󵄨
󵄨󵄨(Φ𝜙 − Φ𝜁)𝑖 (𝑡)󵄨󵄨󵄨
󵄨󵄨 𝑡
󵄨󵄨
󵄨
󵄨
= 󵄨󵄨󵄨∫ 𝑒−𝑎𝑖 (𝑡, 𝜎 (𝑠)) (𝐹𝑖(1) (𝑠, 𝜓, 𝜂) + 𝐹𝑖(2) (𝑠, 𝜓, 𝜂)) Δ𝑠󵄨󵄨󵄨
󵄨󵄨 −∞
󵄨󵄨
𝑡
≤∫
−∞
𝑗 = 1, 2, . . . , 𝑚,
󵄨
󵄨󵄨
󵄨󵄨(Φ𝜙 − Φ𝜁)Δ (𝑡)󵄨󵄨󵄨 ≤ (1 +
󵄨󵄨
󵄨󵄨
𝑛+𝑗
𝑒−𝑎𝑖 (𝑡, 𝜎 (𝑠))
𝑏𝑗+
)
𝑏𝑗−
𝑛
󵄨
󵄨
× ∑ (𝑏𝑖𝑗+ 𝐻𝑖ℎ + 𝑞𝑖𝑗+ 𝐻𝑖𝑘 ) 󵄨󵄨󵄨𝜑 − 𝜉󵄨󵄨󵄨1 ,
𝑚
󵄨
𝑓󵄨
× ( ∑𝑎𝑗𝑖+ 𝐻𝑗 󵄨󵄨󵄨󵄨𝜓𝑗 (𝑠 − 𝜏𝑗𝑖 (𝑠)) − 𝜂𝑗 (𝑠 − 𝜏𝑗𝑖 (𝑠))󵄨󵄨󵄨󵄨
𝑖=1
𝑗=1
𝑗 = 1, 2, . . . , 𝑚.
𝑚
𝑔 󵄨
+ ∑𝑝𝑗𝑖+ 𝐻𝑗 󵄨󵄨󵄨󵄨𝜓𝑗Δ (𝑠 − 𝜎𝑗𝑖 (𝑠))
(32)
𝑗=1
By (H4 ), we have
󵄨
−𝜂𝑗Δ (𝑠 − 𝜎𝑗𝑖 (𝑠))󵄨󵄨󵄨󵄨 ) Δ𝑠
≤
1 𝑚 + 𝑓
𝑔 󵄨
󵄨
∑ (𝑎 𝐻 + 𝑝𝑗𝑖+ 𝐻𝑗 ) 󵄨󵄨󵄨𝜓 − 𝜂󵄨󵄨󵄨1 ,
𝑎𝑖− 𝑗=1 𝑗𝑖 𝑗
max {max {
1≤𝑖≤𝑛
𝑏𝑗+
1
max { 𝜌𝑗 , (1 + − ) 𝜌𝑗 }} < 1,
1≤𝑗≤𝑚 𝑏−
𝑏𝑗
𝑗
𝑖 = 1, 2, . . . , 𝑛,
󵄨󵄨
󵄨
󵄨󵄨(Φ𝜙 − Φ𝜁)Δ (𝑡)󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨
𝑖
𝑓
󵄨󵄨
󵄨
= 󵄨󵄨󵄨𝐹𝑖(1) (𝑡, 𝜓, 𝜂) + 𝐹𝑖(2) (𝑡, 𝜓, 𝜂) − 𝑎𝑖 (𝑡)
󵄨󵄨
󵄨󵄨
󵄨
× ∫ 𝑒−𝑎𝑖 (𝑡, 𝜎 (𝑠)) (𝐹𝑖(1) (𝑠, 𝜓, 𝜂) + 𝐹𝑖(2) (𝑠, 𝜓, 𝜂)) Δ𝑠󵄨󵄨󵄨
󵄨󵄨
−∞
𝑡
󵄨
󵄨 󵄨
󵄨 󵄨
󵄨
≤ 󵄨󵄨󵄨󵄨𝐹𝑖(1) (𝑡, 𝜓, 𝜂)󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝐹𝑖(2) (𝑡, 𝜓, 𝜂)󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑎𝑖 (𝑡)󵄨󵄨󵄨
𝑡
−∞
󵄨
󵄨
𝑒−𝑎𝑖 (𝑡, 𝜎 (𝑠)) 󵄨󵄨󵄨󵄨𝐹𝑖(1) (𝑠, 𝜓, 𝜂) + 𝐹𝑖(2) (𝑠, 𝜓, 𝜂)󵄨󵄨󵄨󵄨 Δ𝑠
𝑞𝑖𝑗+ 𝐻𝑖𝑘 ), 𝑖 = 1, 2, . . . , 𝑛, 𝑗 = 1, 2, . . . , 𝑚. It implies that ‖Φ𝜙 −
Φ𝜁‖ < ‖𝜑 − 𝜓‖. Hence, Φ is a contraction. Therefore, Φ has
a fixed point in X0 ; that is, (2) has a unique almost periodic
solution in X0 . This completes the proof of Theorem 17.
4. Exponential Stability of Almost
Periodic Solution
In this section, we will study the exponential stability of
almost periodic solution of (2).
(H5 ) 󰜚𝑖 ≥ (𝑎𝑖− − 𝑎𝑖+ 𝑎𝑖− )/(𝑎𝑖+ + 𝑎𝑖− ) and 𝜌𝑗 ≥ (𝑏𝑗− − 𝑏𝑗+ 𝑏𝑗− )/(𝑏𝑗+ +
𝑓
𝑔
+
+
𝑏𝑗− ), where 󰜚𝑖 = ∑𝑚
𝑗=1 (𝑎𝑗𝑖 𝐻𝑗 + 𝑝𝑗𝑖 𝐻𝑗 ) and 𝜌𝑗 =
∑𝑛𝑖=1 (𝑏𝑖𝑗+ 𝐻𝑖ℎ + 𝑞𝑖𝑗+ 𝐻𝑖𝑘 ), 𝑖 = 1, 2, . . . , 𝑛, 𝑗 = 1, 2, . . . , 𝑚.
𝑚
𝑗=1
Then, the almost periodic solution of (2) is exponentially stable.
