Stability results for stochastic delayed recurrent neural networks with discrete and distributed delays

Transcription

1 Sably resuls for sochasc delayed recurren neural neworks wh dscree and dsrbued delays Gulng Chen, Onno van Gaans, Sjoerd Verduyn Lunel Mahemacal Insue, Leden Unversy, P.O.Box 9512, 23 RA, Leden, The Neherlands Reor MI Absrac We resen new condons for asymoc sably and exonenal sably of a class of sochasc recurren neural neworks wh dscree and dsrbued me varyng delays. Our aroaches are based on he mehod usng fxed on heory and he mehod usng an arorae negral neualy, whch do no resor o any Lyaunov funcon. Our resuls neher reure he boundedness, monooncy and dfferenably of he acvaon funcons nor dfferenably of he me varyng delays. In arcular, a class of neural neworks whou sochasc erurbaons s also consdered by usng he wo aroaches. xamles are gven o llusrae our man resuls. Keywords: Fxed on heory, asymoc sably, exonenal sably, sochasc recurren neural neworks, rval soluon, varable delays, Burkholder-Davs-Gundy neualy, Doob s neualy. 1. Inroducon and man resuls Neural neworks have receved an ncreasng neres n varous areas [5, 8]. The sably of neural neworks [6, 16, 37, 38] s crcal for sgnal rocessng, esecally n mage rocessng and solvng some classes of omzaon roblems. For he sochasc effecs o he dynamcal behavors of neural neworks, Lao and Mao [14, 15] naed he sudy of sably and nsably of sochasc neural neworks. Due o he fne swchng seed of neurons and amlfers, me delays whch may lead o nsably and bad erformance n neural rocessng and sgnal ransmsson are commonly encounered n boh bologcal and arfcal neural neworks. In addon, neural neworks usually have a saal exen due o he resence of a mulude of arallel ahways wh a varey of axon szes and lenghs [31]. Thus here wll be a dsrbuon of conducon veloces along hese ahways and a dsrbuon of roagaon delays [39]. In hese crcumsances he sgnal roagaon s no nsananeous and may no be suably modeled wh dscree delays. Therefore, a more arorae way whch ncororaes connuously dsrbued delays n neural nework models has been used. Furher, due o random flucuaons and robablsc causes n he nework, noses do exs n a neural nework. Thus, s necessary and rewardng o sudy sochasc effecs o he sably roery of neural neworks. Many neresng arcles [9, 1, 11, 28, 32] have consdered some classes of he sochasc neural neworks. Hu e al.[9] and Wan and Sun [32] suded a class of sochasc neural neworks wh he delays consan and dscree. The acvaon funcons aearng n [9] are reured o be bounded. Lao and Mao [17] nvesgaed exonenal sably of sochasc delay nerval sysems va Razumkhn-ye heorems develoed n [23], several exonenal sably resuls were rovded. However, he resuls are no only dffcul o verfy bu also resrc o he case of he nerval marces à = B = C =. Sun and Cao [28] nvesgaed he h momen exonenal sably of sochasc dfferenal euaons wh dscree bounded delays by usng he mehod of varaon arameer, neualy echnue and sochasc analyss. Ths mehod was frsly used n [32], whch does no reure he boundedness, monooncy and dfferenably of he acvaon funcons. However, he sably crera n [28] reures ha he delay funcons are bounded, dfferenable and her dervaves are smulaneously reured o be no greaer han 1. Ths may mose a very src consran on model because me delays somemes vary dramacally wh me n real crcus (see [36]). 1

2 Huang e al. [1, 11] nvesgaed he exonenal sably of sochasc dfferenal euaons wh dscree mevaryng delays wh he hel of a Lyaunov funcon and Dn dervave. However, he use of her crera deends very much on he choce of osve numbers k j ec. and a osve dagonal marx M (see Theorem 3.3 n [1] and Theorem 3.3 n [11]). Recenly, Buron [2] has ulzed he fxed on mehod o nvesgae he sably for deermnsc sysems, Luo [19] and Aleby [1] have aled hs mehod o deal wh he sably roblems for sochasc delay dfferenal euaons, and aferwards, a grea number of classes of sochasc delay dfferenal euaons are dscussed by usng fxed on mehod, see, for examle, [5, 2, 21, 25, 26]. I urned ou ha fxed on mehod s a owerful echnue n dealng wh sably roblems for dfferenal euaons wh delays and sochasc dfferenal euaons wh delays, and can yeld he exsence, unueness and sably crera of he consdered sysem n one se by a fxed on argumen, whch s mossble when usng he oher mehods. Chen [4, 3] has aled he mehod by usng an arorae negral neualy o sudy exonenal sably of some classes of sochasc delay dfferenal euaons, and urns ou ha s a convenen way o dscuss exonenal sably of a sysem. The am of hs aer s o sudy a general class of sochasc neural neworks by usng fxed on mehod and he mehod by emloyng an arorae negral neualy. Indeed, we consder he followng class of sochasc neural neworks wh varyng dscree and dsrbued delays whch s descrbed by or dx() = dx () = c x () a j f j (x j ()) b j g j (x j ( τ())) l j r() h j (x j (s)) ds d σ j (, x j (), x j ( τ())) dω j (), (1) [ ] Cx() A f (x()) Bg(x( τ()))w h(x(s)) ds dσ(, x(), x( τ())) dω(), r() for, 2, 3,, n, where x()=(x 1 (), x 2 (), x n ()) T R n s he sae vecor assocaed wh he neurons; C= dag(c 1, c 2, c n )> where c > reresens he rae wh whch he h un wll rese s oenal o he resng sae n solaon when dsconneced from he nework and he exernal sochasc erurbaons; A=(a j ) n n, B=(b j ) n n and W= (l j ) n n reresen he connecon wegh marx, delayed connecon wegh marx and dsrbued delayed connecon wegh marx, resecvely; f j, g j, h j are acvaon funcons, f (x())=( f 1 (x()), f 2 (x()), f n (x())) T R n, g(x()) = (g 1 (x()), g 2 (x()), g n (x())) T R n, h(x()) = (h 1 (x()), h 2 (x()), h n (x())) T R n, whereτ() and r() denoe dscree me varyng delay and dsrbued me varyng delay, resecvely. Moreover, ω() = (ω 1 (),ω 2 (), ω n ()) T R n s an n-dmensonal Brownon moon defned on a comlee robably sace (Ω,F,P) wh naural comlee flraon{f } (.e.f = comleon ofσ{ω(s) : s }) andσ : R R n R n R n n, σ=(σ j ) n n s he dffuson coeffcen marx. Denoeϑ=nf { τ(), r()}. The nal condon for he sysem (1) s gven by x()=φ(), [ϑ, ], (2) whereφ()=(φ 1 (),φ 2 (), φ n ()) T C ( [ϑ, ], L F (Ω;R n ) ) wh he norm defned as φ = su φ (s), ϑ s where denoes execaon wh resec o he robably measure P. To oban our man resuls, we suose he followng condons are sasfed: (A1) he delays τ(), r() are connuous funcons such ha τ() and r() as ; 2

