Existence and exponential stability of periodic solution for fuzzy BAM neural networks with periodic coefficient
A class of fuzzy bidirectional associated memory (BAM) networks with periodic coefficients is studied. Some sufficient conditions are established for the existence and global exponential stability of a periodic solution of such fuzzy BAM neural networks by using a continuation theorem based on the...
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Institute of Mathematics, NAS of Ukraine
2011
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508817681285120 |
|---|---|
| author | Dai-xi, Liao Li-hui, Yang Zhang, Qian-hong Дай-сі, Ляо Лі-гуей, Ян Чжан, Цянь-хун |
| author_facet | Dai-xi, Liao Li-hui, Yang Zhang, Qian-hong Дай-сі, Ляо Лі-гуей, Ян Чжан, Цянь-хун |
| author_sort | Dai-xi, Liao |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:37:39Z |
| description | A class of fuzzy bidirectional associated memory (BAM) networks with periodic coefficients is studied.
Some sufficient conditions are established for the existence and global exponential stability of a periodic solution
of such fuzzy BAM neural networks by using a continuation theorem based on the coincidence degree and the Lyapunov-function method.
The sufficient conditions are easy to verify in pattern recognition and automatic control. Finally, an example is given to show the feasibility and efficiency of our results. |
| first_indexed | 2026-03-24T02:31:14Z |
| format | Article |
| fulltext |
UDC 517.9
Qian-hong Zhang (Guizhou College Finance and Economics, China),
Li-hui Yang (Hunan City Univ., China),
Dai-xi Liao (Hunan Inst. Technology, China)
EXISTENCE AND EXPONENTIAL STABILITY
OF PERIODIC SOLUTION FOR FUZZY BAM NEURAL
NETWORKS WITH PERIODIC COEFFICIENT *
IСНУВАННЯ ТА ЕКСПОНЕНЦIАЛЬНА СТIЙКIСТЬ
ПЕРIОДИЧНОГО РОЗВ’ЯЗКУ ДЛЯ НЕЧIТКИХ НЕЙРОННИХ
МЕРЕЖ КОСКО З ПЕРIОДИЧНИМИ КОЕФIЦIЄНТАМИ
A class of fuzzy bidirectional associated memory (BAM) networks with periodic coefficients is studied. Some
sufficient conditions are established for the existence and global exponential stability of a periodic solution of
such fuzzy BAM neural networks by using a continuation theorem based on the coincidence degree and the
Lyapunov-function method. The sufficient conditions are easy to verify in pattern recognition and automatic
control. Finally, an example is given to show the feasibility and efficiency of our results.
Вивчено клас нечiтких нейронних мереж Коско з перiодичним коефiцiєнтом. За допомогою теореми про
продовження, що базується на ступенi збiгу та методi функцiй Ляпунова, встановлено достатнi умови
для iснування та глобальної експоненцiальної стiйкостi перiодичного розв’язку таких нечiтких нейрон-
них мереж Коско. Цi достатнi умови легко перевiряються при розпiзнаваннi образiв та автоматичному
керуваннi. Наведено приклад, що демонструє застосовнiсть та ефективнiсть отриманих результатiв.
1. Introduction. Recently, a class of two-layer hetero-associative networks called bidi-
rectional associated memory (BAM) neural networks [1, 2] with or without transmission
delays have been proposed by Kosko and used in many fields such as pattern recognition
and automatic control. Many authors studied the stability of BAM neural networks with
delays or without delays (see, for example, [1, 2, 3 – 12, 17]).
It is well known that fuzzy cellular neural networks (FCNNs) first introduced by
T. Yang and L. B. Yang [13, 14] is another type cellular neural networks model, which
combined fuzzy operations (fuzzy AND and fuzzy OR) with cellular neural networks.
Recently researchers have found that FCNNs are useful in image processing, and some
results have been reported on stability and periodicity of FCNNs [15, 16, 18, 19]. How-
ever, the papers above only consider the FCNNs with constant coefficients. At present,
the investigation of BAM neural networks with periodic coefficients and delays has at-
tracted more and more attention of the researcher [17, 22], to the best of our knowledge,
few author consider the stability of fuzzy BAM neural networks with periodic coeffi-
cients. In this paper, we would like to investigate the fuzzy BAM neural networks with
periodic coefficients by the following system:
*This work is partially supported by the Scientific Research Foundation of Guizhou Science and Technology
Department (No. [2011] J 2096), the Doctoral Foundation of Guizhou College of Finance and Economics
(2010), and the Scientific Research Foundation of Hunan Provincial Education Department (10B023).
c© QIAN-HONG ZHANG, LI-HUI YANG, DAI-XI LIAO, 2011
1672 ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 12
EXISTENCE AND EXPONENTIAL STABILITY OF PERIODIC SOLUTION FOR FUZZY. . . 1673
x′i(t) = −ai(t)xi(t) +
m∧
j=1
αij(t)fj(yj(t)) +
m∧
j=1
Tij(t)uj(t) + Ii(t)+
+
m∨
j=1
βij(t)fj(yj(t)) +
m∨
j=1
Hij(t)uj(t),
y′j(t) = −bj(t)yj(t) +
n∧
i=1
pji(t)gi(xi(t)) +
n∧
i=1
Kji(t)ui(t) + Jj(t)+
+
n∨
i=1
qji(t)gi(xi(t)) +
n∨
i=1
Nji(t)ui(t),
(1.1)
where ai(t) ≥ 0, bj(t) ≥ 0, i = 1, 2, . . . , n, j = 1, 2, . . . ,m. xi(t) and yj(t) are
the activations of the ith neuron in X-layer and the jth neuron in Y -layer at the time
t, respectively.
