Existence and exponential stability of an equilibrium point for fuzzy BAM neural networks with time-varying delays in leakage terms on time scales

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1 L et al Advances n Dfference Equatons 203, 203:28 R E S E A R C H Open Access Exstence and exponental stablty of an equlbrum pont for fuzzy BAM neural networks wth tme-varyng delays n leakage terms on tme scales Yongkun L *, L Yang and Le Sun * Correspondence: ykle@ynueducn Department of Mathematcs, Yunnan Unversty, Kunmng, Yunnan 65009, People s Republc of Chna Abstract In ths paper, by usng a fxed pont theorem and dfferental nequalty technques, we consder the exstence and global exponental stablty of an equlbrum pont for a class of fuzzy bdrectonal assocatve memory neural networks wth tme-varyng delays n leakage terms on tme scales We also present a numercal example to show the feasblty of obtaned results The results of ths paper are completely new and complementary to the prevously known results MSC: 34B37; 34N05 Keywords: fuzzy BAM neural networks; exponental stablty; leakage terms; tme scales Introducton The bdrectonal assocatve memory BAM neural networks were frst ntroduced by Kosto n 988 ] These are specal recurrent neural networks that can store bpolar vector pars and are composed of neurons arranged n two layers The neurons n one layer are fully nterconnected to the neurons n the other layer, whle there are no nterconnectons among neurons n the same layer In recent years, due to ther wde range of applcatons, for example, pattern recognton, assocatve memory, and combnatoral optmzaton, BAM neural networks have receved much attenton There are lots of results on the exstence and stablty of an equlbrum pont, perodc solutons or almost perodc solutons of BAM neural networks 20] Based on tradtonal cellular neural networks, Yang and Yang proposed a fuzzy cellular neural network, whch ntegrates fuzzy logc nto the structure of tradtonal cellular neural networks and mantans local connectedness among cells ] The fuzzy neural network has fuzzy logc between ts template nput and/or output besdes the sum of product operaton Studes have revealed that the fuzzy neural network s very useful for mage processng problems, whch s a cornerstone n mage processng and pattern recognton Besdes, n realty, tme delays often occur due to fnte swtchng speeds of the amplfers and communcaton tme and can destroy a stable network or cause sustaned oscllatons, bfurcaton or chaos Hence, t s mportant to consder both the fuzzy logc and delay effect on dynamcal behavors of neural networks There have been many results on the 203 L et al; lcensee Sprnger Ths s an Open Access artcle dstrbuted under the terms of the Creatve Commons Attrbuton Lcense whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted

2 L et al Advances n Dfference Equatons 203, 203:28 Page 2 of 6 fuzzy neuralnetworks wth tme delays 27] Moreover, tme delay n the leakage term also has a great mpact on the dynamcs of neural networks As ponted out by the author n 8], tme delay n the stablzng negatve feedback term has a tendency to destablze a system Therefore, t s meanngful to consder fuzzy neural networks wth tme delays n the leakage terms 924] In fact, both contnuous and dscrete systems are very mportant n mplementaton and applcatons To avod the trouble of studyng the dynamcal propertes for contnuous and dscrete systems respectvely, t s meanngful to study those on tme scales, whch was ntated by Stefan Hlger n hs PhD thess, n order to unfy contnuous and dscrete analyses Lots of scholars have studed neural networks on tme scales and obtaned many good results 2534] However, to the best of our knowledge, there s no paper publshed on the stablty of fuzzy BAM neural networks wth tme delays n the leakage terms on tme scales Motvated by the above, n ths paper, we ntegrate fuzzy operatons nto BAM neural networks wth tme delays n the leakage terms and study the stablty of consdered neural networks on tme scales By usng a fxed pont theorem and dfferental nequalty technques, we consder the exstence and global exponental stablty of an equlbrum pont for the followng BAM neural network wth tme-varyng delays n leakage terms on tme scales: x t=a x t δ t m c f y t τ t m α f y t τ t m T μ m β f y t τ t m H μ I, t T, =,2,,n, y t=b y t η t n d g x t σ t n p g x t σ t n F ν n q g x t σ t n G ν J, t T, =,2,,m, where T s a tme scale; n, m are the number of neurons n layers; x t andy t denote the actvatons of the th neuron and the th neuron at tme t; a >0andb >0represent therateatwhchtheth neuron andthe th neuron wll reset ther potental to the restng state n solaton when they are dsconnected from the network and the external nputs; 0<δ < δ and 0 < η < η denote the leakage delays; f, g are the nput-output functons the actvaton functons; 0 τ t τ and 0 σ t σ are transmsson delays; t τ t T, t σ t T, t δ t 0, T and t η t 0, T ; c, d are elements of feedback templates; α, p denote the elements of fuzzy feedback MIN templates and β, q are the elements of fuzzy feedback MAX templates; T, F are fuzzy feed-forward MIN templates and H, G are fuzzy feed-forward MAX templates; μ, ν denote the nput of the th neuron and the th neuron; I, J denote bases of the th neuron and the th neuron, =,2,,n, =,2,,m, and denote the fuzzy AND and fuzzy OR operatons, respectvely The ntal condton of softheform { x s=ϕ s, s σ,0] T, σ = max{max, σ, max δ }, =,2,,n, y s=ψ s, s τ,0] T, τ = max{max, τ, max η }, =,2,,m, where ϕ, ψ denote postve real-valued contnuous functons on σ,0] T τ,0] T,respectvely and