3 (A2) f j (x), g j (x), and h j (x) sasfy Lschz condons. Tha s, for each, 2, 3, n, here exss consansα j, β j,γ j such ha for every x, y R n, f j (x) f j (y) α j x y, g j (x) g j (y) β j x y, h j (x) h j (y) γ j x y ; (A3) Assume ha f (), g(), h(),σ(,, ) ; (A4)σ(, x, y) sasfes a Lschz condon. Tha s, here are nonnegave consansµ andν such ha race [ (σ(, x, y) σ(, u, v)) T (σ(, x, y) σ(, u, v)) ] [ µ (x u ) 2 ν (y v ) 2]. I follows from [7, 22] ha under he hyohess (A2), (A3) and (A4), sysem (1) wh nal condon (2) has one unue global soluon whch s denoed by x(,,φ) or x(), and su s x(s,,φ) < for >. Clearly, sysem (1) adms he rval soluon x(,, ). Defnon 1.1. The rval soluon of sysem (1) s sad o be sable n h ( 2) momen f for arbrary gven ǫ>, here exss aδ> such ha φ <δyelds ha x(,,φ) <ǫ,. whereφ() C ( [ϑ, ], L F (Ω;R n ) ). In arcular, when =2, he rval soluon s sad o be mean suare sable. Defnon 1.2. The rval soluon of sysem (1) s sad o be asymocally sable n h ( 2) momen f s sable n h ( 2) momen and here exss a scalarσ>, such ha φ <σmles whereφ() C ( [ϑ, ], L F (Ω;R n ) ). lm x(,,φ) =. Defnon 1.3. The rval soluon of sysem (1) s sad o be h ( 2) momen exonenally sable f here exss a ar of consansλ, C> such ha x(,,φ) C φ e λ,, holds forφ() C ( [ϑ, ], L F (Ω;R n ) ). secally, when =2, we seak of exonenally sable n mean suare. Dfferen choces of norms are defned for sace of sochasc rocesses. The norms we choose should be such ha he sace under consderaon s comlee and he euaon yelds a conracon wh resec o he norm. For he sysem (1) wh nal condon (2), we consder he followng wo dfferen comlee saces whch are defned by usng wo yes of norms. DefneS φ he sace of allf -adaed rocessesϕ(,ω) : [ϑ, ) Ω R n such haϕ C ( [ϑ, ), L F (Ω;R n ) ). Moreover, we seϕ(,ω)=φ() for [ϑ, ] and n ϕ () as,, 2, n. If we defne he norm ϕ := su ϑ ϕ (), (3) hens φ s a comlee merc sace. Usng he conracve mang defned on he saces φ and alyng he conracon mang rncle, we oban our frs resul. Theorem 1.4. Suose ha he assumons (A1)-(A4) hold. If he followng condons are sasfed, () he dsrbued delay r() s bounded by a consan r; 3

4 () α 5 1 c a j α j ( r 5 1 c ) l j γ j / / 5 1 c 5 1 n 1 b j β j / ( c /2 µ /2 ν /2) < 1, whereµ=max{µ 1,µ 2, µ n },ν=max{ν 1,ν 2, ν n }, hen he rval soluon of (1) s h momen asymocally sable. Consder he case when boh he dscree delayτ() and he dsrbued delay r() are bounded by a consanτ. DefneC φ he sace of allf -adaed rocessesϕ(,ω) : [ τ, ) Ω R n such haϕ C ( [ τ, ), L F (Ω;R n ) ). Moreover, we seϕ(,ω)=φ() for [ τ, ] and for, n su ϕ (s). The norm onc φ s defned as ( ϕ = su su ϕ (s) ). (5) henc φ s a comlee merc sace. Usng he conracon defned on he sacec φ and alyng he conracon mang rncle, we oban our second resul. Theorem 1.5. Suose ha he assumons (A1)-(A4) hold. If he followng condons are sasfed, () he dscree delayτ() and he dsrbued delay r() are bounded by a consanτ; () / / α 5 1 e cτ c a j α j 5 1 e cτ c b j β j / 5 1 τ e cτ c l j γ j 5 1 K n e cτ c 1 /2 (2c) 1( µ /2 ν /2) < 1, (6) where c=mn{c 1, c 2, c n },µ=max{µ 1,µ 2, µ n },ν=max{ν 1,ν 2, ν n }, hen he rval soluon of (1) s h momen asymocally sable. Remark 1.6. In Theorem 1.5, we oban ha { [ lm su x(s,,φ) ]}=, ha s, for any funcon s x (s,,φ), we have ha lm x (,,φ) C[ τ,] =, whch mles lm x(,,φ) =. Remark 1.7. In some aers, see, for examle, [18, 19, 33, 34], he norm for he sace of sochasc rocess s defned as { ( )} 1/2 ϕ [,] = su ϕ(s,ω) 2. s [,] As n [19], n order o show P(S) S, we need o esmae su s [,] I 5 (s) 2, where I 5 (s)= s e s z h(u) du [c(z)x(z)e(z)x(z δ(z))] dω(z). However, I 5 (s) s no a local marngale (see Secon 8 for s roof). Hence, Burkholder-Davs-Gundy Ineualy can no be aled drecly. 4 (4)

5 Usng an arorae negral neualy, we oban suffcen condons for exonenal sably of (1) wh nal condon (2), whch s our hrd resul. Theorem 1.8. Suose ha he assumons (A1)-(A4) hold. If he followng condons are sasfed, () he dscree delayτ() and dsrbued delay r() are bounded by a consanτ; () / / 5 1 c a j α j 5 1 c b j β j (7) ( τ ) / 5 1 c l j γ j 5 1 n c /2 (µ /2 ν /2 )<1, where c=mn{c 1, c 2, c n },µ=max{µ 1,µ 2, µ n },ν=max{ν 1,ν 2, ν n }, hen he rval soluon of (1) s exonenally sable n h momen, Remark 1.9. The sably crera we rovded n our man resuls are only n erms of he sysem arameers c, a j, b j, l j, ec. Hence, hese crera can usually be verfed easly n alcaons. Remark 1.1. Many arcles, see, for examle, [27, 28] have suded he case and secal case of sochasc neural nework (1). However, he delays should sasfy he followng condon: (H) he dscree delay τ() s dfferenable funcon and he dsrbued delay r() s non-negave and bounded, ha s, here exs consansτ M,ζ, r M such ha τ() τ M, τ () ζ, r() r M, (8) In Sun and Cao [28], seems ha he consran condon (H) on he dscree delays can be relaxed asτ() s bounded. As an examle, we consder a wo-dmensonal sochascally erurbed Hofeld neural nework wh mevaryng delays, dx()=[ Cx()A f (x()) Bg(x τ ())] dσ(, x(), x τ ()) dω(), where f (x)= 1 5 arcan x, g(x)= 1 5 anh x= 1 5 (ex e x )/(e x e x ),τ()= 1 2 sn 1 2, ( ) ( ) ( ) C=, A= and B= In hs examle, he dscree delay s bounded bu no dfferenable. Consder he case when here are no sochasc effecs on he sysem (1), whch hen comes down o he followng neural nework descrbed by dx () = c x () a j f j (x j ()) b j g j (x j ( τ())) d j h j (x j (s)) ds,, 2, 3,, n, (9) d or dx() d r() = Cx() A f (x()) Bg(x τ())d h(x(s)) ds, r() where x( )=(x 1 ( ), x 2 ( ),, x n ( )) T s he neuron sae vecor of he ransformed sysem (9). The nal condon for he sysem (9) s x()=φ(), [ϑ, ], (1) whereφ s a connuous funcon wh he norm defned by φ =su ϑ s n φ (s). Assume ha (A1) (A3) are sasfed, hen (9) and (1) adm a rval soluon x=. Denoe by x(; s;φ)= (x 1 (; s,φ 1 ),, x n (; s,φ n )) T R n he soluon of (9) wh nal condon (1). 5