∧
and
∨
denote fuzzy AND and fuzzy OR operations, respectively.
fj , j = 1, 2, . . . ,m, gi, i = 1, 2, . . . , n, are signal transmission functions. αij(t) and
βij(t) are respectively the elements of fuzzy feedback MIN and fuzzy feedback MAX
in X-layer at the time t. Tij(t) and Hij(t) are respectively the elements of fuzzy feed-
forward MIN and fuzzy feed-forward MAX in X-layer at the time t. pji(t) and qji(t) are
respectively the elements of fuzzy feedback MIN and fuzzy feedback MAX in Y -layer
at the time t. Kji(t) and Nji(t) are respectively the elements of fuzzy feed-forward
MIN and fuzzy feed-forward MAX in Y -layer at the time t. uj(t) and ui(t) denote the
external inputs at the time t. Ii(t) and Jj(t) denote bias of the ith neurons in X-layer
and bias of the jth neurons in Y -layer at the time t, respectively.
Throughout this paper, we always assume that ai(t), bj(t), αij(t), βij(t), Tij(t),
Hij(t), pji(t), qji(t), Kji(t), Nji(t), ui(t), uj(t), Ii(t), Jj(t) are continuous ω-periodic
functions.
For the sake of convenience, we introduce the following notations: Let r(t) be a
ω-periodic solution defined on R
r− = min
0≤t≤ω
|r(t)|, r+ = max
0≤t≤ω
|r(t)|,
r =
1
ω
ω∫
0
r(t)dt, ‖r‖2 =
ω∫
0
|r(t)|2dt
1/2
.
Throughout this paper, we give the following assumptions:
(A1) fj(·) and gi(·) are Lipschitz continuous on R with Lipschitcz constants Lfj ,
j = 1, 2, . . . ,m, Lgi , i = 1, 2, . . . , n, and fj(0) = gi(0) = 0. That is, for all x, y ∈ R
|fj(x)− fj(y)| ≤ Lfj |x− y|, |gi(x)− gi(y)| ≤ Lgi |x− y|.
(A2) There exist constant Mj > 0, Rj > 0 such that |fj(y)| ≤ Mj , |gj(x)| ≤ Rj
for j = 1, 2, . . . , n, x, y ∈ R.
For any solution
z(t) = (x(t)T , y(t)T )T = (x1(t), . . . , xn(t), y1(t), . . . , ym(t))T
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 12
1674 QIAN-HONG ZHANG, LI-HUI YANG, DAI-XI LIAO
and periodic solution
z∗(t) = (x∗(t)T , y∗(t)T )T = (x∗1(t), . . . , x∗n(t), y∗1(t), . . . , y∗m(t))T
of system (1.1), define ‖(φT , ϕT )T − (x∗T , y∗T )T ‖ as
‖(φT , ϕT )T − (x∗T , y∗T )T ‖ =
n∑
i=1
max
t∈[0,ω]
|φi(t)− x∗i (t)|+
m∑
j=1
max
t∈[0,ω]
|ϕj(t)− y∗j (t)|.
Definition 1.1. The periodic solution (x∗T (t), y∗T (t))T of system (1.1) is said to
be globally exponentially stable, if there exist constants γ > 0 and M ≥ 1 such that
|xi(t)− x∗i (t)| ≤M‖(φT , ϕT )T − (x∗T , y∗T )T ‖e−γt ∀t > 0, i = 1, 2, . . . , n,
|yj(t)− y∗j (t)| ≤M‖(φT , ϕT )T − (x∗T , y∗T )T ‖e−γt ∀t > 0, j = 1, 2, . . . ,m,
for any solution of system (1.1).
Lemma 1.1 [13]. Suppose x and y are two states of system (1.1), then we have∣∣∣∣∣∣
n∧
j=1
αij(t)gj(x)−
n∧
j=1
αij(t)gj(y)
∣∣∣∣∣∣ ≤
n∑
j=1
|αij(t)||gj(x)− gj(y)|
and ∣∣∣∣∣∣
n∨
j=1
βij(t)gj(x)−
n∨
j=1
βij(t)gj(y)
∣∣∣∣∣∣ ≤
n∑
j=1
|βij(t)||gj(x)− gj(y)|.
The rest of this paper is organized as follows. In Section 2, we will prove the
existence of the periodic solution by using the continuation theorem of coincidence
degree theory. In Section 3, we establish the result that the periodic solutions are the
globally exponentially stable by using Lyapunov function method. In Section 4, an
example will be given to illustrate the feasibility and effectiveness of our results. General
conclusion is drawn in Section 5.
2. Existence of periodic solution. In this section, based on Mawhin’s continuation
theorem, we shall study the existence of at least one periodic solution of (1.1). To do so,
we shall make some preparations.
