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Existence and exponential stability of almost periodic solution for shunting inhibitory cellular neural networks with impulses
Chaos, Solitons and Fractals 34 (2007) 1599–1607 www.elsevier.com/locate/chaos
Existence and exponential stability of almost periodic solution for shunting inhibitory cellular neural networks with impulses q Yonghui Xia a
a,*
, Jinde Cao b, Zhenkun Huang
c
College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350002, PR China b Department of Mathematics, Southeast University, Nanjing 210096, PR China c School of Sciences, Jimei University, Xiamen, Fujian 361021, PR China Accepted 3 May 2006
Abstract In this paper, by using the contraction principle and Gronwall–Bellman’s inequality, some sufficient conditions are obtained for checking the existence and exponential stability of almost periodic solution for shunting inhibitory cellular neural networks (SICNNs) with impulse. Our results are essentially new. It is the first time that the existence of almost periodic solutions for the impulsive neural networks are obtained. 2006 Elsevier Ltd. All rights reserved.
1. Introduction Cellular networks (CNNs) have been paid much attention in the past decade due to its applicability in signal processing, image processing, pattern recognition, and so on. The theoretical and applied studies of CNNs have been a new focus of studies worldwide (to see [5–8,20–22], and the references cited therein). Bouzerdout and Pinter [1] have introduced a new class of CNNs, namely the shunting inhibitory CNNs (SICNNs). It is well known that SICNNs have been extensively applied in psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image processing. Consider a two-dimensional grid of processing cells, let Cij denote the cell at the (i, j) position of the lattice, the r-neighborhood Nr(i, j) of Cij is N r ði; jÞ ¼ fC hl : maxðjh ij; jl jjÞ 6 r; 1 6 h 6 m; 1 6 l 6 ng: In SICNNs, neighboring cells exert mutual inhibitory interactions of the shunting type. The dynamics of a cell Cij are described by the following nonlinear ordinary differential equation:
q
This work was supported by the Foundation of Developing Science and Technology of Fuzhou University under the grant 2004XY-12, and the National Natural Science Foundation of China under Grants Nos. 60574043 and 60373067. * Corresponding author. E-mail addresses: [email protected] (Y. Xia), [email protected] (J. Cao). 0960-0779/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.05.003
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Y. Xia et al. / Chaos, Solitons and Fractals 34 (2007) 1599–1607
X dxij ðtÞ C hl ¼ aij ðtÞxij ðtÞ þ ij ðtÞfij ðxhl ðtÞÞxij ðtÞ þ Lij ðtÞ; dt C 2N ði;jÞ hl
r
where xij is the activity of the cell Cij, Lij(t) is the external input to Cij, aij represents the passive decay rate of the cell activity, C hl ij is the connection or coupling strength of postsynaptic activity of the cell transmitted to the cell Cij, and the function fij(xhl) is a continuous function representing the output or firing rate of the cell Chl. It is well known that studies on neural dynamic systems not only involve a discussion of stability properties, but also involve many dynamic behaviors such as periodic oscillatory behavior, almost periodic oscillatory properties, chaos and bifurcation. In applications, if the various constituent components of the temporally nonuniform environment is with incommensurable (nonintegral multiples) periods, then one has to consider the environment to be almost periodic since there is no a priori reason to expect the existence of periodic solutions. If we consider the effects of the environmental factors, the assumption of almost periodicity is more realistic, more important and more general. For significance of almost periodicity, one also can refer to [8,10,12,19]. Recently, Chen and Cao [2], Huang and Cao [3], Li et al. [4] obtained several sufficient conditions for the global exponential stability of almost periodic solutions or periodic solutions for the delayed SICNNs. On the other hand, the theory of impulsive differential equations is now being recognized to be not only richer than the corresponding theory of differential equations without impulse, but also represents a more natural framework for mathematical modelling of many real-world phenomena, such as population dynamic and the neural networks. In recent years, the impulsive differential equations have been extensively studied (see the monographs [9,11,13] and the works [14–18,20,23]). However, to the author’s best knowledge, there is no published paper considering the almost periodic solutions for SICNNs neural networks with impulses. In this paper, we study the impulsive SICNNs neural networks with almost periodic coefficients 8 X dx ðtÞ > < ij ¼ aij ðtÞxij ðtÞ þ C hl t 6¼ tk ; ij ðtÞfij ðxhl ðtÞÞxij ðtÞ þ Lij ðtÞ; dt ð1:1Þ C hl 2N r ði;jÞ > : Dxij ðtk Þ ¼ akij xij ðtk Þ þ I kij ðxij ðtk ÞÞ þ Lkij ; t ¼ tk ; i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; n; where Dxij ðtk Þ ¼ xij ðtþ k Þ xij ðt k Þ are impulses at moments tk and t1 < t2 < is a strictly increasing sequence such that limk!1tk = +1. We denote by x(t) = x(t, t0, z0), x = (x11, . . . , x1n, . . . , xi1, . . . , xin, . . . , xm1, . . . , xmn)T, x0 = (x011, . . . , x01n, . . . , x0i1, . . . , x0in, . . . , x0m1, . . . , x0mn)T 2 X, where X is a domain in RðmnÞ , X 5 /. The system (1.1) is supplemented with initial values problem given by
xðt0 þ 0; t0 ; x0 Þ ¼ x0 :
ð1:2Þ
Denote by PCðJ ; Rmn Þ, J R, the space of all piecewise continuous functions x : J ! Rmn with points of discontinuity of the first kind tk, k = ±1, ±2, . . . and which are continuous from the left, i.e., x(tk 0) = x(tk). The rest of this paper is organized as follows. In next section, we shall introduce some definitions and lemmas. Section 3 is devoted to establishing some criteria for the existence, uniqueness and exponential stability of almost periodic solution of system (1.1).