6 Defnon For he sysem (9) wh nal condon (1), we have ha () he rval soluon of (9) s sad o be sable f for anyǫ>, here exssδ> such ha for any nal condonφ(s) C([ϑ, ],R n ) sasfyng φ <δ, we have for he corresondng soluon ha x(, s,φ) <ǫfor ; () he rval soluon of (9) s sad o be asymocally sable f s sable and for any nal condonφ(s) C([ϑ, ],R n ) we have for he corresondng soluon ha lm x(, s,φ) =; () he rval soluon of (9) s sad o be globally exonenally sable f here exs scalars k> andα> such ha for any nal condonφ(s) C([ϑ, ],R n ), we have for he corresondng soluon ha x(, s,φ) αe k φ for. DefneH φ =H 1φ H 2φ H nφ, whereh φ s he sace conssng of connuous funconsϕ () : R R such haϕ (θ)=φ(θ) forϑ θ andϕ () as,, 2 n. For anyϕ()=(ϕ 1 (),ϕ 2 (),,ϕ n ()) H φ and η()=(η 1 (),η 2 (),,η n ()) H φ, f we defne he merc as d(ϕ,η)=su ϑ n ϕ () η (), henh φ becomes a comlee merc sace. Usng he conracon mang defned on he sace H φ and alyng he conracon mang rncle, we oban our fourh resul. Theorem Suose ha he assumons (A1)-(A3) hold. If he followng condons are sasfed, () he dsrbued delay r() s bounded by a consan r; () α 1 c max a jα j,2,,n 1 c hen he rval soluon of (9) s asymocally sable. max b jβ j,2,,n r c max d jγ j <1, (11),2,,n By esablshng an arorae negral neualy, we oban suffcen condons for exonenal sably of (9), whch s our ffh resul. Theorem Suose ha he assumons (A1)-(A3) hold. If he followng condons are sasfed, () he dscree delayτ() and he dsrbued delay r() are bounded by a consanτ; () 1 c max,2,,n a jα j 1 c max,2,,n b jβ j 1 c hen he rval soluon of (9) wh nal condon (1) s exonenally sable. τ max,2,,n d jγ j <1, c=mn{c 1, c 2, c n }, (12) Remark Several exonenal sably resuls [13, 29, 3] were rovded for he sysem (9), by consrucng an arorae Lyaunov funconal and emloyng lnear marx neualy (LMI) mehod. However, he delays n hose resuls should sasfy he followng condon (H). From our man resuls, we need no know he arcular form relaed o he delays, we only need o know ha he delays are bounded. Furhermore, Theorem 1.12 s an exenson and mrovemen of he resul n La and Zhang [12]. As an examle, we consder a cellular neural nework wh me varyng delays where C= dx() d ( = Cx()Ag(x()) Bg(x τ()), ) (.2, A= ) (.1.1, B=.2.1 ).

7 The acvaon funcon s descrbed by g(x)= x1 x 1 2. The me-varyng delay sτ()= 1 5 cos. I s clear ha he dscree delay s bounded bu no dfferenable. Hence, he resuls n [13, 29, 3] are no alcable. The res of hs aer s organzed as follows. In Secon 2, we resen a roof of Theorem 1.4. The roof of Theorem 1.5 s resened n Secon 3 and he roof of Theorem 1.8 s gven n Secon 4. we resen he roofs of Theorem 1.12 and Theorem 1.13 n Secon 5 and Secon 6, resecvely. Some examles are gven o llusrae our man resuls n Secon 7 and an aendx s gven n Secon Proof of Theorem 1.4 In hs secon, we rove Theorem 1.4. We sar wh some rearaons. Lemma 2.1. ([32]) Ifω()=(ω 1,ω 2,,ω n ) T s a n-dmensonal Brownan moon defned on a comlee robably sace (Ω, F, P), hen we have he followng formula ( ) f (s) dω (s) f j (s) dω j (s) = f (s) f j (s) d ω,ω s, where ω,ω s =δ j s are he cross-varaons,δ j s correlaon coeffcen, 1, j n. If we mully boh sdes of (1) by e c and negrae from o, we oban x () = e c x () for,, 2, 3,, n. e c ( s) s e c ( s) l j s r(s) a j f j (x j (s)) ds h j (x j (u)) du ds e c ( s) e c ( s) b j g j (x j (s τ(s))) ds (13) σ j (s, x j (s), x j (s τ(s))) dω j (s). Lemma 2.2. Defne an oeraor by (Qϕ)()=φ() for [ϑ, ], and for,, 2, 3,, n, (Qϕ )() = e c ϕ () e c ( s) s e c ( s) l j s r(s) a j f j (ϕ j (s)) ds h j (ϕ j (u)) du ds e c ( s) e c ( s) b j g j (ϕ j (s τ(s))) ds (14) σ j (s,ϕ j (s),ϕ j (s τ(s))) dω j (s). Suose ha he assumon (A1)-(A4) holds. If condon (4) holds, hen Q :S φ S φ and Q s a conracon mang. Proof. Denoe (Qϕ )() := J 1 () J 2 () J 3 () J 4 () J 5 (), where J 1 () J 3 () = J 4 () = J 5 () = = e c ϕ (), J 2 ()= e c ( s) e c ( s) e c ( s) e c ( s) b j g j (ϕ j (s τ(s))) ds, l j s s r(s) h j (ϕ j (u)) du ds, a j f j (ϕ j (s)) ds, σ j (s,ϕ j (s),ϕ j (s τ(s))) dω j (s). 7

8 Se1. From he defnon of Banach saces φ, we have ha n ϕ () <, for all,ϕ S φ. Se2. We rove he connuy n h momen of Q on [, ). Le x S φ, 1, r be suffcenly small and ake he lm r. We have 1 J 2 ( 1 r) J 2 ( 1 ) = ( e c ( 1 r s) e ) n c ( 1 s) a j f j (x j (s)) ds 1 r e c ( 1 r s) a j f j (x j (s)) ds as r. Smlarly, we have ha J 3 ( 1 r) J 3 ( 1 ) as r, In he followng, we check he connuy of J 5 (). J 5 ( 1 r) J 5 ( 1 ) = n r e c ( 1 r s) J 4 ( 1 r) J 4 ( 1 ) as r. ( e c ( 1 r s) e ) n c ( 1 s) σ j (s, x j (s), x j (s τ(s))) dω j (s) 1 r e c ( 1 r s) 1 1 (2n) 1 (2n) 1 = (2n) 1 [1 r 1 σ j (s, x j (s), x j (s τ(s))) dω j (s) ( e c ( 1 r s) e c ( 1 s) ) σ j (s, x j (s), x j (s τ(s))) dω j (s) σ j (s, x j (s), x j (s τ(s))) dω j (s) e c ( 1 r s) e c ( 1 s) σ j (s, x j (s), x j (s τ(s))) dω j (s) 1 e c ( 1 r s) σ j (s, x j (s), x j (s τ(s))) dω j (s) 1 e c ( 1 r s) e c ( 1 s) 2 σ 2 j (s, x j(s), x j (s τ(s))) ds 1 r Thus, Q s ndeed connuous n h momen on [, ). ] /2 e 2c ( 1 r s) σ 2 j (s, x j(s), x j (s τ(s))) ds as r. /2 Se3. We rove ha Q(S φ ) S φ. Qϕ () = 5 J j () J j (). (15) 8