Let X = {(xT (t), yT (t)))T ∈ C(R,Rn+m)|x(t + ω) = x(t), y(t + ω) = y(t) for
some ω > 0} and ‖(xT (t), yT (t))T ‖ =
∑n
i=1
max
t∈[0,ω]
|xi(t)| +
∑m
j=1
max
t∈[0,ω]
|yj(t)|, it
can be proved that X is a Banach space.
Consider the following abstract equation in the Banach space X:
Lx = λNx (2.1)
where L : DomL
⋂
X → X is a Fredholm mapping of index zero and λ ∈ [0, 1] is a
parameter. There exist two linear and continuous projectors P and Q
P : X
⋂
DomL→ KerL, Q : X → X/ImL
such that ImP = KerL, KerQ = ImL. Due to dim ImQ = dim KerL, there exists an
algebraical and topological isomorphism J : ImQ→ KerL.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 12
EXISTENCE AND EXPONENTIAL STABILITY OF PERIODIC SOLUTION FOR FUZZY. . . 1675
Lemma 2.1 (see [21]). Let X be a Banach space and L be a Fredholm mapping
of index zero. Assume that N : Ω→ X is a L — compact on Ω with Ω open and bound
in X. Furthermore, suppose that
(a) for each λ ∈ (0, 1), x ∈ ∂Ω
⋂
DomL, Lx 6= λNx;
(b) for each x ∈ ∂Ω
⋂
KerL, QNx 6= 0;
(c) deg {QNx,Ω
⋂
KerL, 0} 6= 0,
then the equation Lx = Nx has at least one solution in Ω, where Ω is the closure to Ω,
∂Ω is the boundary of Ω.
Theorem 2.1. Assume that (A1) and (A2) hold, then system (1.1) has at least one
ω-periodic solution.
Proof. In order to use continuation theorem of coincidence degree theory to establish
the existence of periodic solution. Let
(Nz)i(t) = −ai(t)xi(t) +
m∧
j=1
αij(t)fj(yj(t)) +
m∧
j=1
Tij(t)uj(t) + Ii(t)+
+
m∨
j=1
βij(t)fj(yj(t)) +
m∨
j=1
Hij(t)uj(t), i = 1, 2, . . . , n,
(Nz)n+j(t) = −bj(t)yj(t) +
n∧
i=1
pji(t)gi(xi(t)) +
n∧
i=1
Kji(t)ui(t) + Jj(t)+
+
n∨
i=1
qji(t)gi(xi(t)) +
n∨
i=1
Nji(t)ui(t), j = 1, 2, . . . ,m,
(Lz)(t) = z′(t), P z =
1
ω
ω∫
0
z(t)dt, QU =
1
ω
ω∫
0
U(t)dt
for z(t) = (xT (t), yT (t))T ∈ X
⋂
DomL, U ∈ X. It is easy to prove that L is a
Fredholm mapping of index zero, that P : X
⋂
DomL → KerL and Q : X → X/ImL
are two projector, and N is L compact on Ω for any given open bounded set.
Corresponding to the operator equation Lz = λNz, λ ∈ (0, 1), we have
x′i(t) = λ
−ai(t)xi(t) +
m∧
j=1
αij(t)fj(yj(t)) +
m∧
j=1
Tij(t)uj(t) + Ii(t) +
+
m∨
j=1
βij(t)fj(yj(t)) +
m∨
j=1
Hij(t)uj(t)
,
y′j(t) = λ
[
−bj(t)yj(t) +
n∧
i=1
pji(t)gi(xi(t)) +
n∧
i=1
Kji(t)ui(t) + Jj(t) +
+
n∨
i=1
qji(t)gi(xi(t)) +
n∨
i=1
Nji(t)ui(t)
]
.
(2.2)
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 12
1676 QIAN-HONG ZHANG, LI-HUI YANG, DAI-XI LIAO
Suppose that z(t) = (x1(t), . . . , xn(t), y1(t), . . . , ym(t))T ∈ X is a solution of system
(2.2) for a certain λ ∈ (0, 1). Integrating (2.2) over [0, ω], we obtain
ω∫
0
ai(t)xi(t)dt =
ω∫
0
m∧
j=1
αij(t)fj(yj(t)) +
m∧
j=1
Tij(t)uj(t) + Ii(t) +
+
m∨
j=1
βij(t)fj(yj(t)) +
m∨
j=1
Hij(t)uj(t)
dt. (2.3)
Let ξ ∈ [0, ω] such that xi(ξ) = inft∈[0,ω] xi(t), i = 1, 2, . . . , n. Then by (2.3), we have
ωaixi(ξ) ≤
ω∫
0
∣∣∣∣∣∣
m∧
j=1
αij(t)fj(yj(t))−
m∧
j=1
αij(t)fj(0)
∣∣∣∣∣∣+
∣∣∣∣∣∣
m∧
j=1
Tij(t)uj(t)
∣∣∣∣∣∣+ |Ii(t)| +
+
∣∣∣∣∣∣
m∨
j=1
βij(t)fj(yj(t))−
m∨
j=1
βij(t)fj(0)
∣∣∣∣∣∣+
∣∣∣∣∣∣
m∨
j=1
Hij(t)uj(t)
∣∣∣∣∣∣
dt ≤
≤
ω∫
0
m∑
j=1
|αij(t)||fj(yj(t))|+
m∑
j=1
|βij(t)||fj(yj(t))| +
+
∣∣∣∣∣∣
m∧
j=1
Tij(t)uj(t)
∣∣∣∣∣∣+ |Ii(t)|+
∣∣∣∣∣∣
m∨
j=1
Hij(t)uj(t)
∣∣∣∣∣∣
dt ≤
≤ ω
m∑
j=1
(α+
ij + β+
ij)Mj + (T+
ij +H+
ij )u
+
j + I+i
.