2. Definitions and lemmas In this section, we shall introduce some known definitions and lemmas (to see [9,11,13]). Since the solutions of problem (1.1), (1.2) is a piecewise continuous functions with points of discontinuity of the first kind t = tk, k 2 Z and we adopt the following definitions and Lemmas for almost periodicity. Let B ¼ fftk g1 k¼1 : tk 2 R; tk < t kþ1 ; k 2 Z; limk!1 t k ¼ 1g denote the set of all sequence unbounded and strictly increasing. A matrix or vector D P 0 means that all entries of D are greater than or equal to zero. For matrices or vectors D and E, D P E means D E P 0. Definition 1 [13]. The set of sequences ftjk g; tjk ¼ tkþj tk ; k 2 Z; j 2 Z; ftk g 2 B is said to be uniformly almost periodic if for arbitrary e > 0 there exists relatively dense set of e-almost periods common for any sequences. Definition 2 [13]. A piecewise continuous function u : R ! Rmn with discontinuity of first kind at the points tk is said to be almost periodic, if
Y. Xia et al. / Chaos, Solitons and Fractals 34 (2007) 1599–1607
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(a) the set of sequence ftjk g; tjk ¼ tkþj tk ; k 2 Z; j 2 Z; ftk g 2 B is uniformly almost periodic; (b) for any e > 0 there exists a real number d > 0 such that if the points t 0 and t00 belong to one and the same interval of continuity of u(t) and satisfy the inequality jt 0 t00 j < d, then ju(t 0 ) u(t00 )j < e; (c) for any e > 0 there exists a relatively dense set T such that if s 2 T, then ju(t + s) u(t)j < e for all t 2 R satisfying the condition jt tk j > e; k 2 Z. Together with the system (1) we consider the linear system ( _ ZðtÞ ¼ P ðtÞZðtÞ; t 6¼ tk ; DZðtÞ ¼ P k ZðtÞ; t ¼ tk ; k 2 Z:
ð2:1Þ
Introduce the following conditions: (i) P ðtÞ 2 CðR; Rmn Þ and is almost periodic in the sense of Bohr. (ii) det(E + Pk) 5 0 and the sequence fP k g; k 2 Z is almost periodic, E 2 Rmnmn . (iii) The set of sequences ftjk g; tjk ¼ tkþj tk ; k 2 Z; j 2 Z; ftk g 2 B is uniformly almost periodic and there exists h > 0 such that inf k t1k ¼ h > 0. Recall [9] that if Uk(t, s) is the Cauchy matrix for the system _ ZðtÞ ¼ P ðtÞZðtÞ;
tk1 < t 6 tk ; ftk g 2 B;
then the Cauchy matrix for the system (2.1) is in the form 8 U k ðt; sÞ; tk1 < s 6 t 6 tk ; > > > > < U kþ1 ðt; tk þ 0ÞðE þ P k ÞU k ðt; sÞ; tk1 < s 6 tk < t 6 tkþ1 ; W ðt; sÞ ¼ > U kþ1 ðt; tk þ 0ÞðE þ P k ÞU k ðtk ; tk þ 0Þ ðE þ P i ÞU i ðti ; sÞ; > > > : ti1 < s 6 ti < tk < t 6 tkþ1 : Lemma 1 [13]. In addition to conditions (i)–(iii) are fulfilled. (iv) For the Cauchy matrix W(t, s) of the system (2.1) there exist positive constants K and k such that jW ðt; sÞj 6 KekðtsÞ ;
t P s; t; s 2 R:
Then for any e > 0, t 2 R, S 2 R, t P s, jt tkj > e, js tkj > e, k 2 Z there exists a relatively dense set T of e-almost periods of the matrix P(t) and a positive constant C such that for s 2 T it follows: k
jW ðt þ s; s þ sÞ W ðt; sÞj 6 eCe2ðtsÞ : Lemma 2 [13]. Let the condition (iii) be fulfilled: Then for each p > 0 there exists a positive integer N such that on each interval of length p no more than N elements of the sequence {tk}, i.e., iðt; sÞ 6 N ðt sÞ þ N ; where i(s, t) is the number of the points tk lying in the interval (s, t). Lemma 3 [13]. In addition to condition (iii), let the following conditions be fulfilled: (v) The function u 2 PC(R, X), X RmÆn and it is almost periodic. Then the sequence {u(tk)} is almost periodic. Lemma 4 [13]. In addition to conditions (iii) and (v), let following conditions be fulfilled: (vi) F(y) is uniformly continuous defined in X. Then F(u(t)) is almost periodic function. Lemma 5 [13]. Let g(t), g 2 PC(R, X) and the sequence fgk g; k 2 Z are almost periodic. Then there exists a positive constant C1 such that ! max sup kgðtÞk;
sup
t2R
k¼1;2;...
kgk k
6 C1:
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Y. Xia et al. / Chaos, Solitons and Fractals 34 (2007) 1599–1607
§ Throughout this paper, we always assume i = 1, . . . , m; j = 1, . . . , n, unless otherwise stated. Throughout this paper, we introduce the following conditions: (H1) aij(t), C hl ij ðtÞ and Lij(t) are almost periodic functions in the sense of Bohr, and denote 0 < inf faij ðtÞg ¼ a ij < 1; t2R
e hl ¼ supfjC hl ðtÞjg; C ij ij t2R
e L ij ¼ supfjLij ðtÞjg; t2R
(H2) The condition (iii) holds. (H3) The functions fij(x) is uniformly continuous functions defined on X with 0 < supt2Rjfij(t)j < 1, fij(0) = 0 and further assume that there exist positive constants uij such that for x, y 2 R, jfij(x) fij(y)j 6 uijjx yj. (H4) fakij gk2Z , and fLkij gk2Z are almost periodic sequence and from Lemma 5, there exists strictly positive constants e L ij such that supt2R fjLij ðtÞj; maxk jLkij jg 6 e L ij . (H5) The sequence of functions I kij ðxij ðtk ÞÞ are almost periodic uniformly with respect to x 2 X and there exists vij > 0 such that jI kij ðxÞ I kij ðxÞj 6 vij jx xj; for k 2 Z; x; x 2 X. k (H6) kij :¼ a ij N lnð1 þ maxk jaij jÞ > 0. Now from [13], we have Lemma 6. Assume the conditions (H1), (H2), (H4) hold. Then for each e > 0 there exist e1, 0 < e1 < e and relatively dense sets T of real numbers and Q of whole numbers, such that the following relations are fulfilled: (a) (b) (c) (d) (e) (f) (g)
jP ðt þ sÞ P ðtÞj < e; t 2 R; s 2 T ; jt tk j > e; jP kþq P k j < e; k 2 Z; q 2 Q; jLij ðt þ sÞ Lij ðtÞj < e; t 2 R; s 2 T ; jt tk j > e; k 2 Z; jLkþq Lkij j < e; k 2 Z; q 2 Q; ij hl jC ij ðt þ sÞ C hl ij ðtÞj < e; t 2 R; s 2 T ; jt t k j > e; k 2 Z; jC hl;kþq C hl;k ij ij j < e; k 2 Z; q 2 Q; jtqk sj < e; q 2 Q; s 2 T ; k 2 Z.
Lemma 7. Assume (H1)–(H2), (H4) and (H6) hold. Then 1. For Cauchy matrix W(t, s) of the system (2.1), there exists positive constants K and k such that jW(t, s)j < Kek(ts), t P s, t, s 2 R. 2. For any e > 0, t 2 R; s 2 R, t P s, jt tkj > e > 0, js tkj > 0, k 2 Z, there exists a relatively dense set T of e-almost periods of the matrix P(t) and positive constant C such that for s 2 T it follows k
W ðt þ s; s þ sÞ W ðt; sÞj < eCe2ðtsÞ : Proof. Recall [11] the matrix W(t, s) for system (2.1) is in the form W ðt; sÞ ¼ eP ðtÞðtsÞ one has W ðt; sÞ 6 eP
ðtsÞ
iðs;tÞ ðE þ P þ 6 eP k Þ
ðtsÞ
where P ðtÞ ¼ diagða11 ðtÞ; . . . ; aij ðtÞ; . . . ; amn ðtÞÞðmnÞðmnÞ ; P ¼ diagða 11 ; . . . ; aij ; . . . ; amn ÞðmnÞðmnÞ ; þ k k k ; P k ¼ diag max ja11 j; . . . ; max jaij j; . . . ; max jamn j k k k ðmnÞðmnÞ k nij ¼ exp N ln 1 þ max jakij j ; kij ¼ a N ln 1 þ max ja j ; ij ij k
j ¼ 1; 2; . . . ; n:
iðs;tÞ ðE
þ P k Þ. Then by Lemma 2,
N ðtsÞþN ðE þ P þ k Þ
¼ diagðn11 ek11 ðtsÞ ; . . . ; nij ekij ðtsÞ ; nmn ekmn ðtsÞ ÞðmnÞðmnÞ ;
i ¼ 1; 2; . . . ; m;
Q
k
ð2:2Þ
Y. Xia et al. / Chaos, Solitons and Fractals 34 (2007) 1599–1607
1603
Take K¼
max
16i6m;16j6n
fnij g;
k¼
min
16i6m;16j6n
fkij g:
ð2:3Þ
It follows from (2.2) and condition (H6) that jW ðt; sÞj 6 ekðtsÞ ;
t P s; t; s 2 R:
This completes the Proof of Assertion 1. From Lemma 1, the assertion 2 is immediately proved.
