Research Article Almost Automorphic Solutions for Fuzzy Cohen-Grossberg Neural Networks with Mixed Time Delays

Transcription

1 Hdaw Publhg Corporao Mahemacal Problem Egeerg Volume 15, Arcle ID 8167, 14 page hp://dx.do.org/1.1155/15/8167 Reearch Arcle Almo Auomorphc Soluo for Fuzzy Cohe-Groberg Neural Nework wh Mxed Tme Delay Yogku L, 1 Xaofag Meg, 1 ad Xueme Zhag 1 Deparme of Mahemac, Yua Uvery, Kumg, Yua 6591, Cha School of Mahamac ad Iformao Scece, Qug Normal Uvery, Qug, Yua 65511, Cha Correpodece hould be addreed o Yogku L; ykle@yu.edu.c Receved 7 December 14; Acceped 17 February 15 Academc Edor: Sor K. Nouya Copyrgh 15 Yogku L e al. Th a ope acce arcle drbued uder he Creave Commo Arbuo Lcee, whch perm urerced ue, drbuo, ad reproduco ay medum, provded he orgal work properly ced. Th paper cocered wh he problem of almo auomorphc oluo of a cla of fuzzy Cohe-Groberg eural ework wh mxed me delay ad varable coeffce. Baed o equaly aaly echque ad combg he expoeal dchoomy wh fxed po heorem, ome uffce codo for he exece ad global expoeal ably of almo auomorphc oluo are obaed. Fally, a umercal example gve o how he feably of our reul. 1. Iroduco I rece year, eural ework have bee exevely uded due o her mpora applcao may area uch a fuco approxmao, paer recogo, aocave memory, ad combaoral opmzao. I parcular, he reearch o he dyamcal behavor of he eural ework ha become a mpora opc eural ework heory. Therefore, he dyamc of eural ework, epecally he exece of perodc oluo, aperodc oluo, ad almo perodc oluo o eural ework, ha bee exevely vegaed ad a large umber of crera o he ably of eural ework have bee dcued he leraure 1 5]. Moreover, well kow ha Cohe- Groberg eural ework 6] oe of he mo popular ad ypcal ework model. Some model uch a Hopfeld-ype eural ework, CNN, BAM-ype model are pecal cae of Cohe-Groberg eural ework. For more deal abou Cohe-Groberg eural ework, oe ca ee, for example, 7 1]. I realy, me delay ofe occur due o fe wchg peed of he amplfer ad commucao me. Moreover, oberved boh expermeally ad umercally ha me delay eural ework may duce ably. Therefore, eural ework wh me delay have recely become a opc of reearch ere. O he oher had, mahemacal modelg of real world problem, we ecouer ome coveece bede delay, amely, he complexy ad he uceray or vaguee. Vaguee oppoe o exace ad we argue ha cao be avoded he huma way of regardg he world. Ay aemp o expla a exeve dealed decrpo ecearly lead o ug vague cocep ce prece decrpo coa abuda umber of deal. To uderad, we mu group hem ogeher ad h ca hardly be doe precely. A oubuable role here played by aural laguage. For he ake of akg vaguee o coderao, fuzzy heory vewed a a more uable eg. Baed o radoal CNN, T. Yag ad L.-B. Yag 13] fr roduced he fuzzy cellular eural ework (FCNN), whch egrae fuzzy logc o he rucure of radoal CNN ad maa local coecede amog cell. Ulke prevou CNN, FCNN a very ueful paradgm for mage proceg problem, whch ha fuzzy logc bewee emplae pu ad/or oupu bede he um of produc operao. I a coreroe mage proceg ad paer recogo. Recely, ome reul o ably ad oher behavor have bee derved for fuzzy eural ework wh or whou me delay (ee 14 ]). Alo, he almo perodcy more uveral ha he perodcy. Moreover, almo auomorphc fuco, whch were roduced by Bocher h paper 3 5]