3 L et al Advances n Dfference Equatons 203, 203:28 Page 3 of 6 For the sake of convenence, we ntroduce some notatons For matrx D, D T denotes the transpose of D, ρd denotes the spectral radus of D AmatrxoravectorD 0means that all entres of D are greater than or equal to zero, D > 0 can be defned smlarly For matrces or vectors D and E, D E respectvely D > EmeansthatD E 0respectvely D E >0 Throughout ths paper, we assume that the followng condton holds: H f, g CR, R and there exst postve constants L f, Lg such that f uf v f L u v, g ug v g L u v for all u, v R, =,2,,n, =,2,,m The organzaton of the rest of ths paper s as follows In Secton 2,wentroducesome prelmnary results whch are needed n the later sectons In Secton 3,we establsh some suffcent condtons for the exstence and unqueness of the equlbrum pont of In Secton 4, we prove the equlbrum pont of s globally exponentally stable In Secton 5, we gve an example to llustrate the feasblty and effectveness of our results obtaned n prevous sectons 2 Prelmnares In ths secton, we state some prelmnary results Defnton 2 25] LetT be a nonempty closed subset tme scale of R Theforward and backward ump operators σ, ρ : T T and the granness μ : T R are defned, respectvely, by σ t=nf{s T : s > t}, ρt=sup{s T : s < t} and μt=σ tt Lemma 2 25] Assume that p, q : T R are two regressve functons, then e 0 t, s and e p t, t ; e p t, s= e p s,t = e ps, t; e p t, se p s, r=e p t, r; v e p t, s = pte p t, s Lemma 22 25] Let f, gbe -dfferentable functons on T, then ν f ν 2 g = ν f ν 2 g for any constants ν, ν 2 ; fg t=f tgtf σ tg t=f tg tf tgσ t Lemma 23 35] Assume that pt 0 for t s, then e p t, s Defnton 22 35] A functon r : T R s called regressve f μtrt 0 for all t T k The set of all regressve and rd-contnuous functons r : T R wll be denoted by RWedefnethesetR = {r R :μtrt>0, t T} Lemma 24 35] Suppose that p R, then

4 L et al Advances n Dfference Equatons 203, 203:28 Page 4 of 6 e p t, s>0for all t, s T; f pt qt for all t s, t, s T, then e p t, s e q t, s for all t s Lemma 25 35] If p R and a, b, c T, then ep c, ] =p ep c, ] σ and b a pte p c, σ t t = ep c, ae p c, b Lemma 26 35] Let a T k, b T and assume that f : T T k R s contnuous at t, t, where t T k wth t > a Also assume that f t, s rd-contnuous on a, σ t] Suppose that for each ε >0,there exsts a neghborhood U of τ a, σ t] such that f σ t, τ f s, τf t, τ σ ts ε σ ts, s U, where f denotes the dervatve of f wth respect to the frst varable Then gt:= t a f t, τ τ mples g t:= t a f t, τ τ f σ t, t; ht:= b t f t, τ τ mples h t:= b t f t, τ τ f σ t, t Defnton 23 Apontz =x, x 2,,x n, y, y 2,,y m T R nm s sad to be an equlbrum pont of fzt=z s a soluton of Lemma 27 7] Let f be defned on R, =,2,,m Then, for any a R, =,2,,n, =,2,,m, we have the followng estmatons: m m m a f u a f v a f u f v and m m m a f u a f v a f u f v, where u, v R, =,2,,m Defnton 24 36] ArealmatrxA =a nn s sad to be an M-matrx f a 0,, =,2,,n, and all successve prncpal mnors of A are postve Lemma 28 36] Let A =a nn be a matrx wth nonpostve off-dagonal elements, then the followng statements are equvalent: A s an M-matrx; there exsts a vector η >0such that Aη >0; there exsts a vector ξ >0such that ξ T A >0; v there exsts a postve defnte n n dagonal matrx D such that AD DA T >0