3. Main results Theorem 3.1. In addition to (H1)–(H6) hold, then the following hold: 1. If r < 1, d = d1 + d2 6 1 and
K 1d
6 1, then system (1.1) has a unique almost periodic solution y(t). Where
1 X e hl uij ; d2 ¼ max vij N nij ; d1 ¼ max nij C C hl 2N r ði;jÞ ij ði;jÞ ði;jÞ 1 ekij kij 1 N 2d1 K þ d2 : L ij ; r ¼ n e K ¼ max þ ði;jÞ kij 1 ekij ij 1d n P o e hl uij 2K þ lnð1 þ maxði;jÞ fnij vij gÞ > 0, then the unique solution y(t) is exponentially 2. If k maxði;jÞ nij Chl 2N r ði;jÞ C ij 1d stable. T mÆn Proof of Assertion 1. Let D = {u(t)ju(t) = (u11(t), n.. . , uij(t), . .. , umn(t)) o 2 PC(R, R ) is almost periodic with kuk < K, where kuk = supt2Rmax(i, j)juij(t)j, K ¼ maxði;jÞ k1ij þ 1eNkij nij e L ij . Obviously D PCðR; Rmn Þ. Set Fðt; xðtÞÞ ¼
ðF 11 ðt; xðtÞÞ; . . . ; F ij ðt; xðtÞÞ; . . . ; F mn ðt; xðtÞÞÞT where X C hl F ij ðt; xðtÞÞ ¼ ij ðtÞfij ðxhl ðtÞÞxij ðtÞ; C hl 2N r ði;jÞ
Fk ðxðtk ÞÞ ¼ ðI k11 ðx11 ðtk ÞÞ; . . . ; I kij ðxij ðtk ÞÞ; . . . ; I kmn ðxmn ðtk ÞÞÞT ; H ðtÞ ¼ ðL11 ðtÞ; . . . ; Lij ðtÞ; . . . ; Lmn ðtÞÞT ; H k ¼ ðLk11 ; . . . ; Lkij ; . . . ; Lkmn ÞT : Define an operator S in D Su ¼
Z
t
W ðt; sÞ½Fðs; uðsÞÞ þ H ðsÞ ds þ
1
X
W ðt; tk Þ½Fk ðuðtk ÞÞ þ H k :
ð3:1Þ
tk
Obviously, it is easy to check that Su is a solution of (1.1). Take subset D* D, D ¼
u 2 Dj ku u0 k 6
dK ; 1d
ð3:2Þ
where u0 ¼
Z
t
1
W ðt; sÞH ðsÞ ds þ
X tk
W ðt; tk ÞH k :
ð3:3Þ
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Y. Xia et al. / Chaos, Solitons and Fractals 34 (2007) 1599–1607
From (3.3), it follows Lemma 7 that ( ) Z t X ku0 k ¼ sup max jW ðt; sÞjjLij ðsÞj ds þ max jW ðt; tk Þj max jLkij j ði;jÞ
t2R
Z
kij ðtsÞ
nij e
ði;jÞ
jLij ðsÞj ds þ max
X
ði;jÞ
1
1 N e n :¼ K: þ L ij kij 1 ekij ij
6 max ði;jÞ
k
tk
t
6 sup max t2R
ði;jÞ
1
(
) nij e
kij ðttk Þ
tk
max jLkij j k ð3:4Þ
Then for arbitrary u 2 D*, it follows from (3.1), (3.2) and (3.4) that kuk 6 ku u0 k þ ku0 k 6
d 1 K þK ¼ K: 1d 1d
ð3:5Þ
Now we prove that S is self-mapping from D* to D*. Firstly, we shall show that for arbitrary u 2 D*, then Su 2 D*. In fact, Z t X W ðt; sÞFðs; uðsÞÞ ds þ W ðt; tk ÞFk ðuðtk ÞÞ: Su u0 ¼ 1
ð3:6Þ
tk
1 This,together with (2.2) and kuk 6 1d K 6 1, we have
kSu u0 k ( ¼ sup max ði;jÞ
t2R
(
Z
jW ðt; sÞjj
1
Z ði;jÞ
( 6 max ði;jÞ
( ¼ max ði;jÞ
( 6 max ði;jÞ
1 kij 1 kij 1 kij
C hl ij ðsÞfij ðuhl ðsÞÞuij ðsÞj ds
C hl 2N r ði;jÞ
X
t
6 sup max t2R
X
t
nij e
kij ðtsÞ
1
ði;jÞ
! ) vij N kuk nij 1 ekij C hl 2N r ði;jÞ ! ) X vij N hl e n kuk C ij uij kuk þ 1 ekij ij C hl 2N r ði;jÞ ! ) X e hl uij þ vij N n kuk C ij 1 ekij ij C 2N ði;jÞ hl
ði;jÞ
e hl uij juhl ðsÞjjuij ðsÞj ds þ max C ij
C hl 2N r ði;jÞ
X
þ max
X
) jW ðt; tk Þj
tk
X
nij e
max jI kij ðuij ðtk ÞÞj k )
kij ðttk Þ
vij juij ðtk Þj
tk
e hl uij kuk2 þ C ij
r
d K: :¼ dkuk 6 1d
ð3:7Þ
Secondly, we shall prove that Su is almost periodic. In fact, let s 2 T, q 2 Q, where the sets T and Q are determined in Lemma 6. By Lemmas 6 and 7, we have Z t jW ðt þ s; s þ sÞ W ðt; sÞj kSuðt þ sÞ SuðtÞk 6 sup t2R # "1 X hl C ij ðs þ sÞfij ðuhl ðs þ sÞÞuij ðs þ sÞ þ Lij ðs þ sÞ ds max C 2N ði;jÞ ði;jÞ r hl " Z t X jW ðt; sÞj max C hl ðs þ sÞfij ðuhl ðs þ sÞÞuij ðs þ sÞ þ C 2N ði;jÞ ij ði;jÞ 1 r hl # X hl C ij ðsÞfij ðuhl ðsÞÞuij ðsÞ þ jLij ðs þ sÞ Lij ðsÞj ds C hl 2N r ði;jÞ X kþq þ jW ðt þ s; tkþq Þ W ðt; tk Þj max½jI kþq ij ðuij ðtkþq ÞÞj þ Lij j ði;jÞ
tk
þ
X tk
jW
ðt; tk Þj max½jI kþq ij ðuij ðt kþq ÞÞ ði;jÞ
I kij ðuij ðtk ÞÞj þ jLkþq Lkij j 6 eM; ij
ð3:8Þ
Y. Xia et al. / Chaos, Solitons and Fractals 34 (2007) 1599–1607
where
1605
" ( )#
X 1 2K CN CN K hl e e e max þ L ij þ K þ L ij : M¼ ðC þ KÞ C ij uij þ max vij þ 1 þ k max vij k ði;jÞ C 2N ði;jÞ 1d 1 ek ij 1d 1 e2 ij r
hl
It follows from (3.7) and (3.8) that Su 2 D*. For arbitrary u, w 2 D*, Z t X W ðt; sÞ½Fðs; uðsÞÞ Fðs; wðsÞÞ ds þ W ðt; tk Þ½Fk ðuðtk ÞÞ Fk ðwðtk ÞÞ: Su Sw ¼ 1
ð3:9Þ
tk
It follows from (2.2) and (3.9) that X X hl hl kSu Swk 6 sup max jW ðt;sÞj C ij ðsÞfij ðuhl ðsÞÞuij ðsÞ C ij ðsÞfij ðwhl ðsÞÞwij ðsÞ ds C 2N ði;jÞ ði;jÞ t2R 1 C hl 2N r ði;jÞ r hl )
X k jW ðt;tk Þj max jI kþq þ ij ðuij ðtk ÞÞ I ij ðwij ðt k ÞÞj (
Z
t
(Z
t
ði;jÞ
tk
"
6 sup max þ
nij e
ði;jÞ
t2R
X
1
(Z
"
t
ði;jÞ
þ
1
C hl 2N r ði;jÞ
)
e hl uij jwhl ðsÞjjuij ðsÞ wij ðsÞj þ C ij
C hl 2N r ði;jÞ
X
# e hl uij juhl ðsÞ whl ðsÞjjuij ðsÞj ds C ij
C hl 2N r ði;jÞ
)
nij ekij ðttk Þ vij juij ðtk Þ wij ðtk Þj
tk
(Z
ði;jÞ
( 6 max ði;jÞ
nij ekij ðtsÞ
1
1 kij C
X
t
6 sup max t2R
X
nij ekij ðtsÞ
6 sup max X
C hl 2N r ði;jÞ
# hl e C ij jfij ðuhl ðsÞÞ fij ðwhl ðsÞÞjjuij ðsÞj ds
X
e hl kfij ðwhl ðsÞÞjjuij ðsÞ wij ðsÞj þ C ij
nij ekij ðttk Þ jI kij ðuij ðtk ÞÞ I kij ðwij ðtk ÞÞj
tk
t2R
X
kij ðtsÞ
e hl uij ½kwk þ kukku wkds þ C ij
C hl 2N r ði;jÞ
hl 2N r
) nij ekij ðttk Þ vij ku wk
tk
! )
X
X
2d1 K e hl uij 2K þ vij N C nij ku wk :¼ þ d2 ¼ rku wk: ij kij 1 d 1 e 1d ði;jÞ
ð3:10Þ