2 Mahemacal Problem Egeerg relao o ome apec of dffereal geomery, are much more geeral ha almo perodc fuco. The oo of almo auomorphc fuco wa roduced o avod ome aumpoofuformcovergecehaarewheug almo perodc fuco. I he la everal decade, he bac heore o he almo auomorphc fuco have bee well developed 6, 7] ad have bee appled uccefully o he vegao of almo auomorphc oluo of dffereal equao (ee 8 3] ad referece here). I 31 33], auhor uded almo auomorphc oluo for everal clae of eural ework, repecvely. Recely, coderable effor have bee devoed o he perodc oluo ad almo perodc oluo of Cohe-Groberg eural ework wh me delay (ee, e.g., 34 41]). However, o he be of our kowledge, here o paper publhed o he exece ad ably of almo auomorphc oluo for fuzzy eural ework wh me delay. Movaed by he above dcuo, h paper, we propoe a cla of fuzzy Cohe-Groberg eural ework wh mxed me delay ad varable coeffce a follow: x () =a (x ()) he wchg ad ramo procee; K he delay kerel fuco; I () deoe exeral pu o he h euro a me. The al value of (1) he followg: x () =φ (), (, ], () where φ C((, ], R), =1,,...,. Our ma purpoe of h paper by ulzg equaly aaly echque ad combg he expoeal dchoomy wh fxed po heorem, o udy he exece ad global expoeal ably of almo auomorphc oluo of (1). Our reul of h paper are compleely ew. Throughou h paper, we aume ha he followg codo hold. (H 1 ) The amplfcao fuco a ( ) ( = 1,,...,)are cououly bouded; ha, here ex pove coa a ad a uch ha <a a (x) a, x R, =1,,...,. (3) b (x ()) c () f (x ()) d () g (x ( τ ())) e () μ () α () K () h (x ())d T () μ () β () K () h (x ())d (1) (H )The behaved fuco b ( ) ( = 1,,...,) are couou wh b () = ad here ex coa λ uch ha b (u) b (V) λ uv >, u, V R, u = V, =1,,...,. (H 3 ) f, g, h C(R, R) are Lpchcz couou wh pove Lpchcz coa L f, Lg,adLh uch ha, for all u, V R, f (u) f (V) Lf uv, g (u) g (V) Lg uv, h (u) h (V) Lh uv ; (4) (5) S () μ () I () ], ] where = 1,,...,, correpod o he umber of u eural ework; x () correpod o he ae of he h u a me ; a (x ()) repree a amplfcao fuco a me ; b (x ()) a appropraely behaved fuco a me uch ha he oluo of model (1) rema bouded; α, β, T,adS are he eleme of fuzzy feedback MIN emplae, fuzzy feedback MAX emplae, fuzzy feed forward MIN emplae, ad fuzzy feed forward MAX emplae, repecvely; d wegh he regh of he h euro a he me, c ad e are he eleme of feedback emplae ad feed forward emplae;, deoe he fuzzy AND ad fuzzy OR operao, repecvely; f, g,adh are he acvao fuco; τ () he me-varyg delay caued durg moreover, we uppoed ha f () = g () = h () =, =1,,...,. (H 4 ) c (), d (), e (), α (), β (), T (), S (), I (), ad τ () are all couou almo auomorphc fuco defed o R. (H 5 )ThekerelK ( ) almo auomorphc ad afe ha K () d = 1, e μ K () d <,,,,...,, (6) where μ a real pove umber ad here ex ρ> ad C>uch ha K () ρe C,,,,...,.

3 Mahemacal Problem Egeerg 3 For coveece, we roduce he followg oao: c = up c () R, d = up d () R, e = up e () R, α = up α () R, β = up β () R, T = up T () R, S = up S () R, J () = τ= max up 1, R e () μ () c = f R c (), d = f R d (), e = f R e (), α = f R α (), β = f R β (), T = f R T (), S = f R S (), τ (), T () μ () S () μ () I (), J J=max = up J (), R J 1 ω1 λ a, where ω >,,,,...,. Th paper orgazed a follow. I Seco, we roduce ome defo ad make ome preparao for laer eco. I Seco 3, we pree ome uffce codo for he exece ad global expoeal ably of almo auomorphc oluo of (1). ISeco 4, wegve a example o demorae he feably of our reul.. Prelmary I h eco, we roduce ome defo ad ae ome prelmary reul. Defo 1 (ee 7]). A couou fuco f:r R ad o be almo auomorphc f for every equece of real umber ( ) N here ex a ubequece ( ) N uch ha well defed for each R ad for each R. (7) lm f ( ) =g() (8) lm g( )=f() (9) Remark. Noe ha he fuco g he defo above meaurable bu o ecearly couou. Moreover, f g couou, he f uformly couou. Bede, f he covergece above uform R, he f almo perodc. Deoe by AA(R, R ) he colleco of all almo auomorphc fuco AP (R, R ) AA(R, R ) BC(R, R ), (1) where AP(R, R ) ad BC(R, R ) are he colleco of all almo perodc fuco ad he e of bouded couou fuco from R o R,repecvely. Lemma 3 (ee 7]). Le f, g AA(R, R ).Thewehavehe followg: () f g AA(R, R ); () λf AA(R, R ) for ay calar λ R; () f α AA(R, R ) where f α : R R defed by f α ( ) = f( α); (v) le f AA(R, R ); he he rage R f := f(), R relavely compac R,huf bouded orm; (v) f φ : R R a couou fuco, he he compoe fuco φ f : R R almo auomorphc; (v) (AA(R, R ), ) a Baach pace. Defo 4 (ee 7]). A fuco f C(R R, R ) ad o be almo auomorphc R for each x R f for every equece of real umber ( ) N here ex a ubequece ( ) N uch ha lm f(,x)=g(, x) (11) well defed for each R, x R ad lm g(,x)=f(, x) (1) for each R, x R. The colleco of uch fuco wll be deoed by AA(R R, R ). Lemma 5 (ee 7]). Le f:r R R be a almo auomorphcfuco Rfor each x R ad aume ha f afe a Lpchz codo x uformly R. Le φ:r R be a almo auomorphc fuco. The he fuco almo auomorphc. Defo 6 (ee 7]). Syem φ: φ () =f(,φ()) (13) x () =A() x () (14) ad o adm a expoeal dchoomy f here ex a proeco P ad pove coa α, βo ha he fudameal oluo marx X() afe X () PX1 () βeα(), X ()(IP) X1 () βeα(),,. (15)