Hence
xi(ξ) ≤
1
ai
m∑
j=1
(α+
ij + β+
ij)Mj + (T+
ij +H+
ij )u
+
j + I+i
:= Ui, i = 1, 2, . . . , n.
(2.4)
Similarly, let η ∈ [0, ω] such that yj(η) = inft∈[0,ω], j = 1, 2, . . . ,m, we obtain
yj(η) ≤ 1
bj
{
n∑
i=1
(p+ji + q+ji)Rj + (K+
ji +N+
ji )u
+
i + J+
j
}
:= Vj , j = 1, 2, . . . ,m.
(2.5)
Set t0 = 0, tq+1 = ω, from (2.2), (2.4) and (2.5), we have
ω∫
0
|x′i(t)|dt ≤
q+1∑
k=1
tk∫
tk−1
|x′i(t)|dt ≤
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 12
EXISTENCE AND EXPONENTIAL STABILITY OF PERIODIC SOLUTION FOR FUZZY. . . 1677
≤
ω∫
0
|ai(t)||xi(t)|dt+
ω∫
0
m∑
j=1
(|αij(t)|+ |βij(t)|)|fj(yj(t))|dt+
+
ω∫
0
∣∣∣∣∣∣
m∧
j=1
Tij(t)uj(t)
∣∣∣∣∣∣+
∣∣∣∣∣∣
m∨
j=1
Hij(t)uj(t)
∣∣∣∣∣∣
dt+
ω∫
0
|Ii(t)|dt ≤
≤
ω∫
0
|ai(t)|2dt
1/2 ω∫
0
|xi(t)|2dt
1/2
+
+
m∑
j=1
ω∫
0
|αij(t)|2dt
1/2 ω∫
0
|fj(yj(t))|2dt
1/2
+
+
m∑
j=1
ω∫
0
|βij(t)|2dt
1/2 ω∫
0
|fj(yj(t))|2dt
1/2
+ (T+
ij +H+
ij )u
+
j ω + I+i ω ≤
≤
√
ωa+i ‖xi‖2 +
m∑
j=1
√
ω(α+
ij + β+
ij)Mj + (T+
ij +H+
ij )u
+
j ω + I+i ω. (2.6)
Multiplying both sides of system (2.2) by xi(t) and integrating over [0, ω], we obtain
that
0 =
ω∫
0
xi(t)x
′
i(t)dt = −λ
ω∫
0
ai(t)x
2
i (t)dt+
+λ
ω∫
0
m∧
j=1
αij(t)fj(yj(t)) +
m∨
j=1
βij(t)fj(yj(t))
×
×xi(t)dt+ λ
ω∫
0
m∧
j=1
Tij(t)uj(t) +
m∨
j=1
Hij(t)uj(t)
xi(t)dt+ λ
ω∫
0
Ii(t)xi(t)dt.
(2.7)
From (2.7) and applying Lemma 1.1, it follows that
a−i
ω∫
0
|xi(t)|2dt ≤
ω∫
0
∣∣∣∣∣∣
m∧
j=1
αij(t)fj(yj(t))
∣∣∣∣∣∣+
∣∣∣∣∣∣
m∨
j=1
βij(t)fj(yj(t))
∣∣∣∣∣∣
|xi(t)|dt+
+
ω∫
0
∣∣∣∣∣∣
m∧
j=1
Tij(t)uj(t) +
m∨
j=1
Hij(t)uj(t)
∣∣∣∣∣∣ |xi(t)|dt+
ω∫
0
|Ii(t)||xi(t)|dt ≤
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 12
1678 QIAN-HONG ZHANG, LI-HUI YANG, DAI-XI LIAO
≤
ω∫
0
m∑
j=1
(|αij |+ |βij |)|fj(yj(t))||xi(t)|dt+
+
ω∫
0
m∧
j=1
|Tij(t)||uj(t)|+
m∨
j=1
|Hij(t)||uj(t)|
|xi(t)|dt+
ω∫
0
|Ii(t)||xi(t)|dt ≤
≤
m∑
j=1
(α+
ij + β+
ij)Mj + T+
ij u
+
j +H+
iju
+
j + I+i
√ω
ω∫
0
|xi(t)|2dt
1/2
. (2.8)
It follows from (2.8) that
‖xi‖2 ≤
1
a−i
m∑
j=1
(α+
ij + β+
ij)Mj + T+
ij u
+
j +H+
iju
+
j + I+i
√ω := Gi. (2.9)
Substituting (2.9) into (2.6), we obtain that
ω∫
0
|x′i(t)|dt ≤
√
ωa+i Gi +
m∑
j=1
√
ω(α+
ij + β+
ij)Mj + (T+
ij +H+
ij )u
+
j ω + I+i ω. (2.10)
From (2.4) and (2.10), there exists positive constant Bi, i = 1, 2, . . . , n, such that for
t ∈ [0, ω],
|xi(t)| ≤ Bi, i = 1, 2, . . . , n.