Then from (3.10) and condition (H7), it follows that S is contraction mapping in D*. Therefore, there exists a unique y 2 D* such that S y = y, so there exists a unique almost periodic solution y(t) of (1.1). h Proof of Assertion 2. Let x(t) be arbitrary solution of (1.1) with the initial condition (1.2), and y(t) = (y11(t), . . . , yij(t), . . . , ymn(t))T be the unique almost periodic solution of (1.1) with the initial condition y(t0 + 0, t0, y0) = y0. Then from (3.1), we have Z t X W ðt; sÞ½Fðs; xðsÞÞ Fðs; yðsÞÞ ds þ W ðt; tk Þ½Fk ðxðtk ÞÞ Fk ðyðtk ÞÞ: xðtÞ yðtÞ ¼ W ðt; t0 Þðx0 y 0 Þ þ t0
t0
ð3:11Þ It follows from Lemma 7, (2.2) and (3.11) that (Z X t X hl kxðtÞ yðtÞk 6 Kekðtt0 Þ kx0 y 0 k þ sup max jW ðt; sÞj C hl ðsÞf ðx ðsÞÞx ðsÞ C ðsÞf ðy ðsÞÞy ðsÞ ds ij hl ij ij hl ij ij ij C 2N ði;jÞ t2R ði;jÞ t0 C hl 2N r ði;jÞ r hl ) X jW ðt; tk Þj½jI ijkþq ðxij ðtk ÞÞ I kij ðy ij ðtk ÞÞj þ t0 6tk
6 Kekðtt0 Þ kx0 y 0 k þ sup max t2R
þ
X
t0
"
t
(
þ max nij
X
nij ekij ðtsÞ #
e hl uij jxhl ðsÞ y hl ðsÞjjxij ðsÞj ds þ C ij
kij ðttk Þ
nij e
) vij jxij ðtk Þ y ij ðtk Þj 6 Kekðtt0 Þ kx0 y 0 k
)Z t X e hl uij 2K C ekðtsÞ kxðtÞ yðtÞkds þ maxfnij vij g ekðttk Þ kxðtk Þ yðtk Þk: ij ði;jÞ 1 d t 0 tk
X C hl 2N r
e hl uij jy hl ðsÞjjxij ðsÞ y ij ðsÞj C ij
C hl 2N r ði;jÞ
X tk
C hl 2N r ði;jÞ
ði;jÞ
ði;jÞ
(Z
1606
Y. Xia et al. / Chaos, Solitons and Fractals 34 (2007) 1599–1607
Set z(t) = kx(t) y(t)kekt and from Gronwall–Bellman’s Lemma [13], we have (" ( iðt0 ;tÞ X exp k þ max nij kxðtÞ yðtÞk 6 Kkx0 y 0 k 1 þ maxfnij vij g ði;jÞ
ði;jÞ
C hl 2N r ði;jÞ
e hl uij C ij
) )# 2K ðt t0 Þ : 1d
By Lemma 2, then one has kxðtÞ yðtÞk 6 K exp N ln 1 þ maxfnij vij g kx0 y 0 k ði;jÞ
(
( "
exp k max nij ði;jÞ
) # ) 2K hl e þ lnð1 þ maxfnij vij gÞ ðt t0 Þ : C ij uij ði;jÞ 1d ði;jÞ
X C hl 2N r
From the assumption of assertion 2, the solution y(t) is exponentially stable.
h
Remark 1. When system (1.1) n o without impulse, the conditions in Theorem 3.1 reduces to r < 1, d1 6 1 and 1 e where K 0 ¼ maxði;jÞ kij nij L ij .
K0 1d
6 1,
Remark 2. Our results and the method used in the proof are essentially new. To our best knowledge, there is no published paper considering the almost periodic solutions for shunting inhibitory cellular neural networks with impulses. Moreover, it should be pointed out that the precision of Theorem 3.1 is very high due to the high precision computation of each nij and kij in the proof of Lemma 7.
References [1] Bouzerdoum A, Pinter RB. Shunting inhibitory cellular neural networks: derivation and stability analysis. IEEE Trans Circ Syst – I: Fundam Theory Appl 1993;40(3):215–21. [2] Chen A, Cao J. Almost periodic solution of shunting inhibitory CNNs with delays. Phys Lett A 2002;298:161–70. [3] Huang X, Cao J. Almost periodic solution of shunting inhibitory cellular neural networks with time-varying delay. Phys Lett A 2003;314:222–31. [4] Li Y, Liu C, Zhu L. Global exponential stability of periodic solution for shunting inhibitory CNNs with delays. Phys Lett A 2005;337:40–54. [5] Xia YH, Cao J, Lin M. Existence and exponential stability of almost periodic solution for BAM neural networks with impulse, dynamics of continuous, discrete and impulsive systems A, DCDIS proceeding, vol. 3, 2005. p. 248–55. [6] Cao J, Liang J, Lam J. Exponential stability of high-order bidirectional associative memory neural networks with time delays. Physica D 2004;199(3–4):425–36. [7] Cao J, Lu J. Adaptive synchronization of neural networks with or without time-varying delays. Chaos 2006;16:013133 (6 pages). [8] Cao J, Chen A, Huang X. Almost periodic attractor of delayed neural networks with variable coefficients. Phys Lett A 2005;340(1–4):104–20. [9] Bainov DD, Simeonov PS. Theory of impulsive differential equations: periodic solutions and applications. Harlow: Longman; 1993. [10] Xia YH, Cao J. Almost periodicity in an ecological mode with M-predators and N-preys by ‘‘pure-delay tpye’’ system. Nonlinear Dynam 2005;39(3):275–304. [11] Lakshmikantham V, Bainov DD, Simeonov P. Theory of impulsive differential equations. Singapore: World Scientific; 1989. [12] Xia YH, Cao J. Almost periodic solutions for an ecological model with infinite delays. Proc Edinburgh Math Soc, in press. [13] Samoilenko AM, Perestyuk NA. Differential equations with impulses effect. Visca Skola: Kiev; 1987 [in Russian]. [14] Liu X, Ballinger G. Existence and continuability of solutions for differential equations with delays and state-dependent impulses. Nonlinear Anal 2002;51:633–47. [15] Liu X, Ballinger G. Uniform asymptotic stability of impulsive delay differential equations. Comput Math Appl 2001;41:903–15. [16] Yan JR, Zhao A, Yan W. Existence and global attractivity of periodic solution for an impulsive delay differential equation with Allee effect. J Math Anal Appl 2005;309:489–504. [17] Shen JH. Global existence and uniqueness, oscillation and nonoscillation of impulsive delay differential equations. Acta Math Sinica 1997;40(1):53–9 [in Chinese]. [18] Yan JR. Existence and global attractivity of positive periodic solution for an impulsive Lasota–Wazewska model. J Math Anal Appl 2003;279:111–20. [19] Xia YH, Cao J, Lin M. New results on the existence and uniqueness of almost periodic solutions for BAM neural networks with continuously distributed delays. Chaos, Solitons & Fractals, in press, doi:10.1016/j.chaos.2005.10.043.