4 4 Mahemacal Problem Egeerg Coder he followg almo auomorphc yem: x () =A() x () f(), (16) where A() a almo auomorphc marx fuco ad f() a almo auomorphc vecor fuco. Lemma 7 (ee 7]). If he lear yem (14) adm a expoeal dchoomy, he yem (16) ha a uque almo auomorphc oluo: x () = X () PX 1 () f () d X ()(IP) X 1 () f () d, where X() he fudameal oluo marx of (14). (17) Lemma 8 (ee 33]). Le c () be a almo auomorphc fuco o R ad Mc ]= lm T he he lear yem T 1 T c () d>, =1,,...,; (18) x () = dag (c 1 (),c (),...,c ())x() (19) adm a expoeal dchoomy o R. Defo 9. The almo auomorphc oluo x () = (x 1 (), x (),...,x ())T of yem (1) ad o be globally expoeally able, f, for ay oluo x() = (x 1 (), x (),...,x ()) T, here ex coa M > ad μ>uch ha, for all R, 3. Ma Reul x () x () Meμ. () I h eco, we eablh ome reul for he exece ad uquee of almo auomorphc oluo of (3). Le R be he -dmeoal Eucldea pace. Se x = x =(x 1,x,...,x ) T.Foreveryx R, orm defed by x =max 1 ω 1 x,whereω >, =1,,...,. From (H 1 ), he adervave of 1/a (x ) ex. We may chooe a adervave R (x ) of 1/a (x ) wh R () =. Obvouly, R (x ) = 1/a (x ).Bya (x ) >,weoba ha R (x ) creag abou x adheverefuco R 1 (x ) of R (x ) exeal, couou, ad dffereable. So, (R 1 ) (x ) = a (x ),where(r 1 ) (x ) he dervave of R 1 (x ) abou x, ad compoo fuco b (R 1 (x )) dffereable. Deoe y () = R (x ()). Ieayoeeha y () = R (x ())x () = x ()/a (x ()) ad x () = R 1 (y ()). Subug hee equale o yem (1),wecageha y () =b (R 1 (y ())) c () f (R 1 (y ())) d () g (R 1 (y ( τ ()))) e () μ () α () K () h (R 1 (y ())) d T () μ () β () K () h (R 1 (y ())) d S () μ () I (), >, =1,,...,. The al codo of (1) a follow: y () =R (φ ()) =φ (), The yem (1) cabewrea y () = b (y ())y () (1) (, ], =1,,...,. () c () f (R 1 (y ())) d () g (R 1 (y ( τ ()))) e () μ () α () K () h (R 1 (y ())) d T () μ () β () K () h (R 1 (y ())) d S () μ () I (), >, (3) where b (y ()) b (R 1 (y ()))/y (), =1,,...,. Syem (1) ha a almo auomorphc oluo whch globally expoeally able f ad oly f yem (3) ha a

5 Mahemacal Problem Egeerg 5 almo auomorphc oluo whch globally expoeally able. I eay o ee ha R1 (u) R 1 where ξ bewee u ad V. (V) = (R1 ) (ξ)(uv) = a (ξ) uv a uv, =1,,...,, (4) Lemma 1. Suppoehaaumpo(H 3 ), (H 4 ),ad(h 5 ) hold ad φ AA(R, R);he φ: K () h (R 1 (φ ())) d (5) belog o AA(R, R). Proof. By Lemma 3, wehaveha,forφ AA(R, R), he fuco ψ: h (R 1 (φ ()) belog o AA(R, R). Now, le be a equece of real umber. By (H 5) we ca exrac aubequece of uch ha, for all, R, Smlarly, for each R, lm φ1 ( )=φ(), (3) whch mple ha φ: K ( )h (R 1 (φ ()))d belog o AA(R, R). The proof complee. Lemma 11. Suppoe ha aumpo (H 1 ) (H 4 ) hold. Defe he olear operaor Φ by Φφ = ((Φφ) 1,(Φφ),...,(Φφ) ) (31) for each φ AA(R, R ),wherefor=1,,...,, (Φφ) () = exp ( b (y (θ))dθ) Se lm K ( )=K 1 (), lm K1 ( )=K (), lm ψ( )=ψ 1 (), Clearly, φ( )φ 1 () = = lm ψ1 ( )=ψ(). (6) φ 1 : K () ψ 1 () d. (7) K ( )ψ() d K () ψ 1 () d K (u) ψ(u )du K () ψ 1 () d = K () ψ( )ψ 1 () d ρe ()C ψ( )ψ 1 () d. (8) By he well-kow Lebegue Domaed Covergece Theorem ad (H 4 ),wehave,forall R, lm φ ( ) =φ 1 (). (9) c () f (R 1 (φ ())) d () g (R 1 (φ ( τ ()))) e () μ () α () K (u) h (R 1 (φ (u))) du T () μ () β () K (u) h (R 1 (φ (u))) du S () μ () I () ] d. ] The Φ map AA(R, R ) o elf. (3)