𝑎𝑖+ 𝑚 + 𝑓
𝑔 󵄨
󵄨
∑ (𝑎 𝐻 + 𝑝𝑗𝑖+ 𝐻𝑗 ) 󵄨󵄨󵄨𝜓 − 𝜂󵄨󵄨󵄨1
𝑎𝑖− 𝑗=1 𝑗𝑖 𝑗
= (1 +
𝑔
Theorem 18. Let (H1 )–(H4 ) hold. Suppose further that
𝑓
𝑔 󵄨
󵄨
≤ ∑ (𝑎𝑗𝑖+ 𝐻𝑗 + 𝑝𝑗𝑖+ 𝐻𝑗 ) 󵄨󵄨󵄨𝜓 − 𝜂󵄨󵄨󵄨1
+
(33)
𝑛
+
+
+ ℎ
where 󰜚𝑖 = ∑𝑚
𝑗=1 (𝑎𝑗𝑖 𝐻𝑗 + 𝑝𝑗𝑖 𝐻𝑗 ) and 𝜌𝑗 = ∑𝑖=1 (𝑏𝑖𝑗 𝐻𝑖 +
󵄨󵄨 𝑡
Δ 󵄨󵄨
󵄨󵄨
󵄨󵄨
= 󵄨󵄨󵄨(∫ 𝑒−𝑎𝑖 (𝑡, 𝜎 (𝑠)) (𝐹𝑖(1) (𝑠, 𝜓, 𝜂) + 𝐹𝑖(2) (𝑠, 𝜓, 𝜂)) Δ𝑠) 󵄨󵄨󵄨
󵄨󵄨 −∞
𝑡 󵄨󵄨󵄨
󵄨
×∫
𝑎𝑖+
1
− 󰜚𝑖 , (1 + − ) 󰜚𝑖 } ,
𝑎𝑖
𝑎𝑖
𝑎𝑖+ 𝑚 + 𝑓
𝑔 󵄨
󵄨
) ∑ (𝑎 𝐻 + 𝑝𝑗𝑖+ 𝐻𝑗 ) 󵄨󵄨󵄨𝜓 − 𝜂󵄨󵄨󵄨1 ,
𝑎𝑖− 𝑗=1 𝑗𝑖 𝑗
𝑖 = 1, 2, . . . , 𝑛.
(31)
Proof. By Theorem 17, (2) has an almost periodic solution
𝜔(𝑡) = (𝛼1 (𝑡), 𝛼2 (𝑡), . . . , 𝛼𝑛 (𝑡), 𝛽1 (𝑡), 𝛽2 (𝑡), . . . , 𝛽𝑚 (𝑡))𝑇 with
initial condition 𝜙∗ (𝑠) = (𝜑1∗ (𝑠), 𝜑2∗ (𝑠), . . . , 𝜑𝑛∗ (𝑠), 𝜓1∗ (𝑠),
∗
(𝑠))𝑇 . Suppose that 𝑧(𝑡) = (𝑥1 (𝑡), 𝑥2 (𝑡), . . . ,
𝜓2∗ (𝑠), . . . , 𝜓𝑚
𝑥𝑛 (𝑡), 𝑦1 (𝑡), 𝑦2 (𝑡), . . . , 𝑦𝑚 (𝑡))𝑇 is an arbitrary solution of (2)
with initial condition 𝜙(𝑠) = (𝜑1 (𝑠), 𝜑2 (𝑠), . . . , 𝜑𝑛 (𝑠), 𝜓1 (𝑠),
𝜓2 (𝑠), . . . , 𝜓𝑚 (𝑠))𝑇 . Denote V(𝑡) = (𝑢1 (𝑡), 𝑢2 (𝑡), . . . , 𝑢𝑛 (𝑡),
]1 (𝑡), ]2 (𝑡), . . . , ]𝑚 (𝑡))𝑇 , where 𝑢𝑖 (𝑡) = 𝑥𝑖 (𝑡) − 𝛼𝑖 (𝑡),
Journal of Applied Mathematics
7
]𝑗 (𝑡) = 𝑦𝑗 (𝑡) − 𝛽𝑗 (𝑡), 𝑖 = 1, 2, . . . , 𝑛, 𝑗 = 1, 2, . . . , 𝑚. Then,
it follows from (2) that
𝑢𝑖Δ (𝑡) = − 𝑎𝑖 (𝑡) 𝑢𝑖 (𝑡)
Similarly, multiplying both sides of (35) by 𝑒−𝑏𝑗 (𝑡, 𝜎(𝑠)) and
integrating on [𝑡0 , 𝑡]T , we get
]𝑗 (𝑡)
𝑚
+ ∑ 𝑎𝑗𝑖 (𝑡) (𝑓𝑗 (𝑦𝑗 (𝑡 − 𝜏𝑗𝑖 (𝑡)))
= ]𝑗 (𝑡0 ) 𝑒−𝑏𝑗 (𝑡, 𝑡0 )
𝑗=1
−𝑓𝑗 (𝛽𝑗 (𝑡 − 𝜏𝑗𝑖 (𝑡))))
𝑚
𝑡
+ ∫ 𝑒−𝑏𝑗 (𝑡, 𝜎 (𝑠))
(34)
𝑡0
+ ∑ 𝑝𝑗𝑖 (𝑡) (𝑔𝑗 (𝑦𝑗Δ (𝑡 − 𝜎𝑗𝑖 (𝑡)))
𝑛
× {∑𝑏𝑖𝑗 (𝑠) (ℎ𝑖 (𝑥𝑖 (𝑠 − 𝜁𝑖𝑗 (𝑠)))
𝑗=1
𝑖=1
−𝑔𝑗 (𝛽𝑗Δ (𝑡 − 𝜎𝑗𝑖 (𝑡)))) ,
−ℎ𝑖 (𝛼𝑖 (𝑠 − 𝜁𝑖𝑗 (𝑠))))
𝑖 = 1, 2, . . . , 𝑛,
𝑛
+ ∑𝑞𝑖𝑗 (𝑠) (𝑘𝑖 (𝑥𝑖Δ (𝑠 − 𝜍𝑖𝑗 (𝑠)))
]Δ𝑗 (𝑡) = − 𝑏𝑗 (𝑡) ]𝑗 (𝑡)
𝑖=1
𝑛
+ ∑𝑏𝑖𝑗 (𝑡) (ℎ𝑖 (𝑥𝑖 (𝑡 − 𝜁𝑖𝑗 (𝑡)))
− 𝑘𝑖 (𝛼𝑖Δ (𝑠 − 𝜍𝑖𝑗 (𝑠)))) } Δ𝑠,
𝑖=1
− ℎ𝑖 (𝛼𝑖 (𝑡 − 𝜁𝑖𝑗 (𝑡))))
𝑛
𝑗 = 1, 2, . . . , 𝑚.
(38)
(35)
+ ∑𝑞𝑖𝑗 (𝑡) (𝑘𝑖 (𝑥𝑖Δ (𝑡 − 𝜍𝑖𝑗 (𝑡)))
For positive constant 𝛼 < min{min1≤𝑖≤𝑛 𝑎𝑖− , min1≤𝑗≤𝑚 𝑏𝑗− } with
−𝛼 ∈ R+ , we have 𝑒⊖𝛼 (𝑡, 𝑡0 ) > 1, where 𝑡 ∈ [−𝜃, 𝑡0 ]T . Take
𝑖=1
− 𝑘𝑖 (𝛼𝑖Δ (𝑡 − 𝜍𝑖𝑗 (𝑡)))) ,
𝑀 > max {𝜖1 , 𝜖2 } ,
𝑗 = 1, 2, . . . , 𝑚.
The initial condition of (34) and (35) is
𝑢𝑖 (𝑠) = 𝜑𝑖 (𝑠) −
]𝑗 (𝑠) = 𝜓𝑗 (𝑠) − 𝜓𝑗∗ (𝑠) ,
𝜑𝑖∗
where
(𝑠) ,
𝑠 ∈ [−𝜃, 0]T ,
(39)
(36)
where 𝑖 = 1, 2, . . . , 𝑛, 𝑗 = 1, 2, . . . , 𝑚.