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[20] Li Y, Xing W, Lu L. Existence and global exponential stability of periodic solution of a class of neural networks with impulses. Chaos, Solitons & Fractals 2006;27:437–45. [21] Cao J, Li P, Wang W. Global synchronization in arrays of delayed neural networks with constant and delayed coupling. Phys Lett A 2006;353(4):318–25. [22] Cao J, Li X. Stability in delayed Cohen–Grossberg neural networks: LMI optimization approach. Physica D: Nonlinear Phenom 2005;212(1–2):54–65. [23] Xia YH. Positive periodic solutions for a neutral impulsive delayed Lotka–Volterra competition systems with the effect of toxic substance. Nonlinear Anal: RWA, in press, doi:10.1016/j.nonrwa.2005.07.002.
Existence and exponential stability of almost periodic solution for shunting inhibitory cellular neural networks with impulses q Yonghui Xia a
a,*
, Jinde Cao b, Zhenkun Huang
c
College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350002, PR China b Department of Mathematics, Southeast University, Nanjing 210096, PR China c School of Sciences, Jimei University, Xiamen, Fujian 361021, PR China Accepted 3 May 2006
Abstract In this paper, by using the contraction principle and Gronwall–Bellman’s inequality, some sufficient conditions are obtained for checking the existence and exponential stability of almost periodic solution for shunting inhibitory cellular neural networks (SICNNs) with impulse. Our results are essentially new. It is the first time that the existence of almost periodic solutions for the impulsive neural networks are obtained. 2006 Elsevier Ltd. All rights reserved.
1. Introduction Cellular networks (CNNs) have been paid much attention in the past decade due to its applicability in signal processing, image processing, pattern recognition, and so on. The theoretical and applied studies of CNNs have been a new focus of studies worldwide (to see [5–8,20–22], and the references cited therein). Bouzerdout and Pinter [1] have introduced a new class of CNNs, namely the shunting inhibitory CNNs (SICNNs). It is well known that SICNNs have been extensively applied in psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image processing. Consider a two-dimensional grid of processing cells, let Cij denote the cell at the (i, j) position of the lattice, the r-neighborhood Nr(i, j) of Cij is N r ði; jÞ ¼ fC hl : maxðjh ij; jl jjÞ 6 r; 1 6 h 6 m; 1 6 l 6 ng: In SICNNs, neighboring cells exert mutual inhibitory interactions of the shunting type. The dynamics of a cell Cij are described by the following nonlinear ordinary differential equation:
q
This work was supported by the Foundation of Developing Science and Technology of Fuzhou University under the grant 2004XY-12, and the National Natural Science Foundation of China under Grants Nos. 60574043 and 60373067. * Corresponding author. E-mail addresses: [email protected] (Y. Xia), [email protected] (J. Cao). 0960-0779/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.05.003
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Y. Xia et al. / Chaos, Solitons and Fractals 34 (2007) 1599–1607
X dxij ðtÞ C hl ¼ aij ðtÞxij ðtÞ þ ij ðtÞfij ðxhl ðtÞÞxij ðtÞ þ Lij ðtÞ; dt C 2N ði;jÞ hl
r
where xij is the activity of the cell Cij, Lij(t) is the external input to Cij, aij represents the passive decay rate of the cell activity, C hl ij is the connection or coupling strength of postsynaptic activity of the cell transmitted to the cell Cij, and the function fij(xhl) is a continuous function representing the output or firing rate of the cell Chl. It is well known that studies on neural dynamic systems not only involve a discussion of stability properties, but also involve many dynamic behaviors such as periodic oscillatory behavior, almost periodic oscillatory properties, chaos and bifurcation. In applications, if the various constituent components of the temporally nonuniform environment is with incommensurable (nonintegral multiples) periods, then one has to consider the environment to be almost periodic since there is no a priori reason to expect the existence of periodic solutions. If we consider the effects of the environmental factors, the assumption of almost periodicity is more realistic, more important and more general. For significance of almost periodicity, one also can refer to [8,10,12,19]. Recently, Chen and Cao [2], Huang and Cao [3], Li et al. [4] obtained several sufficient conditions for the global exponential stability of almost periodic solutions or periodic solutions for the delayed SICNNs. On the other hand, the theory of impulsive differential equations is now being recognized to be not only richer than the corresponding theory of differential equations without impulse, but also represents a more natural framework for mathematical modelling of many real-world phenomena, such as population dynamic and the neural networks. In recent years, the impulsive differential equations have been extensively studied (see the monographs [9,11,13] and the works [14–18,20,23]). However, to the author’s best knowledge, there is no published paper considering the almost periodic solutions for SICNNs neural networks with impulses. In this paper, we study the impulsive SICNNs neural networks with almost periodic coefficients 8 X dx ðtÞ > < ij ¼ aij ðtÞxij ðtÞ þ C hl t 6¼ tk ; ij ðtÞfij ðxhl ðtÞÞxij ðtÞ þ Lij ðtÞ; dt ð1:1Þ C hl 2N r ði;jÞ > : Dxij ðtk Þ ¼ akij xij ðtk Þ þ I kij ðxij ðtk ÞÞ þ Lkij ; t ¼ tk ; i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; n; where Dxij ðtk Þ ¼ xij ðtþ k Þ xij ðt k Þ are impulses at moments tk and t1 < t2 < is a strictly increasing sequence such that limk!1tk = +1. We denote by x(t) = x(t, t0, z0), x = (x11, . . . , x1n, . . . , xi1, . . . , xin, . . . , xm1, . . . , xmn)T, x0 = (x011, . . . , x01n, . . . , x0i1, . . . , x0in, . . . , x0m1, . . . , x0mn)T 2 X, where X is a domain in RðmnÞ , X 5 /. The system (1.1) is supplemented with initial values problem given by
xðt0 þ 0; t0 ; x0 Þ ¼ x0 :
ð1:2Þ
Denote by PCðJ ; Rmn Þ, J R, the space of all piecewise continuous functions x : J ! Rmn with points of discontinuity of the first kind tk, k = ±1, ±2, . . . and which are continuous from the left, i.e., x(tk 0) = x(tk). The rest of this paper is organized as follows. In next section, we shall introduce some definitions and lemmas. Section 3 is devoted to establishing some criteria for the existence, uniqueness and exponential stability of almost periodic solution of system (1.1).