6 6 Mahemacal Problem Egeerg Proof. Fr of all, le u check ha Φ well defed. Ideed, by Lemma 3, he pace AA(R, R ) ralao vara. Bede, by Lemma 5 ad 1,hefuco χ : c () f (R 1 (φ ())) d () g (R 1 (φ ( τ ()))) e () μ () α () T () μ () β () K (u) h (R 1 (φ (u))) du K (u) h (R 1 (φ (u))) du S () μ () I () (33) belog o AA(R, R ).Coequelywecawre (Φφ) () = exp ( b (y (θ))dθ)χ () d. (34) Le be a equece of real umber. By he compoo heorem of almo auomorphc fuco, he fuco b (y ()) AA(R, R ).By(H 3 ) we ca exrac a ubequece of uch ha, for all, R, lm b (y ( )) = b 1 (y ()), Se (Φ 1 φ) () = lm b1 (y ( )) = b (y ()), lm χ ( )=χ 1 (), lm χ1 ( )=χ (). exp ( The follow ha (Φφ) ( )(Φ 1 φ) () (35) b1 (y (θ))dθ)χ 1 () d. (36) = = exp ( b (y (σ )) dσ) χ (u )du u exp ( b1 (y (θ))dθ)χ 1 () d exp ( b (y (σ )) dσ) χ (u )du u exp ( b (y (σ )) dσ) χ 1 (u) du u exp ( b (y (σ )) dσ) χ 1 (u) du u exp ( u b1 (y (θ))dθ)χ 1 (u) du = exp ( b (y (σ )) dσ) (χ ( )χ 1 ())d (exp ( b (y (σ )) dσ) exp ( b1 (y (θ))dθ))χ 1 () d. (37) ByheLebegueDomaedCovergeceTheoremweoba mmedaely ha lm (Φφ) ( )=(Φ 1 φ) () (38) for all R. Reaog a mlar way o he fr ep we ca how ealy ha lm (Φ1 φ) ( ) = (Φφ) () (39) for all R. Coequely, he fuco Φφ belog o AA(R, R ). Theorem 1. Aume ha (H 1 ) (H 5 ) hold, he yem (3) ha a uque almo auomorphc oluo he rego B =B(φ,r)=φ AA(R, R ), φφ f he followg codo hold rj 1r, (4) = = exp ( exp ( b (y (θ))dθ)χ () d b1 (y (θ))dθ)χ 1 () d exp ( b (y (σ )) dσ) χ () d exp ( b1 (y (θ))dθ)χ 1 () d r=max 1 (λ a ω1 ) 1 c Lf a ω d Lg a ω (α β )Lh a ω ] <1, ] (41)

7 Mahemacal Problem Egeerg 7 where φ =(φ 1,φ 1,...,φ 1 )T, φ () = J () exp ( b (y (θ))dθ)d, =1,,...,. (4) Proof. Obvouly, B a cloed covex ube of AA(R, R ) ad oe ha φ = max up 1 R up 1 R up 1 R = max ω 1 ω 1 ω 1 J 1 ω1 J () exp ( b (y (θ))dθ)d J () exp ( b (y (θ))dθ)d J exp (λ a ())d λ a =J. Thu, for ay φ B,wehave φ φφ φ (43) rj 1r J= 1 1r. (44) Now we prove ha Φ a elf-mappg from B o B.Ifac, for arbrary φ B,wehave Φφ φ = max up 1 R ω 1 exp ( b (y (θ))dθ) c () f (R 1 (φ ())) d () g (R 1 (φ ( τ ()))) α () K (u) h (R 1 (φ (u))) du β () up 1 R up 1 R up 1 R ω 1 ω 1 ω 1 K (u) h (R 1 (φ (u))) du] d ] exp ( b (y (θ))dθ) c () f (R 1 (φ ())) d () g (R 1 (φ ( τ ()))) α () K (u) h (R 1 (φ (u))) du β () K (u) h (R 1 (φ (u))) du ] d ] exp ( b (y (θ))dθ) c Lf a d Lg a φ () (α β )Lh a φ ( τ ()) K (u) φ (u) du ] ] exp ( b (y (θ))dθ) d