Multiplying both sides of (34) by 𝑒−𝑎𝑖 (𝑡, 𝜎(𝑠)) and integrating on [𝑡0 , 𝑡]T , where 𝑡0 ∈ [−𝜃, 0]T , we get
𝑢𝑖 (𝑡)
𝜖1 = max {
1≤𝑖≤𝑛
𝑎𝑖+ 𝑎𝑖−
𝑎𝑖−
,
},
𝑎𝑖− − (𝑎𝑖+ + 𝑎𝑖− ) 󰜚𝑖 𝑎𝑖− − 󰜚𝑖
𝑏𝑗− }
𝑏𝑗+ 𝑏𝑗−
{
𝜖2 = max {
, −
}.
1≤𝑗≤𝑚
𝑏𝑗− − (𝑏𝑗+ + 𝑏𝑗− ) 𝜌𝑗 𝑏𝑗 − 𝜌𝑗
{
}
(40)
By (H4 ), we have
= 𝑢𝑖 (𝑡0 ) 𝑒−𝑎𝑖 (𝑡, 𝑡0 )
𝑎𝑖− > (𝑎𝑖+ + 𝑎𝑖− ) 󰜚𝑖 ,
𝑡
+ ∫ 𝑒−𝑎𝑖 (𝑡, 𝜎 (𝑠))
𝑏𝑗− > (𝑏𝑗+ + 𝑏𝑗− ) 𝜌𝑗 ,
𝑡0
{𝑚
× { ∑𝑎𝑗𝑖 (𝑠) (𝑓𝑗 (𝑦𝑗 (𝑠 − 𝜏𝑗𝑖 (𝑠)))
{𝑗=1
− 𝑓𝑗 (𝛽𝑗 (𝑠 − 𝜏𝑗𝑖 (𝑠))))
𝑚
+ ∑𝑝𝑗𝑖 (𝑠) (𝑔𝑗 (𝑦𝑗Δ (𝑠 − 𝜎𝑗𝑖 (𝑠)))
𝑗=1
}
− 𝑔𝑗 (𝛽𝑗Δ (𝑠 − 𝜎𝑗𝑖 (𝑠)))) } Δ𝑠,
}
𝑖 = 1, 2, . . . , 𝑛.
(37)
𝑖 = 1, 2, . . . , 𝑛,
𝑎𝑖− > 󰜚𝑖 ,
𝑏𝑗− > 𝜌𝑗 ,
(41)
𝑗 = 1, 2, . . . , 𝑚.
In view of (H5 ), we have 𝑀 > 1. Hence, it is obvious that
󵄩
󵄩
|V (𝑡)|1 ≤ 𝑀𝑒⊖𝛼 (𝑡, 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩 ,
∀𝑡 ∈ [−𝜃, 𝑡0 ]T .
(42)
∀𝑡 ∈ (𝑡0 , +∞)T .
(43)
We claim that
󵄩
󵄩
|V (𝑡)|1 ≤ 𝑀𝑒⊖𝛼 (𝑡, 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩 ,
To prove this claim, we show that for any 𝑝 > 1, the following
inequality holds:
󵄩
󵄩
|V (𝑡)|1 < 𝑝𝑀𝑒⊖𝛼 (𝑡, 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩 ,
∀𝑡 ∈ (𝑡0 , +∞)T , (44)
8
Journal of Applied Mathematics
which means that, for 𝑖 = 1, 2, . . . , 𝑛, we have
󵄨
󵄩
󵄨󵄨
∗󵄩
󵄨󵄨𝑢𝑖 (𝑡)󵄨󵄨󵄨 < 𝑝𝑀𝑒⊖𝛼 (𝑡, 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙 󵄩󵄩󵄩 ,
󵄨󵄨 Δ 󵄨󵄨
󵄩
∗󵄩
󵄨󵄨󵄨𝑢𝑖 (𝑡)󵄨󵄨󵄨 < 𝑝𝑀𝑒⊖𝛼 (𝑡, 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙 󵄩󵄩󵄩 ,
∀𝑡 ∈ (𝑡0 , +∞)T ,
{𝑚
× { ∑𝑎𝑗𝑖0 (𝑠) (𝑓𝑗 (𝑦𝑗 (𝑠 − 𝜏𝑗𝑖0 (𝑠)))
{𝑗=1
− 𝑓𝑗 (𝛽𝑗 (𝑠 − 𝜏𝑗𝑖0 (𝑠))))
(45)
𝑚
∀𝑡 ∈ (𝑡0 , +∞)T , (46)
+ ∑𝑝𝑗𝑖0 (𝑠) (𝑔𝑗 (𝑦𝑗Δ (𝑠 − 𝜎𝑗𝑖0 (𝑠)))
𝑗=1
and for 𝑗 = 1, 2, . . . , 𝑚, we have
󵄨
󵄨󵄨
󵄨󵄨]𝑗 (𝑡)󵄨󵄨󵄨 < 𝑝𝑀𝑒⊖𝛼 (𝑡, 𝑡0 ) 󵄩󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩󵄩 ,
󵄨
󵄨
󵄨󵄨 Δ 󵄨󵄨
󵄨󵄨]𝑗 (𝑡)󵄨󵄨 < 𝑝𝑀𝑒⊖𝛼 (𝑡, 𝑡0 ) 󵄩󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩󵄩 ,
󵄨
󵄨
−
∀𝑡 ∈ (𝑡0 , +∞)T , (47)
∀𝑡 ∈ (𝑡0 , +∞)T . (48)
󵄩
󵄩
≤ 𝑒−𝑎𝑖 (𝑡1 , 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩
󵄨󵄨
} 󵄨󵄨󵄨
(𝑠 − 𝜎𝑗𝑖0 (𝑠)))) } Δ𝑠󵄨󵄨󵄨
󵄨󵄨
} 󵄨󵄨
0
𝑡1
+ ∫ 𝑒−𝑎𝑖 (𝑡1 , 𝜎 (𝑠))
𝑡0
0
𝑚
𝑓
× ( ∑𝑎𝑗𝑖+ 0 𝐻𝑗
𝑗=1
By way of contradiction, assume that (44) does not hold.
Firstly, we consider the following four cases.
Case 1. Equation (45) is not true and (46)–(48) are all true.
Then, there exists 𝑡1 ∈ (𝑡0 , +∞)T and 𝑖0 ∈ {1, 2, . . . , 𝑛} such
that
𝑔𝑗 (𝛽𝑗Δ
󵄨
󵄨󵄨
󵄨󵄨]𝑗 (𝑠 − 𝜏𝑗𝑖0 (𝑠))󵄨󵄨󵄨
󵄨
󵄨
𝑛
󵄨
𝑔󵄨
+ ∑𝑝𝑗𝑖+ 0 𝐻𝑗 󵄨󵄨󵄨󵄨]Δ𝑗 (𝑠 − 𝜎𝑗𝑖0 (𝑠))󵄨󵄨󵄨󵄨) Δ𝑠
𝑗=1
󵄩
󵄩
≤ 𝑒−𝑎𝑖 (𝑡1 , 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩
0
󵄨󵄨
󵄨
󵄨󵄨𝑢𝑖0 (𝑡1 )󵄨󵄨󵄨 ≥ 𝑝𝑀𝑒⊖𝛼 (𝑡1 , 𝑡0 ) 󵄩󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩󵄩 ,
󵄨
󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨𝑢𝑖0 (𝑡)󵄨󵄨 < 𝑝𝑀𝑒⊖𝛼 (𝑡, 𝑡0 ) 󵄩󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩󵄩 , 𝑡 ∈ (𝑡0 , 𝑡1 )T ,
󵄨
󵄨
󵄨
󵄨󵄨
󵄩
∗󵄩
󵄨󵄨𝑢𝑙 (𝑡)󵄨󵄨󵄨 < 𝑝𝑀𝑒⊖𝛼 (𝑡, 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙 󵄩󵄩󵄩 ,
𝑡1
+ ∫ 𝑒−𝑎𝑖 (𝑡1 , 𝜎 (𝑠))
𝑡0
0
𝑚
(49)
𝑓
󵄩
󵄩
× ( ∑ 𝑎𝑗𝑖+ 0 𝐻𝑗 𝛿1 𝑝𝑀𝑒⊖𝛼 (𝑠 − 𝜏𝑗𝑖0 (𝑠) , 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩
𝑗=1
𝑚
for 𝑙 ≠ 𝑖0 , 𝑡 ∈ (𝑡0 , 𝑡1 ]T , 𝑙 = 1, 2, . . . , 𝑛.