9 Now, we esmae he erms on he rgh sdes of he above neualy. J 2 () = e c ( s) a j f j (ϕ j (s)) ds e c ( s) e c ( s) a j f j (ϕ j (s)) ds [ ] / e c( s) ds e c ( s) a j f j (ϕ j (s)) ds c / e c ( s) a j α j ϕ j (s) ds / c / a j α j e c( s) ϕ j (s) ds. (16) Snceϕ() S φ, we have ha lm n ϕ () =. Thus for anyǫ>, here exss T 1 > such ha T 1 mles n ϕ () <ǫ, combnng wh (16), we oban ha / T1 J 2 () c / a j α j e c( s) ϕ j (s) ds / c / a j α j e c( s) ϕ j (s) ds T 1 / / < c e c (e c T 1 1) a j α j su ϕ j (s) ǫ c a j α j Hence, from he fac ha c > (, 2,, n), we oban ha n J 2 () as. Wh he smlar comuaon as (16), we oban ha / J 3 () c / b j β j e c( s) ϕ j (s τ(s))) ds / J 4 () c / l j γ j e c( s) s ϕ j (u) du s r(s) ds ( r / s c ) / l j γ j e c ( s) ϕ j (u) du ds. (17) s T 1 s r(s) Usng Lemma 2.1, we oban ha J 5 () = e c ( s) σ j (s,ϕ j (s),ϕ j (s τ(s))) dω j (s) [ ] 2 n 1 e c( s) σ j (s,ϕ j (s),ϕ j (s τ(s))) dω j (s) /2 [ ] /2 = n 1 e 2c( s) σ 2 j (s,ϕ j(s),ϕ j (s τ(s))) ds 9 (18)

10 [ n 1 e ( ] /2 2c ( s) µ j ϕ 2 j (s)ν jϕ 2 j (s τ(s))) ds ( ) /2 ( ) /2 n 1 2 /2 1 e 2c( s) µ j ϕ 2 j (s) ds e 2c( s) ν j ϕ 2 j (s τ(s)) ds ( ) /2 1 n 1 2 /2 1 e 2c( s) ds e 2c( s) µ /2 j ϕ j (s) ds ( ) /2 1 n 1 2 /2 1 e 2c( s) ds e 2c( s) ν /2 j ϕ j (s τ(s)) ds n 1 c 1 /2 µ/2 e 2c( s) ϕ j (s) dsν/2 e 2c( s) ϕ j (s τ(s)) ds n 1 c 1 /2 µ/2 e c( s) ϕ j (s) dsν/2 e c( s) ϕ j (s τ(s)) ds. Snce n ϕ (), τ() and r() as, for eachǫ>, here exss T 2 > such ha T 2 mles n ϕ ( τ(s)) <ǫ and n ϕ ( r()) <ǫ. From (17), we oban ha J 3 () / T2 c / b j β j e c( s) ϕ j (s τ(s))) ds / c / b j β j e c( s) ϕ j (s τ(s))) ds T 2 ( ) / / 1 T2 < e c e c s ds b j β j su ϕ j (s τ(s))) ǫ c ϑ s T 2 c b j β j / and J 4 () < ( r c ) / l j γ j ( r c ) / l j γ j ( re c r c ) / l j γ j / T2 s e c ( s) s r(s) / s e c ( s) T 2 s r(s) / su ϑ u T 2 ϕ j (u) du ds ϕ j (u) du ds T2 ϕ j (u) e cs ds ( ǫr r c c ) / l j γ j /. Furher, from (18), we oban J 5 () 1

11 n 1 n 1 c 1 /2 c 1 /2 n 1 < n 1 c 1 /2 n 1 µ/2 e c( s) ϕ j (s) dsν/2 e c( s) ϕ j (s τ(s)) ds T2 T2 µ/2 e c( s) ϕ j (s) dsν/2 e c( s) ϕ j (s τ(s)) ds µ/2 e c( s) ϕ j (s) dsν/2 e c( s) ϕ j (s τ(s)) ds T 2 T 2 T2 µ/2 su ϕ j (s) ν/2 su ϕ j (s) e s T 2 ϑ s T 2 c( s) ds ( ǫ(µ /2 ν /2 ) ). c 1 /2 c 1 /2 Hence, le, from he fac ha c > (, 2,, n), we oban ha c J 3 (), J 4 (), and J 5 (). Thus, combnng wh (15), we oban ha n Qϕ () as n ϕ (). Therefore, Q :S φ S φ. Se4. We rove ha Q s a conracon mang. For anyϕ,ψ S φ, from (16)-(18), we oban su s ϑ Qϕ (s) Qψ (s) 4 1 su s ϑ s ( ) e c (s u) a j f j (x j (u)) f j (y j (u)) du 4 1 su s ϑ s ) e c (s u) b j (g j (x j (u τ(u))) g j (y j (u τ(u))) du 4 1 su s ϑ s s ( ) e c (s u) l j h j (ϕ j (v)) h j (ψ j (v)) dv du s r(s) 4 1 su s ϑ s ( ) e c (s u) σ j (s, x j (s), x j (u τ(u))) σ j (s, y j (s), y j (s τ(u))) dω j (u) / s 4 1 su c / a j α j e c(s u) ϕ j (u) ψ(u) du s ϑ / s 4 1 su c / b j β j e c(s u) ϕ j (u τ(u))) ψ j (u τ(u))) du s ϑ ( τ / s u 4 1 su s ϑ c ) / l j γ j e c (s u) ϕ j (v) ψ j (v) dv du u r(u) 4 1 n 1 s su c 1 /2 s ϑ µ/2 e c(s u) ϕ j (u) ψ j (u) du 11

12 s ν /2 e c(s u) ϕ j (u τ(u)) ψ j (u τ(u)) du / / 4 1 c a j α j c b j β j ( / r ( c ) l j γ j n 1 c /2 µ /2 ν /2) su s ϑ ϕ j (s) ψ j (s) =α su s ϑ ϕ j (s) ψ j (s). From (4), we oban ha Q :S φ S φ s a conracon mang. We are now ready o rove Theorem 1.4. Proof. From Lemma 2.2, by he conracon mang rncle, we oban ha Q has a unue fxed on x(), whch s a soluon of (1) wh x()=φ() as [ϑ, ] and n x () as. Now, we rove ha he rval soluon of (1) s h momen sable. Leǫ> be gven and chooseδ> (δ<ǫ) such ha 5 1 δ<(1 α)ǫ. If x()=(x 1 (), x 2 (), x n ()) T s a soluon of (1) wh he nal condon (2) sasfyng n φ () <δ, hen x()=(qx)() defned n (14). We clam ha n x () <ǫ for all. Noce ha n x () <ǫ for [ϑ, ], we suose ha here exss > such ha n x ( ) =ǫand n x () <ǫforϑ <, hen follows from (4), we oban ha / x ( ) 5 1 e c x () 5 1 c / a j α j e c ( s) x j (s) ds / 5 1 c / b j β j e c( s) x j (s τ(s))) ds ( r / s 5 1 c ) / l j γ j e c ( s) x j (u) du ds s r(s) 5 1 n 1 c 1 /2 µ/2 e c ( s) x j (s) ds ν /2 e c( s) x j (s τ(s)) ds / / 5 1 c a j α j 5 1 c b j β j ( r 5 1 c ) l j γ j < (1 α)ǫαǫ=ǫ. / 5 1 n 1 ( c /2 µ /2 ν /2) ǫ 5 1 δ whch s a conradcon. Therefore, he rval soluon of (1) s asymocally sable n h momen. Corollary 2.3. Suose ha he assumons (A1)-(A4) hold. If he followng condons are sasfed, () he dsrbued delay r() s bounded by a consan r; 12