8 8 Mahemacal Problem Egeerg φ up 1 R φ ω 1 = max 1 (λ a ω1 ) 1 φ c Lf a ω exp (λ a ()) d Lg a ω (α β )Lh a ω ] ] d c Lf a ω c Lf a ω d Lg a ω (α β )Lh a ω ] ] d d Lg a ω (α β )Lh a ω ] ] =r φ rm 1r, (45) whch mple ha Φφ B, ohemappgφ a elfmappg from B o B. Nex, we wll prove ha Φ a coraco mappg. For ay φ, ψ B,wehaveha Φφ Φψ = max up 1 R ω 1 exp ( b (y (θ))dθ) c () f (R 1 (φ ())) f (R 1 (ψ ()))] d () up 1 R up 1 R ω 1 ω 1 g (R 1 (φ ( τ ()))) g (R 1 (ψ ( τ ())))] α () K (u) h (R 1 (φ (u))) du K (u) h β () (R 1 (ψ (u))) du] K (u) h (R 1 (φ (u))) du K (u) h (R 1 (ψ (u))) du] ] d ] exp ( b (y (θ))dθ) c Lf a d Lg a φ () ψ () φ ( τ ())ψ ( τ ()) (α β )Lh a K (u) φ (u) ψ (u) du ] d ] exp ( b (y (θ))dθ) c Lf a ω d Lg a ω

9 Mahemacal Problem Egeerg 9 φψ up 1 R ω 1 φψ = max 1 (λ a ω1 ) 1 (α β )Lh a ω ] ] d exp (λ a ()) c Lf a ω c Lf a ω d Lg a ω (α β )Lh a ω ] ] d d Lg a ω (α β )Lh a ω ] φψ ] =r φψ. (46) Becaue r<1,oφ a coraco mappg. Therefore, Φ ha a uque fxed po x B uch ha Φ(x )=x ;ha, (3) ha a uque almo auomorphc oluo x B. Th complee he proof of Theorem 1. Theorem 13. Aume ha (H 1 ) (H 5 ) hold, he almo auomorphc oluo of (3) globally expoeally able f, for =1,,...,, he followg codo hold λ a ω > (c Lf a ω d Lg a ω (α β )Lh a ω K () d). (47) Proof. For = 1,,...,,coderhefucoΘ (γ) gve by Θ (γ) = γ λ a ω 1 ω 1 ω 1 d Lg a ω e γτ c Lf a ω (α β )Lh a ω K () e γ d. (48) I clear ha he fuco Θ (γ) couou o R ad by (47),wehave Θ () =λ a ω 1 <, ω 1 ω 1 d Lg a ω c Lf a ω (α β )Lh a ω K () d =1,,...,. Thu, here ex a uffcely mall coa μ uch ha (49) Θ (μ)<, =1,,...,. (5) From Theorem 1, weeehayem(3) ha a lea oe almo auomorphc oluo y () = y () =1 wh al codo y () = φ (), (,], = 1,,...,. Suppoe ha y() = y () =1 a arbrary oluo wh al codo y () = φ (), (,], = 1,,...,.Se z() = y() y (); he follow from yem (3) ha z () =b (R 1 (z () y ())) b (R 1 (y ()))] c () f (R 1 (z () y ())) f (R 1 (y ()))] d () g (R 1 (z ( τ ())y ( τ ()))) α () β () g (R 1 (y ( τ ())))] K ()h (R 1 (z () y ())) h (R 1 (y ()))] d K ()h (R 1 (z () y ())) h (R 1 (y ()))] d, >, (51)

10 1 Mahemacal Problem Egeerg where =1,,...,. Thealcodoof(51) z () =ψ () =φ () φ (), (, ], =1,,...,. (5) Smlar o yem (3), wee b (z ()) (b (R 1 (z () y ())) b (R 1 (y ())))/z (). Ivewof(H ) ad (4), we have b (R 1 ( )) λ a >. Now, we coder fuco u () = ω 1 e μ y ()y () = e μ z (), μ>, =1,,...,.The ω 1 D u () =ω 1 =ω 1 e μ μ z () g (z ()) b (z ())z () f (R 1 ω 1 e μ d () e μ μ z () d z () d c () (z () y ())) f (R 1 (y ()))] g (R 1 (z ( τ ())y ( τ ()))) g (R 1 (y ( τ ())))] α () β () μ z () λ a c K ()h (R 1 (z () y ())) h (R 1 (y ()))] d K ()h (R 1 (z () y ())) f (R 1 z () h (R 1 (y ()))] d ] ] (z () y ())) f (R 1 (y ())) d α β ω 1 e μ g (R 1 (z ( τ ())y ( τ ()))) g (R 1 (y ( τ ()))) K () h (R 1 (z () y ())) h (R 1 (y ())) d K () h (R 1 (z () y ())) μ z () λ a c Lf a (λ a ω 1 ω 1 h (R 1 (y ())) d z () z () (α β )Lh a μ)u () ω 1 d Lg a z ( τ ()) K () z () d c Lf a ω u () d Lg a ω e μτ u ( τ ()) (α β )Lh a ω K () e μ() u () d (λ a ω 1 ω 1 μ)u () ω 1 d Lg a ω e μτ up τ (α β )Lh a ω K () e μ d up u (), =1,,...,. c Lf a ω u () u () (53)