Similarly, we have
|yj(t)| ≤ Bn+j , j = 1, 2, . . . ,m.
Clearly, Bi, i = 1, 2, . . . , n + m, is independent of λ. Denote B∗ =
∑n+m
i=1
Bi + δ,
where δ > 0 is taken sufficiently large such that
min
1≤i≤n
aiB
∗ > max
1≤i≤n
m∑
j=1
(|αij |+ |βij |)Mj + T+
ij u
+
j +H+
iju
+
j + |Ii|
,
min
1≤j≤m
biB
∗ > max
1≤j≤m
(
n∑
i=1
(|pji|+ |qji|)Ri +K+
jiu
+
i +N+
jiu
+
i + |Jj |
)
.
Now we take
Ω = {z = (x1(t), . . . , xn(t), y1(t), . . . , ym(t))T ∈ Rn+m|‖z‖ =
= ‖(x1, . . . , xn, y1, . . . , ym)T ‖ < B∗}.
Thus the condition (a) of Lemma 2.1 is satisfied.
When z = (x1, . . . , xn, y1, . . . , ym)T ∈ ∂Ω
⋂
Rn+m, z = (x1, . . . , xn, y1, . . . , ym)T
is a constant vector in Rn+m with |x1|+ . . .+ |xn|+ |y1|+ . . .+ |ym| = B∗. Then
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 12
EXISTENCE AND EXPONENTIAL STABILITY OF PERIODIC SOLUTION FOR FUZZY. . . 1679
QNz = QN(x1, . . . , xn, y1, . . . , ym)T =
Θ1
...
Θn
Θn+1
...
Θn+m
,
where
Θi = −aixi +
m∧
j=1
αijfj(yj) +
m∨
j=1
βijfj(yj)+
+
1
ω
ω∫
0
m∧
j=1
Tij(t)uj(t)dt+
1
ω
ω∫
0
m∨
j=1
Hij(t)uj(t)dt+ Ii,
Θn+j = −bjyj +
n∧
i=1
pjigi(xi) +
n∨
i=1
qjigi(xi)+
+
1
ω
ω∫
0
n∧
i=1
Kji(t)ui(t)dt+
1
ω
ω∫
0
n∨
i=1
Nji(t)ui(t)dt+ Jj .
Therefore,
‖QNz‖ =
n∑
i=1
∣∣∣∣∣∣aixi −
m∧
j=1
αijfj(yj)−
m∨
j=1
βijfj(yj)−
1
ω
ω∫
0
m∧
j=1
Tij(t)uj(t)dt −
− 1
ω
ω∫
0
m∨
j=1
Hij(t)uj(t)dt− Ii
∣∣∣∣∣∣+
+
m∑
j=1
∣∣∣∣∣∣bjyj −
n∧
i=1
pjigi(xi)−
n∨
i=1
qjigi(xi)−
1
ω
ω∫
0
n∧
i=1
Kji(t)ui(t)dt −
− 1
ω
ω∫
0
n∨
i=1
Nji(t)ui(t)dt− Jj
∣∣∣∣∣∣ ≥
n∑
i=1
ai|xi| −
n∑
i=1
∣∣∣∣∣∣
m∧
j=1
αijfj(yj)−
m∧
j=1
αijfj(0)
∣∣∣∣∣∣−
−
n∑
i=1
∣∣∣∣∣∣
m∨
j=1
βijfj(yj)−
m∨
j=1
βijfj(0)
∣∣∣∣∣∣−
n∑
i=1
T+
ij u
+
j −
n∑
i=1
H+
iju
+
j −
n∑
i=1
|Ii|+
+
m∑
j=1
bi|yj | −
m∑
j=1
∣∣∣∣∣
n∧
i=1
pjigi(xi)−
n∧
i=1
pjigi(0)
∣∣∣∣∣−
m∑
j=1
∣∣∣∣∣
n∨
i=1
qjigi(xi)−
n∨
i=1
qjigi(0)
∣∣∣∣∣−
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 12
1680 QIAN-HONG ZHANG, LI-HUI YANG, DAI-XI LIAO
−
m∑
j=1
K+
jiu
+
i −
m∑
j=1
N+
jiu
+
i −
m∑
j=1
|Jj | ≥
≥
n∑
i=1
ai|xi| −
n∑
i=1
m∑
j=1
(|αij |+ |βij |)Mj + T+
ij u
+
j +H+
iju
+
j + |Ii|
+
+
m∑
j=1
bj |yj | −
m∑
j=1
(
n∑
i=1
(|pji|+ |qji|)Rji+K+
jiu
+
i +N+
jiu
+
i + |Jj |
)
≥
≥ min
1≤i≤n
ai|xi| − max
1≤i≤n
m∑
j=1
(|αij |+ |βij |)Mj + T+
ij u
+
j +H+
iju
+
j + |Ii|
+
+ min
1≤j≤m
bj |yj | − max
1≤j≤m
(
n∑
i=1
(|pji|+ |qji|)Ri +K+
jiu
+
i +N+
jiu
+
i + |Jj |
)
> 0.