2. Definitions and lemmas In this section, we shall introduce some known definitions and lemmas (to see [9,11,13]). Since the solutions of problem (1.1), (1.2) is a piecewise continuous functions with points of discontinuity of the first kind t = tk, k 2 Z and we adopt the following definitions and Lemmas for almost periodicity. Let B ¼ fftk g1 k¼1 : tk 2 R; tk < t kþ1 ; k 2 Z; limk!1 t k ¼ 1g denote the set of all sequence unbounded and strictly increasing. A matrix or vector D P 0 means that all entries of D are greater than or equal to zero. For matrices or vectors D and E, D P E means D E P 0. Definition 1 [13]. The set of sequences ftjk g; tjk ¼ tkþj tk ; k 2 Z; j 2 Z; ftk g 2 B is said to be uniformly almost periodic if for arbitrary e > 0 there exists relatively dense set of e-almost periods common for any sequences. Definition 2 [13]. A piecewise continuous function u : R ! Rmn with discontinuity of first kind at the points tk is said to be almost periodic, if
Y. Xia et al. / Chaos, Solitons and Fractals 34 (2007) 1599–1607
1601
(a) the set of sequence ftjk g; tjk ¼ tkþj tk ; k 2 Z; j 2 Z; ftk g 2 B is uniformly almost periodic; (b) for any e > 0 there exists a real number d > 0 such that if the points t 0 and t00 belong to one and the same interval of continuity of u(t) and satisfy the inequality jt 0 t00 j < d, then ju(t 0 ) u(t00 )j < e; (c) for any e > 0 there exists a relatively dense set T such that if s 2 T, then ju(t + s) u(t)j < e for all t 2 R satisfying the condition jt tk j > e; k 2 Z. Together with the system (1) we consider the linear system ( _ ZðtÞ ¼ P ðtÞZðtÞ; t 6¼ tk ; DZðtÞ ¼ P k ZðtÞ; t ¼ tk ; k 2 Z:
ð2:1Þ
Introduce the following conditions: (i) P ðtÞ 2 CðR; Rmn Þ and is almost periodic in the sense of Bohr. (ii) det(E + Pk) 5 0 and the sequence fP k g; k 2 Z is almost periodic, E 2 Rmnmn . (iii) The set of sequences ftjk g; tjk ¼ tkþj tk ; k 2 Z; j 2 Z; ftk g 2 B is uniformly almost periodic and there exists h > 0 such that inf k t1k ¼ h > 0. Recall [9] that if Uk(t, s) is the Cauchy matrix for the system _ ZðtÞ ¼ P ðtÞZðtÞ;
tk1 < t 6 tk ; ftk g 2 B;
then the Cauchy matrix for the system (2.1) is in the form 8 U k ðt; sÞ; tk1 < s 6 t 6 tk ; > > > > < U kþ1 ðt; tk þ 0ÞðE þ P k ÞU k ðt; sÞ; tk1 < s 6 tk < t 6 tkþ1 ; W ðt; sÞ ¼ > U kþ1 ðt; tk þ 0ÞðE þ P k ÞU k ðtk ; tk þ 0Þ ðE þ P i ÞU i ðti ; sÞ; > > > : ti1 < s 6 ti < tk < t 6 tkþ1 : Lemma 1 [13]. In addition to conditions (i)–(iii) are fulfilled. (iv) For the Cauchy matrix W(t, s) of the system (2.1) there exist positive constants K and k such that jW ðt; sÞj 6 KekðtsÞ ;
t P s; t; s 2 R:
Then for any e > 0, t 2 R, S 2 R, t P s, jt tkj > e, js tkj > e, k 2 Z there exists a relatively dense set T of e-almost periods of the matrix P(t) and a positive constant C such that for s 2 T it follows: k
jW ðt þ s; s þ sÞ W ðt; sÞj 6 eCe2ðtsÞ : Lemma 2 [13]. Let the condition (iii) be fulfilled: Then for each p > 0 there exists a positive integer N such that on each interval of length p no more than N elements of the sequence {tk}, i.e., iðt; sÞ 6 N ðt sÞ þ N ; where i(s, t) is the number of the points tk lying in the interval (s, t). Lemma 3 [13]. In addition to condition (iii), let the following conditions be fulfilled: (v) The function u 2 PC(R, X), X RmÆn and it is almost periodic. Then the sequence {u(tk)} is almost periodic. Lemma 4 [13]. In addition to conditions (iii) and (v), let following conditions be fulfilled: (vi) F(y) is uniformly continuous defined in X. Then F(u(t)) is almost periodic function. Lemma 5 [13]. Let g(t), g 2 PC(R, X) and the sequence fgk g; k 2 Z are almost periodic. Then there exists a positive constant C1 such that ! max sup kgðtÞk;
sup
t2R
k¼1;2;...
kgk k
6 C1:
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Y. Xia et al. / Chaos, Solitons and Fractals 34 (2007) 1599–1607
§ Throughout this paper, we always assume i = 1, . . . , m; j = 1, . . . , n, unless otherwise stated. Throughout this paper, we introduce the following conditions: (H1) aij(t), C hl ij ðtÞ and Lij(t) are almost periodic functions in the sense of Bohr, and denote 0 < inf faij ðtÞg ¼ a ij < 1; t2R
e hl ¼ supfjC hl ðtÞjg; C ij ij t2R
e L ij ¼ supfjLij ðtÞjg; t2R
(H2) The condition (iii) holds. (H3) The functions fij(x) is uniformly continuous functions defined on X with 0 < supt2Rjfij(t)j < 1, fij(0) = 0 and further assume that there exist positive constants uij such that for x, y 2 R, jfij(x) fij(y)j 6 uijjx yj. (H4) fakij gk2Z , and fLkij gk2Z are almost periodic sequence and from Lemma 5, there exists strictly positive constants e L ij such that supt2R fjLij ðtÞj; maxk jLkij jg 6 e L ij . (H5) The sequence of functions I kij ðxij ðtk ÞÞ are almost periodic uniformly with respect to x 2 X and there exists vij > 0 such that jI kij ðxÞ I kij ðxÞj 6 vij jx xj; for k 2 Z; x; x 2 X. k (H6) kij :¼ a ij N lnð1 þ maxk jaij jÞ > 0. Now from [13], we have Lemma 6. Assume the conditions (H1), (H2), (H4) hold. Then for each e > 0 there exist e1, 0 < e1 < e and relatively dense sets T of real numbers and Q of whole numbers, such that the following relations are fulfilled: (a) (b) (c) (d) (e) (f) (g)
jP ðt þ sÞ P ðtÞj < e; t 2 R; s 2 T ; jt tk j > e; jP kþq P k j < e; k 2 Z; q 2 Q; jLij ðt þ sÞ Lij ðtÞj < e; t 2 R; s 2 T ; jt tk j > e; k 2 Z; jLkþq Lkij j < e; k 2 Z; q 2 Q; ij hl jC ij ðt þ sÞ C hl ij ðtÞj < e; t 2 R; s 2 T ; jt t k j > e; k 2 Z; jC hl;kþq C hl;k ij ij j < e; k 2 Z; q 2 Q; jtqk sj < e; q 2 Q; s 2 T ; k 2 Z.