𝑔
󵄩
󵄩
+∑𝑝𝑗𝑖+ 0 𝐻𝑗 𝛿1 𝑝𝑀𝑒⊖𝛼 (𝑠−𝜎𝑗𝑖0 (𝑠) , 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩) Δ𝑠
𝑗=1
󵄩
󵄩
= 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩 𝑒⊖𝛼 (𝑡1 , 𝑡0 ) 𝑒−𝑎𝑖 ⊕𝛼 (𝑡1 , 𝑡0 )
0
󵄩
󵄩
󵄩
+ 𝛿1 𝑝𝑀𝑒⊖𝛼 (𝑡1 , 𝑡0 ) 󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩
Therefore, there must be a constant 𝛿1 ≥ 1 such that
󵄨󵄨
󵄨
󵄨󵄨𝑢𝑖0 (𝑡1 )󵄨󵄨󵄨 = 𝛿1 𝑝𝑀𝑒⊖𝛼 (𝑡1 , 𝑡0 ) 󵄩󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩󵄩 ,
󵄨
󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨𝑢𝑖0 (𝑡)󵄨󵄨 < 𝛿1 𝑝𝑀𝑒⊖𝛼 (𝑡, 𝑡0 ) 󵄩󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩󵄩 , 𝑡 ∈ (𝑡0 , 𝑡1 )T ,
󵄨
󵄨
󵄨
󵄨󵄨
󵄩
∗󵄩
󵄨󵄨𝑢𝑙 (𝑡)󵄨󵄨󵄨 < 𝛿1 𝑝𝑀𝑒⊖𝛼 (𝑡, 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙 󵄩󵄩󵄩 ,
𝑡1
× ∫ 𝑒−𝑎𝑖 (𝑡1 , 𝜎 (𝑠))
𝑡0
(50)
0
𝑚
𝑓
× ( ∑ 𝑎𝑗𝑖+ 0 𝐻𝑗 𝑒⊖𝛼 (𝑠 − 𝜏𝑗𝑖0 (𝑠) , 𝑡1 )
𝑗=1
for 𝑙 ≠ 𝑖0 , 𝑡 ∈ (𝑡0 , 𝑡1 ]T , 𝑙 = 1, 2, . . . , 𝑛.
𝑚
𝑔
+ ∑𝑝𝑗𝑖+ 0 𝐻𝑗 𝑒⊖𝛼 (𝑠 − 𝜎𝑗𝑖0 (𝑠) , 𝑡1 )) Δ𝑠
𝑗=1
Note that, in view of (37), we have
󵄩
󵄩
≤ 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩 𝑒⊖𝛼 (𝑡1 , 𝑡0 ) 𝑒−𝑎𝑖− +𝛼 (𝑡1 , 𝑡0 )
0
󵄨󵄨
󵄨
󵄨󵄨𝑢𝑖0 (𝑡1 )󵄨󵄨󵄨
󵄨
󵄨
󵄨󵄨
𝑡1
󵄨󵄨
= 󵄨󵄨󵄨󵄨𝑢𝑖0 (𝑡0 ) 𝑒−𝑎𝑖 (𝑡1 , 𝑡0 ) + ∫ 𝑒−𝑎𝑖 (𝑡1 , 𝜎 (𝑠))
0
0
󵄨󵄨
𝑡0
󵄨
󵄩
󵄩
+ 𝛿1 𝑝𝑀𝑒⊖𝛼 (𝑡1 , 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩
𝑡1
× ∫ 𝑒−𝑎𝑖− (𝑡1 , 𝜎 (𝑠))
𝑡0
0
Journal of Applied Mathematics
𝑚
9
𝑚
𝑓
Hence, there must be a constant 𝛿2 ≥ 1 such that
𝑔
𝜏} + ∑𝑝𝑗𝑖+ 0 𝐻𝑗 exp {−𝛼̃
𝜎}) Δ𝑠
× ( ∑𝑎𝑗𝑖+ 0 𝐻𝑗 exp {−𝛼̃
𝑗=1
𝑗=1
󵄨󵄨 Δ
󵄨
󵄨󵄨𝑢𝑖 (𝑡2 )󵄨󵄨󵄨 = 𝛿2 𝑝𝑀𝑒⊖𝛼 (𝑡2 , 𝑡0 ) 󵄩󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩󵄩 ,
󵄨 1
󵄨
󵄩
󵄩 1
󵄩
󵄩
≤ 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩 𝑒⊖𝛼 (𝑡1 , 𝑡0 ) + 𝛿1 𝑝𝑀𝑒⊖𝛼 (𝑡1 , 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩 −
−𝑎𝑖0
𝑡1
×∫
𝑡0
(−𝑎𝑖−0 ) 𝑒−𝑎𝑖−
0
𝑚
󵄨󵄨 Δ 󵄨󵄨
󵄨󵄨𝑢𝑖 (𝑡)󵄨󵄨 < 𝛿2 𝑝𝑀𝑒⊖𝛼 (𝑡, 𝑡0 ) 󵄩󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩󵄩 ,
󵄨 1 󵄨
(𝑡1 , 𝜎 (𝑠)) Δ𝑠
󵄨󵄨 Δ 󵄨󵄨
󵄨󵄨𝑢𝑙 (𝑡)󵄨󵄨 < 𝛿2 𝑝𝑀𝑒⊖𝛼 (𝑡, 𝑡0 ) 󵄩󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩󵄩 ,
󵄨
󵄨
𝑚
𝑓
𝑔
× ( ∑𝑎𝑗𝑖+ 0 𝐻𝑗 exp {−𝛼̃
𝜏} + ∑𝑝𝑗𝑖+ 0 𝐻𝑗 exp {−𝛼̃
𝜎})
𝑗=1
Note that, in view of (37), we have
󵄩
󵄩
× 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩 (𝑒−𝑎𝑖− (𝑡1 , 𝑡0 ) − 1)
𝑢𝑖Δ (𝑡)
0
×
𝑓
( ∑𝑎𝑗𝑖+ 0 𝐻𝑗
𝑗=1
exp {−𝛼̃
𝜏} +
𝑚
𝑔
∑𝑝𝑗𝑖+ 0 𝐻𝑗
𝑗=1
= −𝑎𝑖 (𝑡) 𝑢𝑖 (𝑡0 ) 𝑒−𝑎𝑖 (𝑡, 𝑡0 )
exp {−𝛼̃
𝜎})
𝑡
+ ∫ (−𝑎𝑖 (𝑡)) 𝑒−𝑎𝑖 (𝑡, 𝜎 (𝑠))
𝑡0
󵄩
󵄩
< 𝛿1 𝑝𝑀 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩 𝑒⊖𝛼 (𝑡1 , 𝑡0 )
{𝑚
× { ∑𝑎𝑗𝑖 (𝑠) (𝑓𝑗 (𝑦𝑗 (𝑠 − 𝜏𝑗𝑖 (𝑠)))
{𝑗=1
𝑚
1
1
𝑓
×[
( ∑ 𝑎+ 𝐻 exp {−𝛼̃
𝜏}
+
𝛿1 𝑝𝑀 𝑎𝑖−0 𝑗=1 𝑗𝑖0 𝑗
[
𝑚
− 𝑓𝑗 (𝛽𝑗 (𝑠 − 𝜏𝑗𝑖 (𝑠))))
𝑚
𝑔
+ ∑𝑝𝑗𝑖+ 0 𝐻𝑗 exp {−𝛼̃
𝜎})]
𝑗=1
]
+ ∑𝑝𝑗𝑖 (𝑠) (𝑔𝑗 (𝑦𝑗Δ (𝑠 − 𝜎𝑗𝑖 (𝑠)))
𝑗=1
󵄩
󵄩
< 𝛿1 𝑝𝑀 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩 𝑒⊖𝛼 (𝑡1 , 𝑡0 )
×(
(53)
for 𝑙 ≠ 𝑖1 , 𝑡 ∈ (𝑡0 , 𝑡2 ]T , 𝑙 = 1, 2, . . . , 𝑛.