13 () 5 c 2 a 2 j α2 j 5 c 2 b 2 j β2 j ( r 5 c ) 2 l j γ j 2n c 1 (µν) α, where c,µ,ν are defned as n Theorem 1.4, hen he rval soluon of (1) s asymocally sable n mean suare Consder he sochasc neural neworks whou dsrbued delays dx ()= c x () a j f j (x j ()) b j g j (x j ( τ())) d σ j (, x j (), x j ( τ())) dω j () (19) for, 2, 3,, n. Corollary 2.4. Suose ha he assumons (A1)-(A4) hold. The rval soluon of (19) s asymocally sable n h momen f he followng neualy holds, 4 1 c a j α j / 4 1 whereµ,ν are defned as n Theorem 1.4. c b j β j / 4 1 n 1 Remark 2.5. Noe ha he dscree delayτ() n Corollary 2.4 can be unbounded. ( c /2 µ /2 ν /2) α, (2) 3. Proof of Theorem 1.5 In hs secon, we rove Theorem 1.5. We sar wh some rearaons. Lemma 3.1. Defne an oeraor by (Pϕ)()=φ() for [ τ, ], and for, (Pϕ)() s defned as (14), f here s α (, 1) such ha (6) holds, hen P :C φ C φ s a conracon mang. Proof. From he roof of Theorem 1.4, we oban ha P s connuous n h momen on [, ). Now, we rove ha P(C φ ) C φ. [ su Qϕ (s) ]= su 5 5 [ ] J j (s) 5 1 su J j (s). We esmae he erms on he rgh-hand sde of he above neualy. Le c=mn{c 1, c 2, c 3,, c n }, su J 2 (s) s = su e c (s u) a j f j (ϕ j (u)) du s c / su e c(s u) a j α j ϕ j (u) du / s c / a j α j su e c(s u) ϕ j (u) du / n { [ s ]} c / a j α j e c(s u) ϕ j (u) du 13 su

14 / { c / a j α j / e cτ c / a j α j / ( e cτ c a j α j su [ [ s ( e c(s u) su e c( u) ( su u τ v u u τ v u ϕ j (v) ) ϕ j (v) ) ] du ]} du ) su x j (s). (21) Snce n su ϕ j (s) as, hen for anyǫ>, here exss such ha T 1 mles [ su ϕ j (s) ]<ǫ. Then, combnng wh (21), we oban ha / J 2 (s) < ecτ c a j α j ǫ. su Hence, we oban ha n su J 2 (s) as. Smlarly, we oban ha / s su J 3 (s) c / b j β j su e c(s u) ϕ j (u τ(u)) du / { [ s ]} c / b j β j su e c(s u) ϕ j (u τ(u)) du / { [ s ]} c / b j β j su e c(s u) su ϕ j (v) du u τ v u / [ ] e cτ c / b j β j e c( u) su ϕ j (v) du u τ v u / [ ] e cτ c b j β j su ϕ j (s). (22) Snce n [ su ϕ j (s) ] as, ha s, for anyǫ >, here exss such ha T 2 mles n [ su ϕ j (s) ] <ǫ, so combnng wh (22) we have su Hence, n su J 3 (s) as. su J 4 (s) c / l j γ j c / l j γ j / J 3 (s) < ecτ c b j β j ǫ. / / 14 su n { su s e c(s u) u u r(u) [ s u e c(s u) u r(u) ϕ j (v) dv ϕ j (v) dv du ]} du

15 / { τ c / l j γ j / τ e cτ c / l j γ j / n [ τ e cτ c l j β j su [ [ s e c(s u) su u τ v u e c( u) su u τ v u ] su ϕ j (s). ]} ϕ j (v) du ] ϕ j (v) du Hence, we have ha [ n su J 4 (s) ] as. Leµ=max{µ 1,µ 2,,µ n },ν=max{ν 1,ν 2,,ν n }, [ s J 5 (s) n 1 su e c(s u) σ j (u,ϕ j (u),ϕ j (u τ(u))) dω j (u) ] su n 1 n 1 { n 1 e cτ su su [ su τ r { [ su su s τ r s [ e c(r u) σ j (u,ϕ j (u),ϕ j (u τ(u))) dω j (u) ]} s e c(r u) σ j (u,ϕ j (u),ϕ j (u τ(u))) dω j (u) ]} e c( u) σ j (u,ϕ j (u),ϕ j (u τ(u))) dω j (u) ( s /2 K n 1 e cτ su e 2c( u) σ 2 j (u,ϕ j(u),ϕ j (u τ(u))) du) ( s K n 1 e cτ 2 /2 1 su e 2c( u)( /2 µ j ϕ 2 j du) (u)) ( s K n 1 e cτ 2 /2 1 su e 2c(T u)( /2 ν j ϕ 2 j du) (u τ(u))) ( s ) /2 1 K n 1 e cτ 2 /2 1 su e 2c( u) du ( s s )]} e 2c( u) µ /2 j ϕ j (u) du e 2c(T u) ν /2 j ϕ j (u τ(u)) du [ ] K n e cτ c 1 /2 (µ /2 ν /2 ) e 2c(T u) su ϕ j (v) du u τ v u [ ] K n e cτ c 1 /2 (2c) 1 (µ /2 ν /2 ) su ϕ j (s). (23) Snce n [ su ϕ j (s) ] as, ha s, for anyǫ>, here exss T 3 > such ha T 3 mles n [ su ϕ j (s) ] <ǫ, so combnng wh (23) we have su J 5 (s) < K n e cτ c 1 /2 (2c) 1 (µ /2 ν /2 )ǫ. Hence, [ n su J 5 (s) ] as. Thus, P(C φ ) C φ. ] 15

16 Fnally, we rove ha Q s a conracon mang. For anyϕ,ψ C φ, from (21)-(23), we oban ha su su Qϕ (s) Qψ (s) 4 1 su s ( ) su e c (s u) a j f j (ϕ j (u)) f j (ψ j (u)) du 4 1 su s ) su e c (s u) b j (g j (ϕ j (u τ(u))) g j (ψ j (u τ(u))) du 4 1 su s s ( ) su e c (s u) l j h j (ϕ j (v)) h j (ψ j (v)) dv du s r(s) 4 1 su s su e c (s u) [σ j (u,ϕ j (u),ϕ j (u τ(u))) σ j (u,ψ j (u),ψ j (u τ(u)))] dω j (u) / / / 4 1 ecτ c a j α j e cτ c b j β j τ e cτ c l j γ j K n e cτ c 1 /2 (2c) 1( µ /2 ν /2) [ ] su su ϕ j (s) ψ j (s) =α su [ ] su ϕ j (s) ψ j (s). From (6), we oban ha Q :C φ C φ s a conracon mang. We are now ready o rove Theorem 1.5 Proof. From Lemma 3.1, by he conracon mang rncle, we oban ha P has a unue fxed on x(), whch s a soluon of (1) wh x()=φ() as [ τ, ] and n [ su x (s) ] as. We rove ha he rval soluon of (1) s h momen sable. Leǫ > be gven and chooseδ> (δ<ǫ) sasfyng 5 1 e c δ<(1 α)ǫ. (24) If x()=(x 1 (), x 2 (), x n ()) T s a soluon of (1) wh he nal condon sasfyng φ <δ, hen x()=(px)() defned n (14). We clam ha x <ǫfor all. Noce ha φ() <ǫfor [ τ, ], we suose ha here exss > such ha n [ su τ s x (s) ] =ǫand n [ su x (s) ] <ǫfor τ <, hen follows from (4) and (24), we oban ha [ ] su x (s) τ s [ su J j (s) τ s ] / 5 1 e c δ5 1 ecτ c a j α j e cτ c b j β j / τ e cτ c l j γ j K n e cτ c 1 /2 (2c) 1( µ /2 ν /2) ǫ < (1 α)ǫαǫ=ǫ. whch s a conradcon. Therefore, he rval soluon of (1) s asymocally sable n h momen. 16 /

