STABILITY OF NONLINEAR NEUTRAL DELAY DIFFERENTIAL EQUATIONS WITH VARIABLE DELAYS

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1 Elecronic Journal of Differenial Equaions, Vol , No. 118, pp ISSN: URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu STABILITY OF NONLINEAR NEUTRAL DELAY DIFFERENTIAL EQUATIONS WITH VARIABLE DELAYS GUILING CHEN, DINGSHI LI, ONNO VAN GAANS, SJOERD VERDUYN LUNEL Absrac. We presen new crieria for asympoic sabiliy of wo classes of nonlinear neural delay differenial equaions. By using wo auxiliary funcions on a conracion condiion, we exend he resuls in [12]. Also we give wo examples ha illusrae our resuls. 1. Inroducion In recen years here has been an increasing ineres in sabiliy resuls for neural delay differenial equaions involving erms of he form cx rx r see [3, 15]. In his aricle, we consider he following wo classes of nonlinear neural delay differenial equaions x cx rx r = ax bgx r, 1.1 x cx rx r = ax r K, sgxs ds, 1.2 where a, b : [, R are coninuous funcions, c : [, R is coninuously differeniable funcion and r : [,, is a coninuous funcion, K, s : [, [r, R is a coninuous funcion, r = inf{ r : }, gx = x γ, γ 1 is a consan, hen g saisfies a locally Lipschiz condiion; ha is, here exiss L > and l > such ha g saisfies gx gy L x y for x, y [ l, l]. 1.3 Ahcene and Rabah [12] sudied he special case of 1.1 and 1.2 when gx = x 2. c The resuls in [12] mainly dependen on he consrain 1 r < 1. However, here are ineresing examples where he consrain is no saisfied. I is our aim in his paper o remove his consrain condiion and sudy he sabiliy properies of 1.1 and 1.2. Recen work of Buron and many ohers [1, 2, 3, 5, 6, 7, 8, 9, 11, 12, 14, 15, 16] has shown he power of he fixed poin mehod in sudying sabiliy properies of funcional differenial equaions. The idea of using fixed poin mehod o sudy properies of soluions seems o have emerged independenly several imes by differen schools of auhors. In addiion o Buron s work, we would like o menion 21 Mahemaics Subjec Classificaion. 34K2, 34K25, 34K4. Key words and phrases. Asympoic sabiliy; fixed poin heory; variable delay; nonlinear neural differenial equaions. c 217 Texas Sae Universiy. Submied December 16, 216. Published May 3,

2 2 G. CHEN,D. LI, O. VAN GAANS, S. VERDUYN LUNEL EJDE-217/118 Corduneann [1] and Azbelev and his co-workers [4]. In his paper, we will use he fixed poin mehod o sudy sabiliy properies of 1.1 and 1.2. In paricular, we inroduce wo auxiliary coninuous funcions v and p o define an appropriae mapping, and presen new crieria for asympoic sabiliy of equaions 1.1 and c 1.2 which can be applied in he case 1 r 1 as well. An iniial condiion for he differenial equaion 1.1 is defined by x = φ for [r, ], 1.4 where φ C[r, ], R. Here C[r, ], R denoes he se of all coninuous funcions ϕ : [r, ] R wih he supremum norm. For φ C[r, ], R, we call a coninuous funcion x, φ o be a soluion of 1.1 wih iniial condiion 1.4 if x : [r, a R for some posiive consan a > saisfies d c x d 21 r x r2 = ax bgx r d d c 21 r x 2 r 1.5 on [, a and x = φ on [r, ]. We denoe such a soluion by x := x, φ. Noe ha equaion 1.5 is in he sandard form d d D, x = f, x as sudied in [13]. According o [13, Theorems 8.1 and 8.3], for each φ C[r, ], R, here exiss a unique soluion x = x, φ of 1.1 defined on [,. Definiion 1.1. The zero soluion of 1.1 is said o be sable, if for every ɛ >, here exiss a δ = δɛ > such ha φ : [r, ] δ, δ implies ha x < ɛ for. Definiion 1.2. The zero soluion of 1.1 is said o be asympoically sable, if i is sable and here exiss a δ > such ha for any iniial funcion φ : [r, ] δ, δ, he soluion x wih x = φ on [r, ] ends o zero as. By inroducing wo auxiliary funcions v and p o consruc a conracion mapping on a complee meric space, we obain Theorem 1.3 and Theorem 1.4, which will be proved in Secion 2 and Secion 3, respecively. Theorem 1.3. Consider he neural delay differenial equaion 1.1 and suppose he following condiions are saisfied: i r is wice differeniable wih r 1 and r as ; ii here exiss a bounded funcion p : [r,, wih p = 1 for [r, ] such ha p exiss on [r, and here exiss a consan α, 1 and