𝑗=1
1
󵄩
󵄩
= 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩 𝑒⊖𝛼 (𝑡1 , 𝑡0 ) − − 𝛿1 𝑝𝑀𝑒⊖(−𝛼) (𝑡1 , 𝑡0 )
𝑎𝑖0
𝑚
𝑡 ∈ (𝑡0 , 𝑡2 )T ,
}
−𝑔𝑗 (𝛽𝑗Δ (𝑠 − 𝜎𝑗𝑖 (𝑠)))) } Δ𝑠 (54)
}
1
1 𝑚
𝑓
𝑔
+ − ∑ (𝑎𝑗𝑖+ 0 𝐻𝑗 + 𝑝𝑗𝑖+ 0 𝐻𝑗 ))
𝑀 𝑎𝑖0 𝑗=1
󵄩
󵄩
< 𝛿1 𝑝𝑀𝑒⊖𝛼 (𝑡1 , 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩 ,
+ 𝑒−𝑎𝑖 (𝜎 (𝑡) , 𝜎 (𝑡))
(51)
where 𝜏̃ = min(𝑖,𝑗) 𝜏𝑗𝑖− , 𝜎̃ = min(𝑖,𝑗) 𝜎𝑗𝑖− . In the proof, we use the
inequality 𝑒−𝑎𝑖− (𝑡1 , 𝑡0 ) < 1. Thus, we get a contradiction.
0
Case 2. Equation (46) is not true and (45), (47), and (48) are
all true. Then, there exists 𝑡2 ∈ (𝑡0 , +∞)T and 𝑖1 ∈ {1, 2, . . . , 𝑛}
such that
󵄨󵄨 Δ
󵄨
󵄨󵄨𝑢𝑖 (𝑡2 )󵄨󵄨󵄨 ≥ 𝑝𝑀𝑒⊖𝛼 (𝑡2 , 𝑡0 ) 󵄩󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩󵄩 ,
󵄨 1
󵄨
󵄨󵄨 Δ 󵄨󵄨
󵄨󵄨𝑢𝑖 (𝑡)󵄨󵄨 < 𝑝𝑀𝑒⊖𝛼 (𝑡, 𝑡0 ) 󵄩󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩󵄩 , 𝑡 ∈ (𝑡0 , 𝑡2 )T ,
󵄨 1 󵄨
󵄨󵄨 Δ 󵄨󵄨
󵄨󵄨𝑢𝑙 (𝑡)󵄨󵄨 < 𝑝𝑀𝑒⊖𝛼 (𝑡, 𝑡0 ) 󵄩󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩󵄩 ,
󵄨
󵄨
for 𝑙 ≠ 𝑖1 , 𝑡 ∈ (𝑡0 , 𝑡2 ]T , 𝑙 = 1, 2, . . . , 𝑛.
{𝑚
× {∑ 𝑎𝑗𝑖 (𝑡) (𝑓𝑗 (𝑦𝑗 (𝑡 − 𝜏𝑗𝑖 (𝑡)))
{𝑗=1
−𝑓𝑗 (𝛽𝑗 (𝑡 − 𝜏𝑗𝑖 (𝑡))))
𝑚
+ ∑ 𝑝𝑗𝑖 (𝑡) (𝑔𝑗 (𝑦𝑗Δ (𝑡 − 𝜎𝑗𝑖 (𝑡)))
𝑗=1
}
−𝑔𝑗 (𝛽𝑗Δ (𝑡 − 𝜎𝑗𝑖 (𝑡)))) } ,
}
𝑖 = 1, 2, . . . , 𝑛.