17 4. Proof of Theorem 1.8 In hs secon, we rove Theorem 1.8. We sar wh a lemma resenng an negral neualy. Lemma 4.1. Consderγ>, osve consansλ 1,λ 2,λ 3 and a funcon y : [ τ, ) [, ). Ifλ 1 λ 2 τλ 3 < c and he followng neualy holds, y e c λ 1 e c( s) y(s) dsλ 2 e c( s) y(s τ(s)) dsλ 3 e c( s) s y(u) du ds, s r(s) y() (25) y e c, [ τ, ], ( ) hen we have y() y e γ ( τ), whereγs a osve roo of he algebrac euaon 1 c γ λ1 e γτ λ 2 eγτ 1 γ λ 3 = 1. ( ) Proof. Le F(γ)= 1 c γ λ1 e γτ λ 2 eγτ 1 γ λ 3 1. We have F()F(c )<, ha s, here exss a osve consan γ (, c) such ha F(γ)=. For anyǫ>, le To rove he lemma, we clam ha (25) mles C ǫ =ǫ y. y() C ǫ e γ, τ. (26) I s easly shown ha (26) holds for [ τ, ]. Assume ha here exss 1 > such ha Combnng wh (25), we have y()<c ǫ e γ, [ τ, 1 ), y( 1 )=C ǫe γ 1. (27) y(1 ) y 1 e c 1 λ1 e c( 1 s) 1 y(s) dsλ 2 e c( 1 s) 1 s y(s τ(s)) dsλ 3 e c( 1 s) 1 < y e c 1 Cǫ λ 1 e c( 1 s) e γs 1 ds C ǫ λ 2 e c( 1 s) e γ(s τ(s)) 1 s ds C ǫ λ 3 e c( 1 s) = [ y C ( )] ǫ λ 1 e γτ λ 2 eγτ 1 C λ 3 e c 1 ǫ c γ γ c γ From he defnon of C ǫ, we have y C ǫ c γ ( ( ) λ 1 e γτ λ 2 eγτ 1 λ 3 e γ 1. γ λ 1 e γτ λ 2 eγτ 1 λ 3 )=y C ǫ <. γ s r(s) y(u) du ds s r(s) e γu du ds Then, ogeher wh he defnon ofγ, we oban ha y( 1 )< C ǫe γ 1, whch conradcs (27), so (26) holds. Asǫ> s arbrarly small, n vew of (26), follows ha y() y e γ, for τ. Proof. For he reresenaon (13), usng (16)-(18), we oban ha / x () 5 1 e c φ () 5 1 c / a j α j e c( s) x j (s) ds / 5 1 c / b j β j e c( s) x j (s τ(s))) ds ( τ ) / / s 5 1 c l j γ j e c( s) x j (u) du ds s r(s) 5 1 n c 1 /2 µ/2 e c( s) x j (s) dsν/2 e c( s) x j (s τ(s)) ds. 17

18 Hence, by usng Lemma 4.1 and (7), we oban ha he rval soluon of (1) s exonenally sable n h momen. Corollary 4.2. Suose ha he assumons (A1)-(A4) hold. If he followng condons are sasfed, () he dscree delayτ() and dsrbued delay r() are bounded by a consanτ; () 5c 2 a 2 j α2 j 5c 2 where c,µ,ν are defned as n Theorem 1.4, b 2 j β2 j 5c 2 τ 2 hen he rval soluon of (1) s exonenally sable n mean suare, l 2 j γ2 j 2n2 c 1 (µν)<1, Corollary 4.3. Le 2. Suose ha he assumons (A1)-(A4) hold. If he followng condons are sasfed, () he dscree delayτ() and dsrbued delay r() are bounded by a consanτ; () / / 4 1 c a j α j 4 1 c b j β j 4 1 n c /2 (µ /2 ν /2 )<1, where c,µ,ν are defned as n Theorem 1.4, hen he rval soluon of (19) s exonenally sable n h momen. 5. Proof of Theorem 1.12 In hs secon, we rove Theorem We sar wh some rearaons. Mully boh sdes of (9) by e c and negrae from o, we oban ha for, x () = e c x () e c ( s) a j g j (x j (s)) ds e c ( s) b j g j (x j (s τ(s))) ds e c ( s) d j s s r(s) g j (x j (u)) du ds,, 2, 3,, n. (28) Lemma 5.1. Defne an oeraor by (Px)(θ) = φ(θ), for ϑ θ, and for, Px () = e c x () e c ( s) a j g j (x j (s)) ds e c ( s) b j g j (x j (s τ(s))) ds e c ( s) d j s s r(s) g j (x j (u)) du ds := I 1 () I 2 () I 3 () I 4 (). (29) If here exssα (, 1) such ha (11) holds, hen P :H φ H φ and P s a conracon mang. Proof. Frs, we rove ha PH φ H φ. In vew of (29), we have ha, for fxed me 1, s easy o check ha lm r [(Px )( 1 r) (Px )( 1 )]=. Thus, P s connuous on [, ). Noe ha (Px )(θ)=φ(θ) forϑ θ, we oban ha P s ndeed connuous on [ϑ, ). 18

19 Nex, we rove ha lm (Px )()= for x () H φ. Snce x () H φ, we have ha lm x ()=. Then for anyǫ>, here exss T > such ha s T mles x (s) <ǫ. Choose T= max,2,,n {T }, combnng wh condon (A2), I 2 () = e c ( s) a j f j (x j (s)) ds T e c ( s) a j k j x j (s) ds e c ( s) a j α j x j (s) ds e c a j α j su x j (s) s T T a j α j su x j (s) s T T e c ( s) dsǫ T a j α j e c s ds ǫ c T e c ( s) ds a j α j. (3) From he fac ha c > (, 2,, n) and esmae (3), we have ha I 2 () as. Snce x () and τ() as, for eachǫ >, here exss T > such ha s T mles x (s τ(s)) <ǫ for, 2,, n. Choose T = max,2, n {T }, we oban I 3 () = e c ( s) b j g j (x j (s τ(s))) ds T e c ( s) b j β j x j (s τ(s)) ds e c ( s) b j k j x j (s τ(s)) ds T e c b j β j su ϑ s T x j (s) T e c s ds ǫ c b j β j. (31) From he fac ha c > (, 2,, n) and esmae (31), we have ha I 3 () as. Snce x () and r() as, for eachǫ >, here exss T > such ha s T mles x (s r(s)) <ǫ for, 2,, n. Choose T = max,2, n {T }, we oban s I 4 () = e c ( s) d j h j (x j (u)) du ds s r(s) T s e c ( s) d j γ j x j (u) du dsǫr e c ( s) d j γ j ds s r(s) T T r d j γ j su x j (u) e c( s) ds ǫr d j γ j. (32) ϑ u T c From he fac ha c > (, 2,, n) and esmae (32), we have ha I 4 () as. From he above esmae, we conclude ha lm (Px )()= for x () H φ. Therefore, P :H φ H φ. Now, we rove ha P s a conracon mang. For any x(), y() H φ, from (3) and (32), we oban ha (Px )() (Py )() max a jα j,2,,n max b jβ j,2,,n e c ( s) e c ( s) x j (s) y j (s) ds x j (s τ(s)) y j (s τ(s)) ds 19