5 L et al Advances n Dfference Equatons 203, 203:28 Page 5 of 6 Lemma 29 36] Let A 0 be an l lmatrxwthρa <,then E l A 0, where ρa denotes the spectral radus of A and E l s the dentty matrx of sze l Defnton 25 Let z =x, x 2,,x n, y, y 2,,y m T be an equlbrum pont of If there exsts a postve constant λ wth λ R such that for ϑ,0] T,thereexsts M > such that for an arbtrary soluton zt=x t, x 2 t,,x n t, y t, y 2 t,,y m t T of wth ntal value φs=ϕ s, ϕ 2 s,,ϕ n s, ψ s, ψ 2 s,,ψ m s T satsfes ztz M e λ t,, t, T, t, where ϑ = max{σ, τ}, ztz = max{max n { x tx }, max m{ y ty }}, φ z = max{max n max s σ,0]t { ϕ sx }, max m max s τ,0]t { ψ sy }} Then the equlbrum pont z s sad to be globally exponentally stable 3 Exstence and unqueness of an equlbrum pont In ths secton, we study the exstence and unqueness of an equlbrum pont of Theorem 3 Let H hold Suppose further that ρf<,where F = A BL, A = daga, a 2,,a n, b, b 2,,b m, L = dagl f, Lf 2,,Lf m, L g, Lg 2,,Lg n and B = 0 nn P T Q T 0 mm, P = c α β mn, Q = d p q nm Then has one unque equlbrum pont Proof Let z =x, x 2,,x n, y, y 2,,y m T be an equlbrum pont of, then we have a x = m c f y m α f y m T μ m β f y m H μ I, b y = n d g x n p g x n F ν n q g x n G ν J, where =,2,,n, =,2,,m Defne a mappng : R nm R nm as follows: x, x 2,,x n, y, y 2,,y m =, 2,, n, n,, nm T, where = a m c f y m α f y m T μ m β f y m n = b H μ I ], n d g x n p g x n F ν n q g x n G ν J ] for =,2,,n, =,2,,m Obvously, we need to show that : R nm R nm s a contracton mappng on R nm Infact,forany =h, h 2,,h n, v, v 2,,v m and =

6 L et al Advances n Dfference Equatons 203, 203:28 Page 6 of 6 h, h 2,, h n, v, v 2,, v m R nm,wehave and h h m = c f v f v a a a m β f v f v a m α f v f v m c α β L f v v, =,2,,n n v n v n = d g h g h b b It follows that b n q g h g h b n p g h g h n d p q L g h h, =,2,,m h, h 2,,h n, v, v 2,,v m h, h 2,, h n, v, v 2,, v m a m c α β L f v v h h a m n c n α n β n L f v v b n d p q L g h h = F h n h n v v 3 b n m d m p m q m L g h h v m v m Let N be a postve nteger In vew of 3, we have N h,,h n, v,,v m N h,, h n, v,, v m F N h h,, h n h n, v v,, v m v m T Snce ρf<,weobtanlm N F N = 0, whch mples that there exst a postve nteger M and a postve constant r <suchthat F M = A BL M =l nmnm, nm l r, =,2,,n m