Thus, we have
(52)
󵄨
󵄨󵄨 Δ
󵄨󵄨𝑢𝑖 (𝑡2 )󵄨󵄨󵄨
󵄨
󵄨 1
󵄩
󵄩
≤ 𝑎𝑖+1 𝑒−𝑎𝑖− (𝑡2 , 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩
1
10
Journal of Applied Mathematics
󵄩
󵄩
+ 𝑎𝑖+1 𝛿2 𝑝𝑀𝑒⊖𝛼 (𝑡2 , 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩
𝑡2
+ 𝑎𝑖+1 ∫ 𝑒−𝑎𝑖 (𝑡2 , 𝜎 (𝑠))
1
𝑡0
𝑡2
× ∫ 𝑒−𝑎𝑖− (𝑡2 , 𝜎 (𝑠))
𝑚
󵄨
𝑓󵄨
× ( ∑ 𝑎𝑗𝑖+ 1 𝐻𝑗 󵄨󵄨󵄨󵄨]𝑗 (𝑠 − 𝜏𝑗𝑖1 (𝑠))󵄨󵄨󵄨󵄨
𝑗=1
𝑚
󵄨
𝑔󵄨
+ ∑ 𝑝𝑗𝑖+ 1 𝐻𝑗 󵄨󵄨󵄨󵄨]Δ𝑗 (𝑠 − 𝜎𝑗𝑖1 (𝑠))󵄨󵄨󵄨󵄨) Δ𝑠
𝑗=1
𝑚
󵄨
𝑓󵄨
+ ∑𝑎𝑗𝑖+ 1 𝐻𝑗 󵄨󵄨󵄨󵄨]𝑗 (𝑡2 − 𝜏𝑗𝑖1 (𝑡2 ))󵄨󵄨󵄨󵄨
𝑗=1
𝑓
󵄩
󵄩
󵄩
󵄩
≤ 𝑎𝑖+1 𝑒−𝑎𝑖− (𝑡2 , 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩 + 𝑎𝑖+1 𝛿2 𝑝𝑀𝑒⊖𝛼 (𝑡2 , 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩
1
𝑗=1
󵄩
󵄩
(𝑡2 , 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩
×
𝑡2
𝑡2
1
−
−
− ∫ (−𝑎𝑖1 ) 𝑒−𝑎𝑖1 (𝑡2 , 𝜎 (𝑠)) Δ𝑠
−𝑎𝑖1 𝑡0
𝑚
+ 𝑎𝑖+1 ∫ 𝑒−𝑎𝑖 (𝑡2 , 𝜎 (𝑠))
𝑗=1
𝑚
𝑓
󵄩
󵄩
×( ∑𝑎𝑗𝑖+ 1 𝐻𝑗 𝛿2 𝑝𝑀𝑒⊖(−𝛼) (𝑠−𝜏𝑗𝑖1 (𝑠) , 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩
𝑗=1
𝑔
+ ∑𝑝𝑗𝑖+ 1 𝐻𝑗 𝛿2 𝑝𝑀𝑒⊖𝛼 (𝑠 − 𝜎𝑗𝑖1 (𝑠) , 𝑡0 )
𝑗=1
𝑚
𝑗=1
󵄩
󵄩
= 𝑎𝑖+1 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩 𝑒⊖𝛼 (𝑡2 , 𝑡0 ) 𝑒−𝑎𝑖− ⊕𝛼 (𝑡2 , 𝑡0 )
1
−
󵄩
󵄩
× 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩 ) Δ𝑠
𝑎𝑖+1
𝑎𝑖−1
󵄩
󵄩
𝛿2 𝑝𝑀𝑒⊖𝛼 (𝑡2 , 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩 (𝑒−𝑎𝑖− (𝑡2 , 𝑡0 ) − 1)
1
𝑚
𝑚
𝑓
󵄩
󵄩
+ 𝛿2 𝑝𝑀𝑒⊖𝛼 (𝑡2 − 𝜏𝑗𝑖1 (𝑡2 ) , 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩 ∑𝑎𝑗𝑖+ 1 𝐻𝑗
𝑗=1
𝑚
𝑔
󵄩
󵄩
+ 𝛿2 𝑝𝑀𝑒⊖𝛼 (𝑡2 − 𝜎𝑗𝑖1 (𝑡2 ) , 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩 ∑ 𝑝𝑗𝑖+ 1 𝐻𝑗
𝑗=1
󵄩
󵄩
≤ 𝑎𝑖+1 𝑒−𝑎𝑖− (𝑡2 , 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩
1
󵄩
󵄩
(𝑡2 , 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩
𝑡2
× ∫ 𝑒−𝑎𝑖 (𝑡2 , 𝜎 (𝑠))
1
𝑡0
𝑚
𝑓
𝑗=1
𝑔
+ ∑𝑝𝑗𝑖+ 1 𝐻𝑗 𝑒⊖𝛼 (𝑠 − 𝜎𝑗𝑖1 (𝑠) , 𝑡2 ) ) Δ𝑠
𝑗=1
󵄩
󵄩
+ 𝛿2 𝑝𝑀𝑒⊖𝛼 (𝑡2 , 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩
𝑚
𝑓
× ∑ (𝑎𝑗𝑖+ 1 𝐻𝑗 𝑒⊖𝛼 (𝑡2 − 𝜏𝑗𝑖1 (𝑡2 ) , 𝑡2 )
𝑗=1
𝑔
≤ 𝑎𝑖+1 𝑒−𝑎𝑖−
1
+𝑝𝑗𝑖+ 1 𝐻𝑗 𝑒⊖𝛼 (𝑡2 − 𝜎𝑗𝑖1 (𝑡2 ) , 𝑡2 ))
󵄩
󵄩
(𝑡2 , 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩
𝑚
𝑓
𝑔
𝜏} + ∑𝑝𝑗𝑖+ 1 𝐻𝑗 exp {−𝛼̃
𝜎})
× ( ∑𝑎𝑗𝑖+ 1 𝐻𝑗 × exp {−𝛼̃
𝑗=1
𝑗=1
𝑚
𝑓
𝑔
󵄩
󵄩
+ 𝛿2 𝑝𝑀𝑒⊖𝛼 (𝑡2 , 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩 ∑ (𝑎𝑗𝑖+ 1 𝐻𝑗 + 𝑝𝑗𝑖+ 1 𝐻𝑗 )
𝑗=1
󵄩
󵄩
≤ 𝛿2 𝑝𝑀𝑒⊖𝛼 (𝑡2 , 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩
+
𝑚
{ 𝑎𝑖
𝑓
𝑔
× { 1 + ∑ (𝑎𝑗𝑖+ 1 𝐻𝑗 + 𝑝𝑗𝑖+ 1 𝐻𝑗 )
𝛿2 𝑝𝑀 𝑗=1
{
+
× ( ∑ 𝑎𝑗𝑖+ 1 𝐻𝑗 𝑒⊖(−𝛼) (𝑠 − 𝜏𝑗𝑖1 (𝑠) , 𝑡2 )
𝑚
𝑔
𝑓
𝑔
󵄩
󵄩
+ 𝛿2 𝑝𝑀𝑒⊖𝛼 (𝑡2 , 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩 ∑ (𝑎𝑗𝑖+ 1 𝐻𝑗 + 𝑝𝑗𝑖+ 1 𝐻𝑗 )
𝑗=1
+
𝑚
𝑓
𝜏} + ∑𝑝𝑗𝑖+ 1 𝐻𝑗 exp {−𝛼̃
𝜎})
× ( ∑𝑎𝑗𝑖+ 1 𝐻𝑗 exp {−𝛼̃
1
𝑎𝑖+1 𝛿2 𝑝𝑀𝑒⊖𝛼
𝑔
𝑗=1
󵄨
𝑔󵄨
+ ∑𝑝𝑗𝑖+ 1 𝐻𝑗 󵄨󵄨󵄨󵄨]Δ𝑗 (𝑡2 − 𝜎𝑗𝑖1 (𝑡2 ))󵄨󵄨󵄨󵄨
𝑚
𝑗=1
󵄩
󵄩
+ 𝛿2 𝑝𝑀𝑒⊖𝛼 (𝑡2 , 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩
𝑚
𝑚
𝑡0
𝑔
× ∑ (𝑎𝑗𝑖+ 1 𝐻𝑗 exp {−𝛼̃
𝜏} + 𝑝𝑗𝑖+ 1 𝐻𝑗 exp {−𝛼̃
𝜎})
𝑗=1
≤
𝑚
𝑓
× ( ∑ 𝑎𝑗𝑖+ 1 𝐻𝑗 exp {−𝛼̃
𝜏} + ∑ 𝑝𝑗𝑖+ 1 𝐻𝑗 exp {−𝛼̃
𝜎}) Δ𝑠
𝑚
𝑎𝑖+1 𝑒−𝑎𝑖−
1
1
𝑡0
𝑎𝑖+1
𝑚
𝑓
+
− ∑ (𝑎𝑗𝑖1 𝐻𝑗
𝑎𝑖1 𝑗=1
exp {−𝛼̃
𝜏}
}
𝑔
+ 𝑝𝑗𝑖+ 1 𝐻𝑗 exp {−𝛼̃
𝜎}) }
}
󵄩
󵄩
< 𝛿2 𝑝𝑀𝑒⊖𝛼 (𝑡2 , 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩
+
𝑎𝑖+1 𝑚 + 𝑓
{ 𝑎𝑖1
𝑔 }
× { + (1 + − ) ∑ (𝑎𝑗𝑖1 𝐻𝑗 + 𝑝𝑗𝑖+ 1 𝐻𝑗 )}
𝑀
𝑎𝑖1 𝑗=1
{
}
󵄩󵄩
∗󵄩
< 𝛿2 𝑝𝑀𝑒⊖𝛼 (𝑡2 , 𝑡0 ) 󵄩󵄩𝜙 − 𝜙 󵄩󵄩󵄩 .