Consequently,QNz = QN(x1, . . . , xn, y1, . . . , ym)T 6= (0, 0, . . . , 0)T , for (x1, . . . , xn,
y1, . . . , ym)T ∈ ∂Ω
⋂
KerL. This satisfies condition (b) of Lemma 2.1.
Define Φ: DomL× [0, 1]→ X by
Φ(x1, . . . , xn, y1, . . . , ym, µ)T =
= −µ(x1, . . . , xn, y1, . . . , ym)T + (1− µ)QN(x1, . . . , xn, y1, . . . , ym)T .
When (x1, . . . , xn, y1, . . . , ym)T ∈ ∂Ω
⋂
KerL, (x1, . . . , xn, y1, . . . , ym)T ∈ ∂Ω
⋂⋂
KerL is a constant vector satisfying
∑n
i=1
|xi| +
∑m
j=1
|yj | = B∗. It easily fol-
lows that
Φ(x1, . . . , xn, y1, . . . , ym, µ)T 6= (0, 0, . . . , 0)T .
Hence
deg (QN(x1, . . . , xn, y1, . . . , ym)T ,Ω
⋂
KerL, (0, 0, . . . , 0)T )
= deg ((−x1, . . . ,−xn,−y1, . . . ,−ym)T ,Ω
⋂
KerL, (0, 0, . . . , 0)T ) 6= 0.
This satisfies condition (c) of Lemma 2.1. Thus by Lemma 2.1 it follows that Lx = Nx
has at least one solution in X, namely, system (1.1) has at least one ω-periodic solution.
Theorem 2.1 is proved.
3. Global exponential stability of the periodic solution. In this section, we will
construct some suitable Lyapunov function to study the global exponential stability of
the periodic solution of system (1.1).
Theorem 3.1. If assumptions (A1), (A2) hold, and furthermore assume that
(A3) The following inequalities hold:
a−i −
m∑
j=1
(p+ji + q+ji)L
g
i > 0, i = 1, 2, . . . , n,
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 12
EXISTENCE AND EXPONENTIAL STABILITY OF PERIODIC SOLUTION FOR FUZZY. . . 1681
b−j −
n∑
i=1
(α+
ij + β+
ij)L
f
j > 0, j = 1, 2, . . . ,m.
Then the periodic solution of system (1.1) is globally exponentially stable.
Proof. According to Theorem 2.1, we know that system (1.1) has an ω-periodic
solution z∗(t) = (x∗1(t), . . . , x∗n(t), y∗1(t), . . . , y∗m(t))T . Suppose that z(t) = (x1(t), . . .
. . . , xn(t), y1(t), . . . , ym(t))T is an arbitrary solution of system (1.1), then it follows
from system (1.1) that
d
dt
(xi(t)− x∗i (t)) = −ai(t)(xi(t)− x∗i (t))+
+
m∧
j=1
αij(t)fj(yj(t))−
m∧
j=1
αij(t)fj(y
∗
j (t))+
+
m∨
j=1
βij(t)fj(yj(t))−
m∨
j=1
βij(t)fj(y
∗
j (t)), i = 1, 2, . . . , n,
d
dt
(yj(t)− y∗j (t)) = −bj(t)(yj(t)− y∗j (t)) +
n∧
i=1
pji(t)gi(xi(t))−
n∧
i=1
pji(t)gi(x
∗
i (t))+
+
n∨
i=1
qji(t)gi(xi(t))−
n∨
i=1
qji(t)gi(x
∗
i (t)), j = 1, 2, . . . ,m.
By (A1) and Lemma 2.1, we have
d+
dt
|xi(t)− x∗i (t)| ≤ −ai(t)|xi(t)− x∗i (t)|+
+
∣∣∣∣∣∣
m∧
j=1
αij(t)fj(yj(t))−
m∧
j=1
αij(t)fj(y
∗
j (t))
∣∣∣∣∣∣+
+
∣∣∣∣∣∣
m∨
j=1
βij(t)fj(yj(t))−
m∨
j=1
βij(t)fj(y
∗
j (t))
∣∣∣∣∣∣ ≤
≤ −a−i |xi(t)− x
∗
i (t)|+
m∑
j=1
(α+
ij + β+
ij)L
f
j |yj(t)− y
∗
j (t)|, (3.1)
d+
dt
|yj(t)− y∗j (t)| ≤ −bj(t)|yj(t)− y∗j (t)|+
+
∣∣∣∣∣
m∧
i=1
pji(t)gi(xi(t))−
n∧
i=1
pji(t)gi(x
∗
i (t))
∣∣∣∣∣+
+
∣∣∣∣∣
n∨
i=1
qji(t)gi(xi(t))−
n∨
i=1
qji(t)gi(x
∗
i (t))
∣∣∣∣∣ ≤
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 12
1682 QIAN-HONG ZHANG, LI-HUI YANG, DAI-XI LIAO
≤ −b−j |yj(t)− y
∗
j (t)|+
n∑
i=1
(p+ji + q+ji)L
g
i |xi(t)− x
∗
i (t)|, (3.2)
where d+/dt denotes the upper right derivative.