7 L et al Advances n Dfference Equatons 203, 203:28 Page 7 of 6 Hence, we have M h,,h n, v,,v m M h,, h n, v,, v m h h h h F M h n h n v v F M h n h n v v = v m v m v m v m nm l nm l n nm l n nm l nm whch mples that M M r Sncer <, t s obvous that the mappng M : R nm R nm s a contracton mappng By the fxed pont theorem of a Banach space, possesses a unque fxed pont n R nm, that s, there exsts a unque equlbrum pont of The proof of Theorem 3 s completed 4 Global exponental stablty of an equlbrum pont In ths secton, we study the global exponental stablty of the equlbrum pont of Theorem 4 Let H and ρf<hold Suppose further that H 5 max{ɛ, ɛ 2 } <, where ɛ = max n { a δ δ a m c α β } L f, and ɛ 2 = max m { b η η b n d p q } L g Then the equlbrum pont of s globally exponentally stable Proof By Theorem 3, has a unque equlbrum pont z =x, x 2,,x n, y, y 2,, y m T Supposethatzt=x t, x 2 t,,x n t, y t, y 2 t,,y m t T s an arbtrary soluton of wth the ntal condton φs=ϕ s, ϕ 2 s,,ϕ n s, ψ s, ψ 2 s,,ψ m s T Let wt=u t, u 2 t,,u n t, v t, v 2 t,,v m t T = ztz,whereu t=x tx, v t=y ty, =,2,,n, =,2,,mThencanberewrttenas u t=a u t δ t m c f v t τ t m α f v t τ t y m α f y m β f v t τ t y m β f y, =,2,,n, v t=b v t η t n d g u t σ t n p g u t σ t x n p g x n q g u t σ t x n q g x, =,2,,m, 4 where t T and f v t τ t = f y t τ t f y, g u t σ t = g x t σ t g x

8 L et al Advances n Dfference Equatons 203, 203:28 Page 8 of 6 The ntal condton of 4 s the followng: { u s=ϕ sx, v s=ψ sy, s σ,0] T, =,2,,n, s τ,0] T, =,2,,m We rewrte 4 as follows: and u sa u s=a s sδ s m u θ θ c f v s τ s m α f v s τ s y m m α f y β f v s τ s y m β f y, =,2,,n 42 v sb v s=b s sη s n v θ θ n d g u s σ s p g u s σ s x n n p g x q g u s σ s x n q g x, =,2,,m 43 Multplyng both sdes of 42 bye a t, σ s and ntegratng on, t] T,where ϑ,0] T,weget u t=u e a t, { t s e a t, σ s a sδ s u θ θ m c f v s τ s m α f v s τ s y m m α f y β f v s τ s y m } β f y s, =,2,,n 44 Smlarly, multplyng both sdes of 43bye b t, σ s and ntegratng on, t] T,weget v t=v e b t, { t s e b t, σ s b sη s v θ θ n d g u s σ s

9 L et al Advances n Dfference Equatons 203, 203:28 Page 9 of 6 n p g u s σ s x n n p g x q g u s σ s x n } q g x s, =,2,,m 45 For a postve constant α < mn{mn n a, mn m b } wth α R, we have e α t, >fort < TakeM > max{ ɛ, ɛ 2 }InvewofH 5, t s obvous that M > Hence, we have wt Me α t,, t ϑ, t0 ] T We clam that wt Me α t,, t, T 46 To prove ths clam, we show that for any p >, the followng nequalty holds: wt < pme α t,, t, T, 47 whch means that for =,2,,n,wehave u t < pme α t,, t t0, T 48 and for =,2,,m,wehave v t < pme α t,, t, T 49 By way of contradcton, assume that 47 does not hold Frstly, we consder the followng two cases Case One: 48 s not true and 49 s true Then there exst t, T and 0 {,2,,n} such that u0 t pme α t,, u0 t < pme α t,, t t0, t T, ul t < pme α t, for l 0, t, t ] T, l =,2,,n Hence, there must be a constant c >suchthat u0 t = cpme α t,, u0 t < cpme α t,, t t0, t T, ul t < cpme α t, for l 0, t, t ] T, l =,2,,n Note that n vew of 44, we have u0 t t = u 0 e a0 t, e a0 t, σ s { s m a 0 u 0 θ θ c 0 f v s τ0 s sδ 0 s

10 L et al Advances n Dfference Equatons 203, 203:28 Page 0 of 6 m α 0 f v s τ0 s y m α 0 f y m β 0 f v s τ0 s y m } β 0 f y s t = u 0 e a0 t, e a0 t, σ s { s a 0 a 0 u 0 θ δ0 θ m c 0 f v θ τ0 θ sδ 0 s m α 0 f v θ τ0 θ y m α 0 f y m β 0 f v θ τ0 θ y m ] m β 0 f y θ c 0 f v s τ0 s m α 0 f v s τ0 s y m α 0 f y m β 0 f v s τ0 s y m } β 0 f y s sδ 0 s t e a0 t, e a0 t, σ s { s a 0 a u0 0 θ δ0 θ m c 0 f v θ τ0 θ m α 0 f v θ τ0 θ y m m α 0 f y β 0 f v θ τ0 θ y m β 0 f y m ] θ c 0 f v s τ0 s m α 0 f v s τ0 s y m α 0 f y m β 0 f v s τ0 s y m β 0 f y } s t e a0 t, σ s {a 0 δ 0 a 0 cpme α t, m c0 α 0 β 0 L f cpme αt, ] m c0 α 0 β 0 L f cpme αt, } s