We also get a contradiction.
(55)
Journal of Applied Mathematics
11
Case 3. Equation (47) is not true and (45), (46), and (48)
are all true. Then, there exists 𝑡3 ∈ (𝑡0 , +∞)T and 𝑗0 ∈
{1, 2, . . . , 𝑚} such that
󵄨󵄨
󵄨
󵄨󵄨]𝑗0 (𝑡3 )󵄨󵄨󵄨 ≥ 𝑝𝑀𝑒⊖𝛼 (𝑡3 , 𝑡0 ) 󵄩󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩󵄩 ,
󵄨
󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨]𝑗0 (𝑡)󵄨󵄨 < 𝑝𝑀𝑒⊖𝛼 (𝑡, 𝑡0 ) 󵄩󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩󵄩 , 𝑡 ∈ (𝑡0 , 𝑡3 )T ,
󵄨
󵄨
(56)
󵄨
󵄨󵄨
󵄩
∗󵄩
󵄨󵄨]𝜄 (𝑡)󵄨󵄨󵄨 < 𝑝𝑀𝑒⊖𝛼 (𝑡, 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙 󵄩󵄩󵄩 ,
By the above four cases, for other cases of negative
proposition of (44), we can obtain a contradiction. Therefore,
(44) holds. Let 𝑝 → 1, then (43) holds. We can take −𝜆 = ⊖𝛼,
then 𝜆 > 0 and −𝜆 ∈ R+ . Hence, we have that
󵄩
󵄩
|V (𝑡)|1 ≤ 𝑀 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩 𝑒−𝜆 (𝑡, 𝑡0 ) ,
𝑡 ∈ [−𝜃, ∞)T , 𝑡 ≥ 𝑡0 ,
(62)
for 𝜄 ≠ 𝑗0 , 𝑡 ∈ (𝑡0 , 𝑡3 ]T , 𝜄 = 1, 2, . . . , 𝑚.
Therefore, there must be a constant 𝛿3 ≥ 1 such that
󵄨󵄨
󵄨
󵄨󵄨]𝑗0 (𝑡3 )󵄨󵄨󵄨 = 𝛿3 𝑝𝑀𝑒⊖𝛼 (𝑡3 , 𝑡0 ) 󵄩󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩󵄩 ,
󵄨
󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨]𝑗0 (𝑡)󵄨󵄨 < 𝛿3 𝑝𝑀𝑒⊖𝛼 (𝑡, 𝑡0 ) 󵄩󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩󵄩 , 𝑡 ∈ (𝑡0 , 𝑡3 )T ,
󵄨
󵄨
󵄨
󵄨󵄨
󵄩
∗󵄩
󵄨󵄨]𝜄 (𝑡)󵄨󵄨󵄨 < 𝛿3 𝑝𝑀𝑒⊖𝛼 (𝑡, 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙 󵄩󵄩󵄩 ,
which means that the almost periodic solution 𝜔(𝑡) of (2) is
exponentially stable. This completes the proof of Theorem 18.
(57)
In this section, we present an example to illustrate the
feasibility of our results obtained in previous sections.
for 𝜄 ≠ 𝑗0 , 𝑡 ∈ (𝑡0 , 𝑡3 ]T , 𝜄 = 1, 2, . . . , 𝑚.
Then, in a similar way, in view of (38), we have that
󵄨󵄨
󵄨
󵄨󵄨]𝑗0 (𝑡3 )󵄨󵄨󵄨 < 𝛿3 𝑝𝑀 󵄩󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩󵄩 𝑒⊖𝛼 (𝑡3 , 𝑡0 )
󵄨
󵄨
1
1 𝑛
+ − ∑ (𝑏𝑖𝑗+0 𝐻𝑖ℎ + 𝑞𝑖𝑗+0 𝐻𝑖𝑘 ))
𝑀 𝑏𝑗0 𝑖=1
󵄩
󵄩
< 𝛿3 𝑝𝑀𝑒⊖𝛼 (𝑡3 , 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩 ,
which is also a contradiction.
×(
Example 1. Let 𝑛 = 𝑚 = 2. Consider the following neutraltype BAM neural networks with delays on a time scale T
(58)
𝑗=1
2
for 𝜄 ≠ 𝑗1 , 𝑡 ∈ (𝑡0 , 𝑡4 ]T , 𝜄 = 1, 2, . . . , 𝑚.
×{
𝑀
+ (1 +
𝑏𝑗−1
(60)
𝑛
) ∑ (𝑏𝑖𝑗+0 𝐻𝑖ℎ + 𝑞𝑖𝑗+0 𝐻𝑖𝑘 )}
𝑖=1
(63)
2
𝑦𝑗Δ (𝑡) = − 𝑏𝑗 (𝑡) 𝑦𝑗 (𝑡) + ∑𝑏𝑖𝑗 (𝑡) ℎ𝑖 (𝑥𝑖 (𝑡 − 𝜁𝑖𝑗 (𝑡)))
𝑖=1
2
+ ∑𝑞𝑖𝑗 (𝑡) 𝑘𝑖 (𝑥𝑖Δ (𝑡 − 𝜍𝑖𝑗 (𝑡))) + 𝐽𝑗 (𝑡) ,
𝑖=1
(61)
𝑎2 (𝑡) = 0.93 + 0.01 cos 𝑡,
𝑎11 (𝑡) = 0.003 sin 𝑡,
𝑎12 (𝑡) = 0.002 cos 𝑡,
𝑎21 (𝑡) = 0.004 sin 𝑡,
𝑎22 (𝑡) = 0.001 sin 𝑡,
𝑝11 (𝑡) = 0.001 + 0.003 cos 𝑡,
𝑝12 (𝑡) = 0.001 sin 𝑡,
𝑝21 (𝑡) = 0.006 + 0.001 sin 𝑡,
𝑝22 (𝑡) = 0.003 cos 𝑡,
𝑓1 (𝑢) = 0.2 sin 𝑢,
𝑓2 (𝑢) = 2 cos 𝑢,
𝑔1 (𝑢) = 0.2 cos 𝑢,
𝑔2 (𝑢) = 2 sin 𝑢,
𝑏1 (𝑡) = 0.95 + 0.01 sin √3𝑡,
󵄩
󵄩
< 𝛿4 𝑝𝑀𝑒⊖𝛼 (𝑡4 , 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩 .
It is also a contradiction.