20 { 1 c = α su ϑ s max d jγ j,2,,n max,2,,n a jα j 1 c x j (s) y j (s). From (11), we oban ha P s a conracon mang. We are now ready o rove Theorem s e c ( s) s r(s) max,2,,n b jβ j r c x j (u) y j (u) du ds } max d jγ j su,2,,n ϑ s x j (s) y j (s) Proof. Le P be defned as n Lemma 5.1, by conracon mang rncle, P has a unue fxed on x H φ wh x(θ)=φ(θ) onϑ θ and x() as. To oban asymocally sable, we need o rove ha he rval eulbrum x = of (9) s sable. For any ǫ >, chooseσ> andσ<ǫ sasfyng he condonσǫα<ǫ. If x(, s,φ)=(x 1 (, s,φ), x 2 (, s,φ),, x n (, s,φ)) s he soluon of (9) wh he nal condon φ <σ, he we clam ha x(, s,φ) <ǫ for all. Indeed, we suose ha here exss > such ha x ( ; s,φ) =ǫ, and x (; s,φ) <ǫ for <. (33) From (11) and (28), we oban x ( ; s,φ) e c x () e c ( s) s e c ( s) d j 1 < σǫ max a jα j,2,,n c s r(s) 1 c a j f j (x j (s)) ds e c ( s) h j (x j (u)) du ds max b jβ j,2,,n r whch conradcs (33). Therefore, x(, s,φ) <ǫ for all. Ths comlees he roof. c b j g j (x j (s τ(s))) ds max d jγ j σǫα<ǫ.,2,,n Le d j for, 2, n,, 2, n, he sysem s reduced o dx () = c x () a j f j (x j ()) b j g j (x j ( τ())), (34) d whch s he descron of cellular neural nework wh me-varyng delays. Followng he resul of heorem 1.12, we have he followng corollary. Corollary 5.2. Suose ha he assumons (A1)-(A3) hold. If he followng condon s sasfed, 1 max a 1 jα j max b jβ j <1, (35),2,,n,2,,n hen he rval soluon of (34) s asymocally sable. c Remark 5.3. Noe ha he delay n Corollary 5.2 can be unbounded. La and Zhang [12] suded he asymoc sably (34) as well. However, he addonal condon 1 max a j k j 1 b j k j c < 1 (36) n,2, n c s needed n Theorem 4.1 of [12]. I s clearly ha Corollary 5.2 s an mrovemen of he resul n [12]. c 2

21 6. Proof of Theorem 1.13 Proof. From he reresenon (28), we oban ha x () e c x () max { b jk j },2, n max { d jk j },2, n e c( s) e c( s) s e c( s) s r(s) max { a jk j },2, n x j (s τ(s)) ds x j (s) ds x j (u) du ds. Combnng wh Lemma 4.1, we oban ha he rval soluon of (9) wh nal condon (1) s exonenally sable. For he cellular neural nework (34), we have he followng resul. Corollary 6.1. Suose ha he assumons (A1)-(A3) hold. If he followng condons are sasfed, () he dscree delayτ() and dsrbued delay r() are bounded by a consanτ; () max a jk j,2,,n max b jk j <c, c=mn{c 1, c 2, c 1 },,2,,n hen he rval soluon of (34) wh nal condon (1) s exonenally sable. 7. xamles xamle 7.1. Consder he followng wo-dmensonal cellular neural nework where C= ( ) c1 = c 2 ( 3 3 dx() d = Cx()Ag(x()) Bg(x τ()), ) ( ) ( a11 a, A= 12 = a 21 a 22 6/7 3/7 1/7 1/7 ) ( ) ( b11 b, B= 12 6/7 2/7 = b 21 b 22 3/7 1/7 The acvaon funcon s descrbed by g (x)= x1 x 1 2, for, 2. The me-varyng delayτ() s connuous and τ() τ, whereτs a consan. I s clear haα =β = 1 for, 2. We check he condon (35) n Corollary 5.2, 2 1 c max,2 a jα j 2 1 max c b jβ j 1 ( 6, ) = < 1. Hence, by Corollary 5.2, he rval eulbrum x = of hs cellular neural nework s asymocally sable. However, he condon (36) becomes 1 2 max a j α j 1 2 b j β j,2 c c = > 1. 2 ). Hence, Theorem 4.1 of [12] s no alcable. 21

22 xamle 7.2. Consder he wo-dmensonal sochasc recurren neural nework wh me-varyng delays ( )( ) ( )( ) 6 x1 () anh(x1 ()) dx() = d d 5 x 2 () anh(x 2 ()) ( )( ) ( ) anh(x1 ( τ 1 ())) 1 2 r() d anh(x 2 ( τ 2 ())) anh(x 1(s)) ds.2 anh(x r() 2(s)) ds) d σ(, x(), x( τ())) dw(), (37) whereτ(), r() are connuous funcons such ha τ() as and r() 1,σ : R R 2 R 2 R 2 R 2 sasfes race [ σ T (, x, y)σ(, x, y) ].3(x 2 1 x2 2 y2 1 y2 2 ). We suose =2, and akeµ =ν =.3 for, 2, by smle comuaon, we haveα =.2, for, 2, c=mn{c 1, c 2 }=5,µ=ν=.3. From Corollary 2.3, we have ha 5 2 c 2 2 a 2 j α2 j 5 2 c 2 2 b 2 j α2 j 2 ( τ 2 5 c ) 2 l 2 j α2 j c 1 (µν)<.256<1. Then he rval soluon of (37) s mean suare asymocally sable. If τ() s bounded, from Corollary 4.2, we oban ha c 2 a 2 j α2 j 5c 2 b 2 j α2 j 5c 2 τ 2 l 2 j α2 j 2 4c 1 (µν)<.298. Hence, he rval soluon of (37) s mean suare exonenally sable. xamle 7.3. Consder a wo-dmensonal sochascally erurbed HNN wh me-varyng delays, dx()=[ Cx()A f (x()) Bg(x τ ())] dσ(, x(), x τ ()) dω(), (38) where f (x)= 1 5 arcan x, g(x)= 1 5 anh x= 1 5 (ex e x )/(e x e x ),τ()= 1 2 sn 1 2, C= ( ) (, A= ) and B= ( In hs examle, le =3, akeα j =.2,β j =.2,, 2,σ : R R 2 R 2 R 2 R 2 sasfes race [ σ T (, x, y)σ(, x, y) ].1(x 2 1 x2 2 y2 1 y2 2 ). Noe ha he exonenal sably of (38) has been suded n Sun and Cao [28] by emloyng he mehod of varaon arameer, neualy echnue and sochasc analyss. Now, we check he condon n Corollary 4.3, / / c (1/) a j α j 4 1 c (1/) b j β j c /2 (µ /2 ν /2 )<.18<1. From Corollary 4.3, he rval soluon of (38) s exonenally sable. ). 22