Define a Lyapunov function V (·) by
V (t) =
n∑
i=1
|xi(t)− x∗i (t)|+
m∑
j=1
|yj(t)− y∗j (t)|
for t ≥ 0, by virtue of (3.1) and (3.2), we have
d+V (t)
dt
=
n∑
i=1
d+
dt
|xi(t)− x∗i (t)|+
m∑
j=1
d+
dt
|yj(t)− y∗j (t)| ≤
≤
n∑
i=1
−a−i |xi(t)− x∗i (t)|+ m∑
j=1
(α+
ij + β+
ij)L
f
j |yj(t)− y
∗
j (t)|
+
+
m∑
j=1
(
−b−j |yj(t)− y
∗
j (t)|+
n∑
i=1
(p+ji + q+ji)L
g
i |xi(t)− x
∗
i (t)|
)
=
= −
n∑
i=1
a−i − m∑
j=1
(p+ji + q+ji)L
g
i
|xi(t)− x∗i (t)|−
−
m∑
j=1
(
b−j −
n∑
i=1
(α+
ij + β+
ij)L
f
j
)
|yj(t)− y∗j (t)|.
Since (A3) hold, there exists a real number γ > 0 such that
a−i −
m∑
j=1
(p+ji + q+ji)L
g
i ≥ γ, b−j −
n∑
i=1
(α+
ij + β+
ij)L
f
j ≥ γ.
It follows that
d+V (t)
dt
≤ −γV (t) for t ≥ 0. (3.3)
Using exponential stability theorem [23], (3.3) implies that
V (t) ≤ e−γtV (0) ∀t ≥ 0.
That is
n∑
i=1
|xi(t)− x∗i (t)|+
m∑
j=1
|yj(t)− y∗j (t)| ≤
≤ e−γt
n∑
i=1
|xi(0)− x∗i (0)|+
m∑
j=1
|yj(0)− y∗j (0)|
,
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EXISTENCE AND EXPONENTIAL STABILITY OF PERIODIC SOLUTION FOR FUZZY. . . 1683
therefore the periodic solution of system (1.1) is globally exponentially stable.
Theorem 3.1 is proved.
4. Example. In this section, we consider the following fuzzy BAM neural networks
with periodic coefficient
x′i(t) = −ai(t)xi(t) +
2∧
j=1
αijfj(yj(t)) +
2∧
j=1
Tij(t)uj(t) + Ii(t)+
+
2∨
j=1
βij(t)fj(yj(t)) +
2∨
j=1
Hij(t)uj(t), i = 1, 2,
y′j(t) = −bj(t)yj(t) +
2∧
i=1
pji(t)gi(xi(t)) +
2∧
i=1
Kji(t)ui(t) + Jj(t)+
+
2∨
i=1
qjigi(xi(t)) +
2∨
i=1
Nji(t)ui(t), j = 1, 2,
(4.1)
where a1(t) = 12 − cos 2t, a2(t) = 13 − 2 cos 2t, b1(t) = 13 + sin 2t, b2(t) = 13 −
− 2 sin 2t, α11(t) = α21(t) = 1 + sin 2t, α12(t) = α22(t) = 2 + sin 2t, β11(t) =
= β21(t) = 1 − sin 2t, β12(t) = β22(t) = 2 − sin 2t, p11(t) = p21(t) = 1 + cos 2t,
p12(t) = p22(t) = 2+cos 2t, q11(t) = q21(t) = 1−cos 2t, q12(t) = q22(t) = 2−cos 2t,
Tij(t) = Hij(t) = sin 2t, Kji(t) = Nji(t) = cos 2t, ui(t) = uj(t) = 2 sin 2t, i, j =
= 1, 2, Ii(t) = Jj(t) = 2 cos 2t, i, j = 1, 2. Take fi(x) = gi(x) =
1
2
(|x+ 1| − |x− 1|),
i = 1, 2, we have Lgi = Lfj = 1, i, j = 1, 2. By simple computation, we have
a−1 = 11, a−2 = 11, b−1 = 12, b−2 = 11,
α+
11 = α+
21 = 2, α+
12 = α+
22 = 3, β+
11 = β+
21 = 2,
β12 = β+
22 = 3, p+11 = p+21 = 2, p+12 = p+22 = 3,
q+11 = q21 = 2, q+12 = q+22 = 3.
Obviously, the following inequalities hold
a−i −
2∑
j=1
(p+ji + q+ji)L
g
i > 0, i = 1, 2; b−j −
2∑
i=1
(α+
ij + β+
ij)L
f
i > 0, j = 1, 2.
Hence, it follows that the assumptions (A1) – (A3) are satisfied. Therefore, according
to Theorems 2.1 and 3.1, system (4.1) has one π-periodic solution which is globally
exponentially stable.