11 L et al Advances n Dfference Equatons 203, 203:28 Page of 6 t e α t, e a0 t, σ s {a 0 δ 0 a 0 cpme α t, m c0 α 0 β 0 L f cpme αt, ] m c0 α 0 β 0 L f cpme αt, } s = e α t, cpme α t, t e a0 t, σ s m {a 0 δ 0 a 0 c0 α 0 β 0 ] L f m c0 α 0 β 0 } L f s = e α t, cpme α t, a 0 m {a 0 δ 0 a 0 c0 α 0 β 0 ] L f m c0 α 0 β 0 } t L f a 0 e a0 t, σ s s = e α t, ea0 t, cpme α t, a 0 m {a 0 δ 0 a 0 c0 α 0 β 0 ] L f m c0 α 0 β 0 } L f < cpm e α t, { cpm m a 0 δ 0 a 0 c0 α 0 β 0 L f a 0 m c0 α 0 β 0 ]} L f < cpm e α t, { M m a 0 δ 0 a 0 c0 α 0 β 0 L f a 0 m c0 α 0 β 0 ]} L f

12 L et al Advances n Dfference Equatons 203, 203:28 Page 2 of 6 = cpm e α t, M a 0 δ 0 δ 0 a m c0 0 α 0 β 0 L f < cpme α t,, whch s a contradcton Case Two: 48 strueand49 s not true Then there exst t 2, T and 0 {,2,,n} such that v0 t 2 pme α t 2,, v0 t < pme α t,, t t0, t 2 T, vk t < pme α t, for k 0, t, t 2 ] T, k =,2,,m Hence, there must be a constant c >suchthat v 0 t 2 = c pme α t 2,, v0 t < c pme α t,, t t0, t 2 T, vk t < c pme α t, for k 0, t, t 2 ] T, k =,2,,m Note that n vew of 45, we have v0 t 2 t2 = v 0 e b0 t 2, e b0 t2, σ s { s b 0 b 0 v 0 θ η0 θ n d 0 g u θ σ0 θ sη 0 s n p 0 g u θ σ0 θ x n p 0 g x n q 0 g u θ σ0 θ x n ] q 0 g x θ n d 0 g u s σ0 s n p 0 g u s σ0 s x n p 0 g x n q 0 g u s σ0 s x n } q 0 g x s e b0 t 2, t2 e b0 t2, σ s { s b 0 b 0 v 0 θ η0 θ n d 0 g u θ σ0 θ sη 0 s n p 0 g u θ σ0 θ x n p 0 g x

13 L et al Advances n Dfference Equatons 203, 203:28 Page 3 of 6 n q 0 g u θ σ0 θ x n ] q 0 g x θ n d 0 g u s σ0 s n p 0 g u s σ0 s x n p 0 g x n q 0 g u s σ0 s x n } q 0 g x s t2 e b0 t2, σ s {b 0 η 0 b 0 c pme α t 2, n d0 p 0 q 0 L g c pme α t 2, ] n d0 p 0 q 0 L g c pme α t 2, } s t2 e α t 2, e b0 t2, σ s {b 0 η 0 b 0 c pme α t 2, n d0 p 0 q 0 L g c pme α t 2, ] n d0 p 0 q 0 L g c pme α t 2, } s = e α t 2, c pme α t 2, t2 e b0 t2, σ s n {b 0 η 0 b 0 d0 p 0 q 0 ] L g n d0 p 0 q 0 } L g s = e α t 2, c pme α t 2, b 0 n {b 0 η 0 b 0 d0 p 0 q 0 ] L g n d0 p 0 q 0 } t2 L g e b0 t2, σ s s = e α t 2, eb0 t 2, c pme α t 2, b 0 n {b 0 η 0 b 0 d0 p 0 q 0 ] L g