11 Mahemacal Problem Egeerg 11 Se M = max 1 up (,] ω 1 z () = max 1 up (,] ω 1 φ () φ () ad ε > a arbrary real umber. The, we ca ge u () =ω 1 e μ z () ω1 z () M Mε, (, ], I he followg, we wll how ha =1,,...,. (54) u () <Mε, (, ), =1,,...,. (55) If (55) o rue, whou lo of geeraly, he here ex acoal ad a fr me 1 uch ha Therefore, we have D u l ( 1 ) (λ l a l ω 1 l ω 1 l < μλ l a l u () <Mε, =l, (, 1 ], u l () <Mε, (, 1 ), u l ( 1 )=Mε. μ)u l ( 1 )ω 1 l d l Lg a ω e μτ up u () 1 τ 1 (α l β l )Lh a ω K l () e μ d up u () ω 1 l ω 1 l (Mε) ω 1 l d l Lg a ω e μτ =Θ l (μ) (Mε) <. c l Lf a ω c l Lf a ω u ( 1 ) (α l β l )Lh a ω K l () e μ d (56) (57) Thacoradco;hece(55) hold. Le ε,he we have u () <M, (, ), =1,,...,. (58) Togeher, (54) wh (58)gve he followg: u () <M, R, =1,,...,. (59) The, we have z () = y () y () Mω e μ, =1,,...,, (6) whch mea ha he almo auomorphc oluo of he yem (3) globally expoeally able. The proof of Theorem 13 compleed. 4. A Example I h eco, we gve a example o llurae he feably ad effecvee of our reul obaed Seco 3. Example 1. Le =. Coder he followg yem: x () =a (x ()) b (x ()) c () f (x ()) d () g (x ( τ ())) e () μ () α () K () h (x ())d T () μ () where =1,ad β () K () h (x ())d S () μ () I () ], ] f 1 (x) =f (x) = 1 ( x1 x 1), g 1 (x) =g (x) = a x, h 1 (x) =h(x) = ah x, a 1 (x 1 ()) = (x 1 ()), a (x ()) =co (x 1 ()), b 1 (x 1 ()) =3x 1 (), b (x ()) =4x (), c 11 () =.5.1 co, c 1 () =..1, c 1 () =..1 co, c () =.3.1, d 11 () =.6.1 co, d 1 () =.5.1, (61)

12 1 Mahemacal Problem Egeerg d 1 () =.5., d () =.7., α 11 () =.7.1 co, α 1 () =.3., α 1 () =.6.1, α () =.6., β 11 () =.1.3, β 1 () =.5.3 co, β 1 () =.3.4 co, β () =.1.4, τ () =, K () =e. S 11 () =.3., S 1 () =.7. co, S 1 () =.3.45 co, S () =.14.7, T 11 () =.15.4 co, T 1 () =.17.3, T 1 () =.14.3, T () =.15.7 co, e 11 () =.3. co, e 1 () =.4.3, e 1 () =..4, e () =.4.3 co, μ 1 () =, μ () = co, I 1 () =co (e ), I () = (e ). (6) By calculag, we have ha a 1 =a =1, a 1 =a =3, λ 1 =3, λ =4, L f =Lg =Lh =1, K ()d = 1, adτ=1ad ake he pove real umber ω =1,,;he r=max 1 (λ a ω1 ) 1 =.47 < 1. c Lf a ω d Lg a ω (α β )Lh a ω ] ] (63) Fgure 1: Behavor of he ae compoe for x 1 () Fgure : Behavor of he ae compoe for x (). Moreover, we have (c 1 Lf a ω d 1 Lg a ω e μτ (α 1 β 1 )Lh a ω K 1 () e μ d) = 1.41 < 3 = λ 1 a 1 ω 1, (c Lf a ω d Lg a ω e μτ (α β )Lh a ω K () e μ d) = 1.17 < 4 = λ a ω. (64) All he aumpo Theorem 1 ad 13 are afed. We e he al codo x() = 3 3] T ; he repoe of he ae compoe howed Fgure 1 ad. Therefore, (61) ha a