Lemma 7. Assume (H1)–(H2), (H4) and (H6) hold. Then 1. For Cauchy matrix W(t, s) of the system (2.1), there exists positive constants K and k such that jW(t, s)j < Kek(ts), t P s, t, s 2 R. 2. For any e > 0, t 2 R; s 2 R, t P s, jt tkj > e > 0, js tkj > 0, k 2 Z, there exists a relatively dense set T of e-almost periods of the matrix P(t) and positive constant C such that for s 2 T it follows k
W ðt þ s; s þ sÞ W ðt; sÞj < eCe2ðtsÞ : Proof. Recall [11] the matrix W(t, s) for system (2.1) is in the form W ðt; sÞ ¼ eP ðtÞðtsÞ one has W ðt; sÞ 6 eP
ðtsÞ
iðs;tÞ ðE þ P þ 6 eP k Þ
ðtsÞ
where P ðtÞ ¼ diagða11 ðtÞ; . . . ; aij ðtÞ; . . . ; amn ðtÞÞðmnÞðmnÞ ; P ¼ diagða 11 ; . . . ; aij ; . . . ; amn ÞðmnÞðmnÞ ; þ k k k ; P k ¼ diag max ja11 j; . . . ; max jaij j; . . . ; max jamn j k k k ðmnÞðmnÞ k nij ¼ exp N ln 1 þ max jakij j ; kij ¼ a N ln 1 þ max ja j ; ij ij k
j ¼ 1; 2; . . . ; n:
iðs;tÞ ðE
þ P k Þ. Then by Lemma 2,
N ðtsÞþN ðE þ P þ k Þ
¼ diagðn11 ek11 ðtsÞ ; . . . ; nij ekij ðtsÞ ; nmn ekmn ðtsÞ ÞðmnÞðmnÞ ;
i ¼ 1; 2; . . . ; m;
Q
k
ð2:2Þ
Y. Xia et al. / Chaos, Solitons and Fractals 34 (2007) 1599–1607
1603
Take K¼
max
16i6m;16j6n
fnij g;
k¼
min
16i6m;16j6n
fkij g:
ð2:3Þ
It follows from (2.2) and condition (H6) that jW ðt; sÞj 6 ekðtsÞ ;
t P s; t; s 2 R:
This completes the Proof of Assertion 1. From Lemma 1, the assertion 2 is immediately proved.
3. Main results Theorem 3.1. In addition to (H1)–(H6) hold, then the following hold: 1. If r < 1, d = d1 + d2 6 1 and
K 1d
6 1, then system (1.1) has a unique almost periodic solution y(t). Where
1 X e hl uij ; d2 ¼ max vij N nij ; d1 ¼ max nij C C hl 2N r ði;jÞ ij ði;jÞ ði;jÞ 1 ekij kij 1 N 2d1 K þ d2 : L ij ; r ¼ n e K ¼ max þ ði;jÞ kij 1 ekij ij 1d n P o e hl uij 2K þ lnð1 þ maxði;jÞ fnij vij gÞ > 0, then the unique solution y(t) is exponentially 2. If k maxði;jÞ nij Chl 2N r ði;jÞ C ij 1d stable. T mÆn Proof of Assertion 1. Let D = {u(t)ju(t) = (u11(t), n.. . , uij(t), . .. , umn(t)) o 2 PC(R, R ) is almost periodic with kuk < K, where kuk = supt2Rmax(i, j)juij(t)j, K ¼ maxði;jÞ k1ij þ 1eNkij nij e L ij . Obviously D PCðR; Rmn Þ. Set Fðt; xðtÞÞ ¼
ðF 11 ðt; xðtÞÞ; . . . ; F ij ðt; xðtÞÞ; . . . ; F mn ðt; xðtÞÞÞT where X C hl F ij ðt; xðtÞÞ ¼ ij ðtÞfij ðxhl ðtÞÞxij ðtÞ; C hl 2N r ði;jÞ
Fk ðxðtk ÞÞ ¼ ðI k11 ðx11 ðtk ÞÞ; . . . ; I kij ðxij ðtk ÞÞ; . . . ; I kmn ðxmn ðtk ÞÞÞT ; H ðtÞ ¼ ðL11 ðtÞ; . . . ; Lij ðtÞ; . . . ; Lmn ðtÞÞT ; H k ¼ ðLk11 ; . . . ; Lkij ; . . . ; Lkmn ÞT : Define an operator S in D Su ¼
Z
t
W ðt; sÞ½Fðs; uðsÞÞ þ H ðsÞ ds þ
1
X
W ðt; tk Þ½Fk ðuðtk ÞÞ þ H k :
ð3:1Þ
tk
Obviously, it is easy to check that Su is a solution of (1.1). Take subset D* D, D ¼
u 2 Dj ku u0 k 6
dK ; 1d
ð3:2Þ
where u0 ¼
Z
t
1
W ðt; sÞH ðsÞ ds þ
X tk
W ðt; tk ÞH k :
ð3:3Þ
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Y. Xia et al. / Chaos, Solitons and Fractals 34 (2007) 1599–1607
From (3.3), it follows Lemma 7 that ( ) Z t X ku0 k ¼ sup max jW ðt; sÞjjLij ðsÞj ds þ max jW ðt; tk Þj max jLkij j ði;jÞ
t2R
Z
kij ðtsÞ
nij e
ði;jÞ
jLij ðsÞj ds þ max
X
ði;jÞ
1
1 N e n :¼ K: þ L ij kij 1 ekij ij
6 max ði;jÞ
k
tk
t
6 sup max t2R
ði;jÞ
1
(
) nij e
kij ðttk Þ
tk
max jLkij j k ð3:4Þ
Then for arbitrary u 2 D*, it follows from (3.1), (3.2) and (3.4) that kuk 6 ku u0 k þ ku0 k 6
d 1 K þK ¼ K: 1d 1d
ð3:5Þ
Now we prove that S is self-mapping from D* to D*. Firstly, we shall show that for arbitrary u 2 D*, then Su 2 D*. In fact, Z t X W ðt; sÞFðs; uðsÞÞ ds þ W ðt; tk ÞFk ðuðtk ÞÞ: Su u0 ¼ 1
ð3:6Þ
tk
1 This,together with (2.2) and kuk 6 1d K 6 1, we have
kSu u0 k ( ¼ sup max ði;jÞ
t2R
(
Z
jW ðt; sÞjj
1
Z ði;jÞ
( 6 max ði;jÞ
( ¼ max ði;jÞ
( 6 max ði;jÞ
1 kij 1 kij 1 kij
C hl ij ðsÞfij ðuhl ðsÞÞuij ðsÞj ds
C hl 2N r ði;jÞ
X
t
6 sup max t2R
X
t
nij e
kij ðtsÞ
1
ði;jÞ
! ) vij N kuk nij 1 ekij C hl 2N r ði;jÞ ! ) X vij N hl e n kuk C ij uij kuk þ 1 ekij ij C hl 2N r ði;jÞ ! ) X e hl uij þ vij N n kuk C ij 1 ekij ij C 2N ði;jÞ hl
ði;jÞ
e hl uij juhl ðsÞjjuij ðsÞj ds þ max C ij
C hl 2N r ði;jÞ
X
þ max
X
) jW ðt; tk Þj
tk
X
nij e
max jI kij ðuij ðtk ÞÞj k )
kij ðttk Þ
vij juij ðtk Þj
tk
e hl uij kuk2 þ C ij
r
d K: :¼ dkuk 6 1d
ð3:7Þ