𝑗=1
𝑎1 (𝑡) = 0.975 + 0.025 sin 𝑡,
Similarly, in view of (38), we obtain that
󵄨󵄨 Δ
󵄨
󵄨󵄨]𝑗 (𝑡4 )󵄨󵄨󵄨 < 𝛿4 𝑝𝑀𝑒⊖𝛼 (𝑡4 , 𝑡0 ) 󵄩󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩󵄩
󵄨 1
󵄨
𝑏𝑗+1
+ ∑ 𝑝𝑗𝑖 (𝑡) 𝑔𝑗 (𝑦𝑗Δ (𝑡 − 𝜎𝑗𝑖 (𝑡))) + 𝐼𝑖 (𝑡) ,
where 𝑡 ∈ T, 𝑖, 𝑗 = 1, 2 and the coefficients are as follows:
for 𝜄 ≠ 𝑗1 , 𝑡 ∈ (𝑡0 , 𝑡4 ]T , 𝜄 = 1, 2, . . . , 𝑚.
𝑏𝑗+1
2
𝑥𝑖Δ (𝑡) = − 𝑎𝑖 (𝑡) 𝑥𝑖 (𝑡) + ∑𝑎𝑗𝑖 (𝑡) 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝜏𝑗𝑖 (𝑡)))
Case 4. Equation (48) is not true and (45), (46), and (47)
are all true. Then, there exists 𝑡4 ∈ (𝑡0 , +∞)T and 𝑗1 ∈
{1, 2, . . . , 𝑚} such that
󵄨󵄨 Δ
󵄨
󵄨󵄨]𝑗 (𝑡4 )󵄨󵄨󵄨 ≥ 𝑝𝑀𝑒⊖𝛼 (𝑡4 , 𝑡0 ) 󵄩󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩󵄩 ,
󵄨 1
󵄨
󵄨󵄨 Δ 󵄨󵄨
󵄨󵄨]𝑗 (𝑡)󵄨󵄨 < 𝑝𝑀𝑒⊖𝛼 (𝑡, 𝑡0 ) 󵄩󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩󵄩 , 𝑡 ∈ (𝑡0 , 𝑡4 )T ,
󵄨 1 󵄨
(59)
󵄨󵄨󵄨]Δ (𝑡)󵄨󵄨󵄨 < 𝑝𝑀𝑒 (𝑡, 𝑡 ) 󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩 ,
󵄨󵄨 𝜄
󵄨󵄨
⊖𝛼
0 󵄩
󵄩
Hence, there must be a constant 𝛿4 ≥ 1 such that
󵄨
󵄨󵄨 Δ
󵄨󵄨]𝑗 (𝑡4 )󵄨󵄨󵄨 ≥ 𝛿4 𝑝𝑀𝑒⊖𝛼 (𝑡4 , 𝑡0 ) 󵄩󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩󵄩 ,
󵄨
󵄨 1
󵄨󵄨 Δ 󵄨󵄨
󵄩
∗󵄩
󵄨󵄨󵄨]𝑗1 (𝑡)󵄨󵄨󵄨 < 𝛿4 𝑝𝑀𝑒⊖𝛼 (𝑡, 𝑡0 ) 󵄩󵄩󵄩𝜙 − 𝜙 󵄩󵄩󵄩 , 𝑡 ∈ (𝑡0 , 𝑡4 )T ,
󵄨󵄨 Δ 󵄨󵄨
󵄨󵄨]𝜄 (𝑡)󵄨󵄨 < 𝛿4 𝑝𝑀𝑒⊖𝛼 (𝑡, 𝑡0 ) 󵄩󵄩󵄩󵄩𝜙 − 𝜙∗ 󵄩󵄩󵄩󵄩 ,
󵄨
󵄨
5. An Example
𝑏11 (𝑡) = 0.002 sin √2𝑡,
𝑏21 (𝑡) = 0.001 sin 𝑡,
𝑏2 (𝑡) = 0.92 + 0.01 cos √2𝑡,
𝑏12 (𝑡) = 0.003 cos 𝑡,
𝑏22 (𝑡) = 0.004 cos 𝑡,
12
Journal of Applied Mathematics
𝑞11 (𝑡) = 0.001 cos √3𝑡,
𝑞21 (𝑡) = 0.003 sin 𝑡,
𝑞12 (𝑡) = 0.004 sin 𝑡,
such as shunting inhibitory cellular neural networks, CohenGrossberg neural networks, and so on.
𝑞22 (𝑡) = 0.004 cos 𝑡,
ℎ1 (𝑢) = sin 𝑢,
ℎ2 (𝑢) = 1.5 cos 𝑢,
Acknowledgment
𝑘1 (𝑢) = cos 𝑢,
𝑘2 (𝑢) = 1.5 sin 𝑢,
This work is supported by the National Natural Sciences
Foundation of People’s Republic of China under Grant
10971183.
𝐼1 (𝑡) = 𝐼2 (𝑡) = 0.6 sin √3𝑡,
𝐽1 (𝑡) = 𝐽2 (𝑡) = 0.5 cos 𝑡,
𝜏𝑗𝑖 (𝑡) = 𝜎𝑗𝑖 (𝑡) = 0.12 sin 𝑡,
𝜁𝑖𝑗 (𝑡) = 𝜍𝑖𝑗 (𝑡) = 0.37 sin 𝑡,
𝑖, 𝑗 = 1, 2.
(64)
By calculating, we have
𝑎1+ = 1,
𝑎1− = 0.95,
𝑎2+ = 0.94,
𝑎2− = 0.92,
𝑏1+ = 0.96,
𝑏1− = 0.94,
𝑏2+ = 0.93,
𝑏2− = 0.91,
+
= 0.003,
𝑎11
+
𝑎12
= 0.002,
+
= 0.004,
𝑎21
+
𝑎22
= 0.001,
+
= 0.004,
𝑝11
+
𝑝12
= 0.001,
+
= 0.007,
𝑝21
+
𝑝22
= 0.003,
+
= 0.002,
𝑏11
+
𝑏12
= 0.003,
+
= 0.001,
𝑏21
+
𝑏22
= 0.004,
+
= 0.001,
𝑞11
+
𝑞12
= 0.004,
+
= 0.003,
𝑞21
+
𝑞22
= 0.004,
𝑓
𝑔
𝐻1 = 𝐻1 = 0.2,
𝐻1ℎ = 𝐻1𝑘 = 1,
𝑓
(65)
𝑔
𝐻2 = 𝐻2 = 2,
𝐻2ℎ = 𝐻2𝑘 = 1.5.
If T = R, then 𝜇(𝑡) = 0, and if T = Z, then 𝜇(𝑡) = 1. We can
verify for the above two cases, and all conditions of Theorems
17 and 18 are satisfied. Therefore, whether T = R or T = Z,
(63) has an almost periodic solution, which is exponentially
stable. That is, the continuous-time neural network and its
discrete-time analogue have the same dynamical behaviors.
6. Conclusion
In this paper, we establish some sufficient conditions ensuring
the existence and exponential stability of almost periodic
solutions for a class of neutral-type BAM neural networks
with delays on time scales. Our results obtained in this paper
are completely new even in case of the time scale T = R
or Z and complementary to the previously known results.
Besides, our method used in this paper may be used to
study many other neutral-type neural networks with delays
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