13 Mahemacal Problem Egeerg 13 almo auomorphc oluo, whch globally expoeally able. Coflc of Iere The auhor declare ha here o coflc of ere regardg he publcao of h paper. Ackowledgme The auhor are deeply graeful o he revewer for her careful readg of h paper ad helpful comme, whch have bee very ueful for mprovg he qualy of h paper. Th work uppored by he Naoal Naural Scece Foudao of Cha uder Gra o Referece 1] B. Lu ad L. Huag, Exece ad expoeal ably of almo perodc oluo for cellular eural ework wh me-varyg delay, Phyc Leer A, vol. 341, o. 1 4, pp , 5. ] E. Yılmaz, Almo perodc oluo of mpulve eural ework a o-precrbed mome of me, Neurocompug, vol. 141, pp , 14. 3] L. Peg ad W. Wag, A-perodc oluo for hug hbory cellular eural ework wh me-varyg delay leakage erm, Neurocompug, vol. 111, pp. 7 33, 13. 4] G. Zhag, Y. She, Q. Y, ad J. Su, Global expoeal perodcy ad ably of a cla of memror-baed recurre eural ework wh mulple delay, Iformao Scece, vol. 3,pp ,13. 5] Y. Q. Ke ad C. F. Mao, Sably ad exece of perodc oluo eral BAM eural ework wh me delay, Neural Compug ad Applcao, vol.3,o.3-4,pp , 13. 6]M.A.CoheadS.Groberg, Abolueablyofglobal paer formao ad parallel memory orage by compeve eural ework, IEEE Traaco o Syem, Ma, ad Cyberec,vol.13,o.5,pp , ] Y. K. L, Exece ad ably of perodc oluo for Cohe- Groberg eural ework wh mulple delay, Chao, Solo ad Fracal,vol.,o.3,pp ,4. 8]T.Huag,C.L,adG.Che, SablyofCohe-Groberg eural ework wh ubouded drbued delay, Chao, Solo & Fracal,vol.34,o.3,pp ,7. 9] Z. Che ad J. Rua, Global dyamc aaly of geeral Cohe-Groberg eural ework wh mpule, Chao, Solo ad Fracal,vol.3,o.5,pp ,7. 1] C. Ba, Global expoeal ably ad exece of perodc oluo of Cohe-Groberg ype eural ework wh delay ad mpule, Nolear Aaly: Real World Applcao, vol. 9, o. 3, pp , 8. 11] Z. Yua, D. Hu, L. Huag, ad G. Dog, Exece ad global expoeal ably of perodc oluo for Cohe-Groberg eural ework wh delay, Nolear Aaly. Real World Applcao,vol.7,o.4,pp.57 59,6. 1] T.L,A.Sog,adS.Fe, FurherablyaalyoCohe- Groberg eural ework wh me-varyg ad drbued delay, Ieraoal Joural of Syem Scece,vol.4,o.1, pp ,11. 13] T. Yag ad L.-B. Yag, The global ably of fuzzy cellular eural ework, IEEE Traaco o Crcu ad Syem. I. Fudameal Theory ad Applcao, vol.43,o.1,pp , ] Q. Sog ad J. Cao, Impulve effec o ably of fuzzy Cohe-Groberg eural ework wh me-varyg delay, IEEE Traaco o Syem, Ma, ad Cyberec, Par B: Cyberec,vol.37,o.3,pp ,7. 15] S.Nu,H.Jag,adZ.Teg, Expoealablyadperodc oluo of FCNN wh varable coeffce ad me-varyg delay, Neurocompug, vol. 71, o , pp , 8. 16] Y. H. Xa, Z. Yag, ad M. Ha, Lag ychrozao of ukow chaoc delayed yag-yag-ype fuzzy eural ework wh oe perurbao baed o adapve corol ad parameer defcao, IEEE Traaco o Neural Nework,vol.,o.7,pp ,9. 17] P. Balaubramaam ad M. S. Al, Robu expoeal ably of ucera fuzzy Cohe-Groberg eural ework wh me-varyg delay, Fuzzy Se ad Syem,vol.161,o.4,pp , 1. 18] G. Bao, S. P. We, ad Z. G. Zeg, Robu ably aaly of erval fuzzy Cohe-Groberg eural ework wh pecewe coa argume of geeralzed ype, Neural Nework, vol. 33, pp. 3 41, 1. 19] R. Sakhvel, A. Arukumar, K. Mahyalaga, ad S. M. Aho, Robu pavy aaly of fuzzy Cohe-Grober BAM eural ework wh me-varyg delay, Appled Mahemac ad Compuao,vol.18,o.7,pp ,11. ] K. Mahyalaga, J. H. Park, R. Sakhvel, ad S. M. Aho, Delay fracog approach o robu expoeal ably of fuzzy Cohe-Groberg eural ework, Appled Mahemac ad Compuao,vol.3,pp ,14. 1] R. Sahy ad P. Balaubramaam, Sably aaly of fuzzy Markova umpg Cohe-Groberg BAM eural ework wh mxed me-varyg delay, Commucao Nolear Scece ad Numercal Smulao,vol.16,o.4,pp.54 64, 11. ]C.Z.L,Y.K.L,adY.Ye, Expoealablyoffuzzy Cohe-Groberg eural ework wh me delay ad mpulve effec, Commucao Nolear Scece ad Numercal Smulao,vol.15,o.11,pp ,1. 3] S. Bocher, Uform covergece of moooe equece of fuco, Proceedg of he Naoal Academy of Scece of he Ued Sae of Amerca,vol.47,pp , ] S. Bocher, A ew approach o almo perodcy, Proceedg of he Naoal Academy of Scece of he Ued Sae of Amerca, vol. 48, pp , ] S. Bocher, Couou mappg of almo auomorphc ad almo perodc fuco, Proceedg of he Naoal Academy of Scece of he Ued Sae of Amerca, vol.5,pp.97 91, ] G. M. N Guerekaa, Topc Almo Auomorphy, Sprger, New York, NY, USA, 5. 7] T. Dagaa, Almo Auomorphc Type ad Almo Perodc Type Fuco Abrac Space, Sprger, Berl, Germay, 13. 8] G. M. N Guerekaa, Exece ad uquee of almo auomorphc mld oluo o ome emlear abrac dffereal equao, Semgroup Forum,vol.69,o.1,pp.8 86,4. 9] J. A. Golde ad G. M. N Guerekaa, Almo auomorphc oluo of emlear evoluo equao, Proceedg of he Amerca Mahemacal Socey, vol.133,o.8,pp.41 48, 5.