23 8. Aendx In hs secon, we frs show ha I 5 (s) n [19] s no a local marngale and hen we resen some examles abou Banach saces. Defnon 8.1. A real valuedf -adaed rocess M={M() : } s a marngale f M() < for all and [M() F s ]= M(s), a.s. for all s<<. Lemma 8.2. For connuous funconσ(), e c( s) σ(s) dω(s) s no a marngale. Proof. In fac, for u, [ ] e c( s) σ(s) dω(s) F u [ u = = u [ e c( s) σ(s) dω(s) F u ] e c( s) σ(s) dω(s) u u e c( s) σ(s) dω(s) F u ] e c(u s) σ(s) dω(s). (39) Lemma 8.3. ([24]) If M() s a local marngale and for every, su s [,] M(s) <, hen M() s a marngale. Lemma 8.4. For connuous funconσ(), e c( s) σ(s) dω(s) s no a local marngale. Proof. We suose ha e c( s) σ(s) dω(s) s a local marngale. For every, we have ha s su e c(s u) σ(u) dω(u) s = su e cs e cu σ(u) dω(u) s [,] s [,] s su e cu σ(u) dω(u) s [,] ( ) 1/2 ( 1/2 K 1 e 2cu σ 2 (u) du K 1 e 2cu σ 2 (u) du) <. From Lemma 8.3, we oban ha M s a marngale. However, from Lemma 8.2, we know ha e c( s) σ(s) dω(s) s no a marngale, whch s a conradcon. A normed lnear sace s a merc sace wh resec o he merc d derved from s norm, where d(x, y)= x y. Defnon 8.5. A Banach sace s a normed lnear sace ha s comlee merc sace wh resec o he merc derved from s norm. Here are some examles of Banach saces. xamle 8.6. The sace C([a, b]) of connuous, real-valued (or comlex-valued) funcons on [a, b] wh he sunormed s a Banach sace. More generally, we have he followng examles. () If X s a Banach sace, he sace C([a, b]; X) of connuous, X-valued funcons on [a, b] eued wh he su-norm s a Banach sace. () If X s a Banach sace, he sace BC([a, b]; X) :={ϕ ϕ C([a, b]; X), ϕ < } of bounded connuous, X- valued funcons on [a, b] eued wh he su-norm s a Banach sace. () If X s a Banach sace, he sace{ϕ ϕ C([a, b]; X), lm ϕ()=} and he sace { } ϕ ϕ C([a, b]; X), ϕ = su ϕ(s) s bounded and lmϕ()= s [a,b] are Banach saces wh resec o he su-norm. Clearly, he sace { } C ([a, b]; L (Ω,R n )) := ϕ ϕ C([a, b]; L (Ω,R n )), lm ϕ() = s a Banach saces wh resec o he norm defned by ϕ := su s [ ϕ(s) ]. 23

24 The followng lemma resens a Banach sace ha s used n hs aer. Lemma 8.7. Suose haf s comlee, ha s, conans all null ses. Then he sace s a closed subsace of C ([a, b]; L (Ω,R n )). D :={ϕ C ([a, b]; L (Ω,R n )),ϕ() sf measurable for all } Proof. Leϕ(),ψ() D, henϕ() andψ() aref -measurable for all, soϕ()ψ() andαϕ() (α C) aref - measurable for all. Suose ha he seuenceϕ 1 (),ϕ 2 (), ϕ n () D,ϕ() C ([a, b]; L (Ω,R n )) andϕ n () ϕ() as n for all, we clam haϕ() sf -measurable. In fac, snceϕ n () ϕ() as n, hen [ su ϕn (s) ϕ(s) ] as n. s Ω So, for every, we oban ha ϕ n (s) ϕ(s) as n, whch mles ha here exss a subseuence (ϕ nk ()) k such haϕ nk () ϕ() a.e. onω. On he oher hand,f s comlee. Hence, we oban haϕ() sf -measurable, whch mles ha D s a closed subsace of he sace C ([a, b]; L (Ω,R n )). References [1] J.A.D. Aleby, Fxed ons, sably and harmless sochasc erurbaons, Prern, 28. [2] T.A. Buron, Sably by fxed on heory for funconal dfferenal euaons. Dover Publcaon, New York, 26. [3] H.B. Chen, Inegral neualy and exonenal sably for neural sochasc aral dfferenal euaons wh delays, Journal of neuales and alcaons 29, Ar. ID , 15 ages. [4] H.B. Chen, Imulsve-negral neualy and exonenal sably for sochasc aral dfferenal euaons wh delays, Sascs and Probably Leers 8 (21), [5] A. Djoud, R. Khems, Fxed ons echnues and sably for neural nonlnear dfferenal euaons wh unbounded delays, Georgan Mahemacal Journal 13(26), No.1, [6] M. For, A. Tes, New condons for global sably of neural neworks wh alcaon o lnear and uadrac rogrammng roblems, I Trans. Crc. Sys. I, 42 (1995), [7] A. Fredman, Sochasc dfferenal euaons and alcaons, Academc ress, New York, [8] S. Haykn Neural neworks, Prence-Hall, nglewood Clffs, [9] J. Hu, S. Zhong, L. Lang, xonenal sably analyss of sochasc delayed cellular neural nework, Chaos Solon Frac. 27 (26), [1] C.X. Huang, Y.G. He, H.N. Wang, Mean suare exonenal sably of sochasc recurren neural neworks wh me-varyng delays, Comuers and Mahemacs wh Alcaons 56 (28), [11] C.X. Huang, Y.G. He, L.H. Huang, W.J. Zhu, h momen sably analyss of sochasc recurren neural neworks wh me-varyng delays, Informaon Scence 178 (28), [12] X.H. La, Y.T. Zhang, Fxed on and asymoc analyss of cellular neural neworks, Journal of Aled Mahemacs 212 (212), Arcle ID , 12 ages do:1.1155/212/ [13] T. L, Q. Luo, C.Y. Sun, B.Y. Zhang, xonenal sably of recurren neural neworks wh me-varyng dscree and dsrbued delays, Nonlnear Analyss: Real World Alcaons 1 (29), [14] X. Lao, X. Mao, xonenal sably and nsably of sochasc neural neworks, Sochasc Analyss and Alcaon 14 (2) (1996), [15] X. Lao, X. Mao, Sably of sochasc neural neworks, Neural, Parallel and Scenfc Comuaons 4 (2) (1996), [16] X.X. Lao, J. Wang, Algebrac crera for global exonenal sably of cellular neural neworks wh mulle me delays, I TARANS. Crc. Sya. I. 5 (23), [17] X. Lao, X. Mao. xonenal sably of sochasc delay nerval sysems, Sys. Conrol. Le., 2, 4: [18] D.Z. Lu, G.Y. Yang and W. Zhang, The sably of neural sochasc delay dfferenal euaons wh osson jums by fxed ons, Journal of Comuaonal and Aled Mahemacs 235(211), [19] J.W. Luo, Fxed ons and sably of neural sochasc delay dfferenal euaons. J. Mah. Anal. Al., 334(27), [2] J.W. Luo and T. Tanguch, Fxed ons and sably of sochasc neural aral dfferenal euaons wh nfne delays, Sochasc Analyss and Alcaons 27 (29), [21] J.W. Luo, Fxed ons and exonenal sably of mld soluons of sochasc aral dfferenal euaons wh delays, J.Mah.Anal.Al. 342 (28), [22] S..A. Mohammed, Sochasc funconal dfferenal euaons, Longman Scenfc and Techncal, [23] X.R. Mao, Razumkhn-ye heorems on exonenal sably of sochasc funconal dfferenal euaons, Sochas. Proc. Al., 1996, 65: [24] P.. Proer, Sochasc Inegraon and Dfferenal uaons, 2nd don, 24. [25] R. Sakhvel, J. Luo, Asymoc sably of nonlnear mulsve sochasc dfferenal euaons, Sas. Probab. Le. 79 (29),

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