5. Conclusion. In this paper, we use the continuation theorem of coincidence degree
theory and Lyapunov function to study the existence and global exponential stability of
periodic solution for fuzzy BAM neural networks with periodic coefficient. The suffi-
cient conditions of existence and global stability of periodic solution are easily verifiable.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 12
1684 QIAN-HONG ZHANG, LI-HUI YANG, DAI-XI LIAO
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|
| id | umjimathkievua-article-2833 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:31:14Z |
| publishDate | 2011 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/70/e72d2454933abe861196afbe2d7c1770.pdf |
| spelling | umjimathkievua-article-28332020-03-18T19:37:39Z Existence and exponential stability of periodic solution for fuzzy BAM neural networks with periodic coefficient Існування та експоненцiальна стiйкiсть перiодичного розв’язку для нечiтких нейронних мереж Коско з перiодичними коефiцiєнтами Dai-xi, Liao Li-hui, Yang Zhang, Qian-hong Дай-сі, Ляо Лі-гуей, Ян Чжан, Цянь-хун A class of fuzzy bidirectional associated memory (BAM) networks with periodic coefficients is studied. Some sufficient conditions are established for the existence and global exponential stability of a periodic solution of such fuzzy BAM neural networks by using a continuation theorem based on the coincidence degree and the Lyapunov-function method. The sufficient conditions are easy to verify in pattern recognition and automatic control. Finally, an example is given to show the feasibility and efficiency of our results. Вивчено клас нечiтких нейронних мереж Коско з перiодичним коефiцiєнтом. За допомогою теореми про продовження, що базується на ступенi збiгу та методi функцiй Ляпунова, встановлено достатнi умови для iснування та глобальної експоненцiальної стiйкостi перiодичного розв’язку таких нечiтких нейронних мереж Коско. Цi достатнi умови легко перевiряються при розпiзнаваннi образiв та автоматичному керуваннi. Наведено приклад, що демонструє застосовнiсть та ефективнiсть отриманих результатiв. Institute of Mathematics, NAS of Ukraine 2011-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2833 Ukrains’kyi Matematychnyi Zhurnal; Vol. 63 No. 12 (2011); 1672-1684 Український математичний журнал; Том 63 № 12 (2011); 1672-1684 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2833/2419 https://umj.imath.kiev.ua/index.php/umj/article/view/2833/2420 Copyright (c) 2011 Dai-xi Liao; Li-hui Yang; Zhang Qian-hong |
| spellingShingle | Dai-xi, Liao Li-hui, Yang Zhang, Qian-hong Дай-сі, Ляо Лі-гуей, Ян Чжан, Цянь-хун Existence and exponential stability of periodic solution for fuzzy BAM neural networks with periodic coefficient |
| title | Existence and exponential stability of periodic solution for fuzzy BAM neural networks with periodic coefficient |
| title_alt | Існування та експоненцiальна стiйкiсть перiодичного розв’язку для нечiтких нейронних мереж Коско з перiодичними коефiцiєнтами |
| title_full | Existence and exponential stability of periodic solution for fuzzy BAM neural networks with periodic coefficient |
| title_fullStr | Existence and exponential stability of periodic solution for fuzzy BAM neural networks with periodic coefficient |
| title_full_unstemmed | Existence and exponential stability of periodic solution for fuzzy BAM neural networks with periodic coefficient |
| title_short | Existence and exponential stability of periodic solution for fuzzy BAM neural networks with periodic coefficient |
| title_sort | existence and exponential stability of periodic solution for fuzzy bam neural networks with periodic coefficient |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2833 |
| work_keys_str_mv | AT daixiliao existenceandexponentialstabilityofperiodicsolutionforfuzzybamneuralnetworkswithperiodiccoefficient AT lihuiyang existenceandexponentialstabilityofperiodicsolutionforfuzzybamneuralnetworkswithperiodiccoefficient AT zhangqianhong existenceandexponentialstabilityofperiodicsolutionforfuzzybamneuralnetworkswithperiodiccoefficient AT dajsílâo existenceandexponentialstabilityofperiodicsolutionforfuzzybamneuralnetworkswithperiodiccoefficient AT líguejân existenceandexponentialstabilityofperiodicsolutionforfuzzybamneuralnetworkswithperiodiccoefficient AT čžancânʹhun existenceandexponentialstabilityofperiodicsolutionforfuzzybamneuralnetworkswithperiodiccoefficient AT daixiliao ísnuvannâtaeksponencialʹnastijkistʹperiodičnogorozvâzkudlânečitkihnejronnihmerežkoskozperiodičnimikoeficiêntami AT lihuiyang ísnuvannâtaeksponencialʹnastijkistʹperiodičnogorozvâzkudlânečitkihnejronnihmerežkoskozperiodičnimikoeficiêntami AT zhangqianhong ísnuvannâtaeksponencialʹnastijkistʹperiodičnogorozvâzkudlânečitkihnejronnihmerežkoskozperiodičnimikoeficiêntami AT dajsílâo ísnuvannâtaeksponencialʹnastijkistʹperiodičnogorozvâzkudlânečitkihnejronnihmerežkoskozperiodičnimikoeficiêntami AT líguejân ísnuvannâtaeksponencialʹnastijkistʹperiodičnogorozvâzkudlânečitkihnejronnihmerežkoskozperiodičnimikoeficiêntami AT čžancânʹhun ísnuvannâtaeksponencialʹnastijkistʹperiodičnogorozvâzkudlânečitkihnejronnihmerežkoskozperiodičnimikoeficiêntami |