14 14 Mahemacal Problem Egeerg 3] K. Ezzb ad G. M. N Guérékaa, Almo auomorphc oluo for ome paral fucoal dffereal equao, Joural of Mahemacal Aaly ad Applcao,vol.38,o. 1,pp ,7. 31] F. Cherf ad Z. B. Naha, Global aracvy ad exece of weghed peudo almo auomorphc oluo for GHNN wh delay ad varable coeffce, GulfJouralofMahemac, vol. 1, pp. 5 4, 13. 3] F. Chérf, Suffce codo for global ably ad exece of almo auomorphc oluo of a cla of RNN, Dffereal Equao ad Dyamcal Syem, vol.,o.,pp.191 7, ] Y. L ad L. Yag, Almo auomorphc oluo for eural ype hgh-order Hopfeld eural ework wh delay leakage erm o me cale, Appled Mahemac ad Compuao, vol. 4, pp , ] J. H. Su ad L. Wa, Global expoeal ably ad perodc oluo of Cohe-Groberg eural ework wh cououly drbued delay, Phyca D: Nolear Pheomea, vol. 8, o. 1-, pp. 1, 5. 35]Y.K.L,X.R.Che,adL.Zhao, Sablyadexeceof perodc oluo o delayed Cohe-Groberg BAM eural ework wh mpule o me cale, Neurocompug, vol. 7,o.7-9,pp ,9. 36] Y. K. L ad X. L. Fa, Exece ad globally expoeal ably of almo perodc oluo for Cohe-Groberg BAM eural ework wh varable coeffce, Appled Mahemacal Modellg: Smulao ad Compuao for Egeerg ad Evromeal Syem,vol.33,o.4,pp.114 1,9. 37] Z. Che, D. H. Zhao, ad J. Rua, Almo perodc aracor for Cohe-Groberg eural ework wh delay, Phyc Leer A,vol.373,o.4,pp ,9. 38] H. J. Xag, J. H. Wag, ad J. D. Cao, Almo perodc oluo o Cohe-Groberg-ype BAM ework wh drbued delay, Neurocompug,vol.7,o.16 18,pp ,9. 39] Y.L,T.Zhag,adZ.Xg, Theexeceofozeroalmo perodc oluo for Cohe-Groberg eural ework wh cououly drbued delay ad mpule, Neurocompug,vol.73,o.16 18,pp ,1. 4] H. J. Xag ad J. D. Cao, Almo perodc oluo of Cohe- Groberg eural ework wh bouded ad ubouded delay, Nolear Aaly: Real World Applcao,vol.1,o. 4, pp , 9. 41] W. G. Yag, Perodc oluo for fuzzy Cohe-Groberg BAM Neural ework wh Boh Tme-Varyg ad drbued delay ad varable coeffce, Neural Proceg Leer, vol. 4, pp , 14.

15 Advace Operao Reearch Hdaw Publhg Corporao hp:// Advace Deco Scece Hdaw Publhg Corporao hp:// Joural of Appled Mahemac Algebra Hdaw Publhg Corporao hp:// Hdaw Publhg Corporao hp:// Joural of Probably ad Sac The Scefc World Joural Hdaw Publhg Corporao hp:// Hdaw Publhg Corporao hp:// Ieraoal Joural of Dffereal Equao Hdaw Publhg Corporao hp:// Subm your maucrp a hp:// Ieraoal Joural of Advace Combaorc Hdaw Publhg Corporao hp:// Mahemacal Phyc Hdaw Publhg Corporao hp:// Joural of Complex Aaly Hdaw Publhg Corporao hp:// Ieraoal Joural of Mahemac ad Mahemacal Scece Mahemacal Problem Egeerg Joural of Mahemac Hdaw Publhg Corporao hp:// Hdaw Publhg Corporao hp:// Hdaw Publhg Corporao hp:// Dcree Mahemac Joural of Hdaw Publhg Corporao hp:// Dcree Dyamc Naure ad Socey Joural of Fuco Space Hdaw Publhg Corporao hp:// Abrac ad Appled Aaly Hdaw Publhg Corporao hp:// Hdaw Publhg Corporao hp:// Ieraoal Joural of Joural of Sochac Aaly Opmzao Hdaw Publhg Corporao hp:// Hdaw Publhg Corporao hp://