14 L et al Advances n Dfference Equatons 203, 203:28 Page 4 of 6 n d0 p 0 q 0 } L g < c pm e α t 2, { c pm n b 0 η 0 b 0 d0 p 0 q 0 L g b 0 n d0 p 0 q 0 ]} L g < c pm e α t 2, { M n b 0 η 0 b 0 d0 p 0 q 0 L g b 0 n d0 p 0 q 0 ]} L g = c pm e α t 2, M b η η b n d p q L g < c pme α t 2,, whch s also a contradcton By the above two cases, for other cases of negatve proposton of 47, we can obtan a contradcton Therefore, 47holdsLetp, then 46 holds Hence, wehavethat wt M e α t,, t ϑ, T, t, whch means that the equlbrum pont z of s globally exponentally stable The proof of Theorem 4 s completed 5 Example In ths secton, we present an example to llustrate the feasblty of our results Example 5 Let n = m = 2 Consder the followng fuzzy BAM system wth delays n leakage terms on a tme scale T: x t=a x t δ t 2 c f y t τ t 2 α f y t τ t 2 T μ 2 β f y t τ t 2 H μ I, t T, =,2, y t=b y t η t 2 d g x t σ t 2 p g x t σ t 2 F ν 2 q g x t σ t 2 G ν J, t T, =,2, wheretmedelaysδ, τ, η, σ,, =, 2 are defned as those n system andthecoeffcents are as follows: a b =, =, 0 a b c 22 =, α 22 =,

15 L et al Advances n Dfference Equatons 203, 203:28 Page 5 of β 22 =, d 22 =, p 22 =, q 22 =, δ t = sn t 0 3, I 2 =, cos t 4 η t = sn 2t 0, sn t J 2 = 2, f u = 002 u, g v =004 v v, μ = ν =,, =,2, and T 22 =H 22 =F 22 =G 22 are dentty matrces By calculatng, we have L f = L g =002WecanverfythatforT = R and T = Z, all the condtons of Theorem 3 and Theorem 4 are satsfed Hence, for T = R or T = Z, 5 always has one unque equlbrum pont, whch s globally exponentally stable Competng nterests The authors declare that they have no competng nterests Authors contrbutons All authors typed, read and approved the fnal manuscrpt Acknowledgements Ths work s supported by the Natonal Natural Scences Foundaton of People s Republc of Chna under Grant Receved: 25 Aprl 203 Accepted: 9 July 203 Publshed: 22 July 203 References Kosko, B: Bdrectonal assocatve memores IEEE Trans Syst Man Cybern 8, Zhao, HY: Global stablty of bdrectonal assocatve memory neural networks wth dstrbuted delays Phys Lett A 297, L, YK: Exstence and stablty of perodc soluton for BAM neural networks wth dstrbuted delays Appl Math Comput 59, Park, JH, Park, CH, Kwon, OM, Lee, SM: A new stablty crteron for bdrectonal assocatve memory neural networks of neutral-type Appl Math Comput 99, Zhou, Q: Global exponental stablty of BAM neural networks wth dstrbuted delays and mpulses Nonlnear Anal, Real World Appl 0, L, YK: Global exponental stablty of BAM neural networks wth delays and mpulses Chaos Soltons Fractals 24, Zheng, B, Zhang, Y, Zhang, C: Global exstence of perodc solutons on a smplfed BAM neural network model wth delays Chaos Soltons Fractals 37, Xa, Y, Cao, JD, Ln, M: New results on the exstence and unqueness of almost perodc soluton for BAM neural networks wth contnuously dstrbuted delays Chaos Soltons Fractals 3, Zhang, L, S, L: Exstence and exponental stablty of almost perodc soluton for BAM neural networks wth varable coeffcents and delays Appl Math Comput 94, Lu, C, L, C, Lao, X: Varable-tme mpulses n BAM neural networks wth delays Neurocomputng 74, Yang, T, Yang, L, Wu, C, Chua, L: Fuzzy cellular neural networks: theory In: Proceedngs of IEEE Internatonal Workshop on Cellular Neural Networks and Applcatons, pp Yuan, K, Cao, JD, Deng, J: Exponental stablty and perodc solutons of fuzzy cellular neural networks wth tme-varyng delays Neurocomputng 69, Zhang, Q, Xang, R: Global asymptotc stablty of fuzzy cellular neural networks wth tme-varyng delays Phys Lett A 372, Song, Q, Cao, JD: Dynamcal behavors of dscrete-tme fuzzy cellular neural networks wth varable delays and mpulses J Frankln Inst 345,

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