Secondly, we shall prove that Su is almost periodic. In fact, let s 2 T, q 2 Q, where the sets T and Q are determined in Lemma 6. By Lemmas 6 and 7, we have Z t jW ðt þ s; s þ sÞ W ðt; sÞj kSuðt þ sÞ SuðtÞk 6 sup t2R # "1 X hl C ij ðs þ sÞfij ðuhl ðs þ sÞÞuij ðs þ sÞ þ Lij ðs þ sÞ ds max C 2N ði;jÞ ði;jÞ r hl " Z t X jW ðt; sÞj max C hl ðs þ sÞfij ðuhl ðs þ sÞÞuij ðs þ sÞ þ C 2N ði;jÞ ij ði;jÞ 1 r hl # X hl C ij ðsÞfij ðuhl ðsÞÞuij ðsÞ þ jLij ðs þ sÞ Lij ðsÞj ds C hl 2N r ði;jÞ X kþq þ jW ðt þ s; tkþq Þ W ðt; tk Þj max½jI kþq ij ðuij ðtkþq ÞÞj þ Lij j ði;jÞ
tk
þ
X tk
jW
ðt; tk Þj max½jI kþq ij ðuij ðt kþq ÞÞ ði;jÞ
I kij ðuij ðtk ÞÞj þ jLkþq Lkij j 6 eM; ij
ð3:8Þ
Y. Xia et al. / Chaos, Solitons and Fractals 34 (2007) 1599–1607
where
1605
" ( )#
X 1 2K CN CN K hl e e e max þ L ij þ K þ L ij : M¼ ðC þ KÞ C ij uij þ max vij þ 1 þ k max vij k ði;jÞ C 2N ði;jÞ 1d 1 ek ij 1d 1 e2 ij r
hl
It follows from (3.7) and (3.8) that Su 2 D*. For arbitrary u, w 2 D*, Z t X W ðt; sÞ½Fðs; uðsÞÞ Fðs; wðsÞÞ ds þ W ðt; tk Þ½Fk ðuðtk ÞÞ Fk ðwðtk ÞÞ: Su Sw ¼ 1
ð3:9Þ
tk
It follows from (2.2) and (3.9) that X X hl hl kSu Swk 6 sup max jW ðt;sÞj C ij ðsÞfij ðuhl ðsÞÞuij ðsÞ C ij ðsÞfij ðwhl ðsÞÞwij ðsÞ ds C 2N ði;jÞ ði;jÞ t2R 1 C hl 2N r ði;jÞ r hl )
X k jW ðt;tk Þj max jI kþq þ ij ðuij ðtk ÞÞ I ij ðwij ðt k ÞÞj (
Z
t
(Z
t
ði;jÞ
tk
"
6 sup max þ
nij e
ði;jÞ
t2R
X
1
(Z
"
t
ði;jÞ
þ
1
C hl 2N r ði;jÞ
)
e hl uij jwhl ðsÞjjuij ðsÞ wij ðsÞj þ C ij
C hl 2N r ði;jÞ
X
# e hl uij juhl ðsÞ whl ðsÞjjuij ðsÞj ds C ij
C hl 2N r ði;jÞ
)
nij ekij ðttk Þ vij juij ðtk Þ wij ðtk Þj
tk
(Z
ði;jÞ
( 6 max ði;jÞ
nij ekij ðtsÞ
1
1 kij C
X
t
6 sup max t2R
X
nij ekij ðtsÞ
6 sup max X
C hl 2N r ði;jÞ
# hl e C ij jfij ðuhl ðsÞÞ fij ðwhl ðsÞÞjjuij ðsÞj ds
X
e hl kfij ðwhl ðsÞÞjjuij ðsÞ wij ðsÞj þ C ij
nij ekij ðttk Þ jI kij ðuij ðtk ÞÞ I kij ðwij ðtk ÞÞj
tk
t2R
X
kij ðtsÞ
e hl uij ½kwk þ kukku wkds þ C ij
C hl 2N r ði;jÞ
hl 2N r
) nij ekij ðttk Þ vij ku wk
tk
! )
X
X
2d1 K e hl uij 2K þ vij N C nij ku wk :¼ þ d2 ¼ rku wk: ij kij 1 d 1 e 1d ði;jÞ
ð3:10Þ
Then from (3.10) and condition (H7), it follows that S is contraction mapping in D*. Therefore, there exists a unique y 2 D* such that S y = y, so there exists a unique almost periodic solution y(t) of (1.1). h Proof of Assertion 2. Let x(t) be arbitrary solution of (1.1) with the initial condition (1.2), and y(t) = (y11(t), . . . , yij(t), . . . , ymn(t))T be the unique almost periodic solution of (1.1) with the initial condition y(t0 + 0, t0, y0) = y0. Then from (3.1), we have Z t X W ðt; sÞ½Fðs; xðsÞÞ Fðs; yðsÞÞ ds þ W ðt; tk Þ½Fk ðxðtk ÞÞ Fk ðyðtk ÞÞ: xðtÞ yðtÞ ¼ W ðt; t0 Þðx0 y 0 Þ þ t0
t0
ð3:11Þ It follows from Lemma 7, (2.2) and (3.11) that (Z X t X hl kxðtÞ yðtÞk 6 Kekðtt0 Þ kx0 y 0 k þ sup max jW ðt; sÞj C hl ðsÞf ðx ðsÞÞx ðsÞ C ðsÞf ðy ðsÞÞy ðsÞ ds ij hl ij ij hl ij ij ij C 2N ði;jÞ t2R ði;jÞ t0 C hl 2N r ði;jÞ r hl ) X jW ðt; tk Þj½jI ijkþq ðxij ðtk ÞÞ I kij ðy ij ðtk ÞÞj þ t0 6tk
6 Kekðtt0 Þ kx0 y 0 k þ sup max t2R
þ
X
t0
"
t
(
þ max nij
X
nij ekij ðtsÞ #
e hl uij jxhl ðsÞ y hl ðsÞjjxij ðsÞj ds þ C ij
kij ðttk Þ
nij e
) vij jxij ðtk Þ y ij ðtk Þj 6 Kekðtt0 Þ kx0 y 0 k
)Z t X e hl uij 2K C ekðtsÞ kxðtÞ yðtÞkds þ maxfnij vij g ekðttk Þ kxðtk Þ yðtk Þk: ij ði;jÞ 1 d t 0 tk
X C hl 2N r
e hl uij jy hl ðsÞjjxij ðsÞ y ij ðsÞj C ij
C hl 2N r ði;jÞ
X tk
C hl 2N r ði;jÞ
ði;jÞ
ði;jÞ
(Z
1606
Y. Xia et al. / Chaos, Solitons and Fractals 34 (2007) 1599–1607
Set z(t) = kx(t) y(t)kekt and from Gronwall–Bellman’s Lemma [13], we have (" ( iðt0 ;tÞ X exp k þ max nij kxðtÞ yðtÞk 6 Kkx0 y 0 k 1 þ maxfnij vij g ði;jÞ
ði;jÞ
C hl 2N r ði;jÞ
e hl uij C ij
) )# 2K ðt t0 Þ : 1d
By Lemma 2, then one has kxðtÞ yðtÞk 6 K exp N ln 1 þ maxfnij vij g kx0 y 0 k ði;jÞ
(
( "
exp k max nij ði;jÞ
) # ) 2K hl e þ lnð1 þ maxfnij vij gÞ ðt t0 Þ : C ij uij ði;jÞ 1d ði;jÞ
X C hl 2N r
From the assumption of assertion 2, the solution y(t) is exponentially stable.
h
Remark 1. When system (1.1) n o without impulse, the conditions in Theorem 3.1 reduces to r < 1, d1 6 1 and 1 e where K 0 ¼ maxði;jÞ kij nij L ij .
K0 1d
6 1,
Remark 2. Our results and the method used in the proof are essentially new. To our best knowledge, there is no published paper considering the almost periodic solutions for shunting inhibitory cellular neural networks with impulses. Moreover, it should be pointed out that the precision of Theorem 3.1 is very high due to the high precision computation of each nij and kij in the proof of Lemma 7.
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