3 EJDE-217/118 STABILITY OF NEUTRAL DELAY DIFFERENTIAL EQUATIONS 3 an arbirary coninuous funcions v : [r, R such ha { cp 2 r l p1 r ks 2b 1 s e R } s vu du ds L where r α, vu du bs ps rsγ s vs as p s ds s vu du vs ds p u vu au du ds pu s vu du vs rs as rs p s rs 1 r s ds ps rs 1.6 ks = [csvs c s]1 r s csr s 1 r s 2, cs = csp2 s rs, 1.7 b 1 s = csps rsp s rs, 1.8 and he consans l, L are defined as in 1.3; iii and such ha lim inf vs ds >. Then he zero soluion of 1.1 is asympoically sable if and only if vs ds as. 1.9 Theorem 1.4. Consider he neural Voerra inegro-differenial equaion 1.2 and suppose he following condiions are saisfied: i r is wice differeniable, r 1, r as ; ii There exiss a bounded funcion p : [r,, wih p = 1 such ha p exiss on [r, and here exiss a consan α, 1 and a coninuous funcions v : [r, R such ha { cp 2 r l p1 r L r α, s vu du vs as p s ds s vu du vs ks 2b 1 s e R } s vu du ds Ks, u p γ u du p u vu au du ds pu s vu du p s rs vs rs as rs 1 r s ds ps rs 1.1

4 4 G. CHEN,D. LI, O. VAN GAANS, S. VERDUYN LUNEL EJDE-217/118 where ks and b 1 s are defined as 1.7 and 1.8, respecively, he consans l, L are defined as in 1.3; iii and such ha lim inf vs ds >. Then he zero soluion of 1.2 is asympoically sable if and only if vs ds as The echnique for consrucing a conracion mapping comes from an idea in [16]. Our work exends and improves he resuls in [12, 16]. Remark 1.5. The mehod applied in his paper can be used o rea more general equaions such as d d x = ah r d Q, x r G, x, x r d sudied by Mesmouli, Ardjouni and Djoudi in [14] o give more general resuls. 2. Proof of Theorem 1.3 We sar wih some preparaions. Define { Sφ l = ϕ C[r,, R : ϕ = sup ϕ l, ϕ = φ r } for [r, ], ϕ as. Then Sφ l is a complee meric space wih meric ρx, y = sup r { x y }. Le z = φ on [r, ], and le x = pz for. If z saisfies z = a p z cp rp r z 2 r p p cp2 r z rz r p bp rγ gz r, p 2.1 hen i can be verified ha x saisfies 1.5. Since p is a posiive bounded funcion, we only have o prove ha he zero soluion of 2.1 is asympoically sable. If we muliply boh sides of 2.1 by e R vs ds and hen inegrae from o, we obain z = φe R vs ds vs as p s s vu du zs ds s s vu du csps rsp s rs z 2 s rs ds s vu du csp2 s rs zs rsz s rs ds vu du bsps rsγ gzs rs ds. 2.2

5 EJDE-217/118 STABILITY OF NEUTRAL DELAY DIFFERENTIAL EQUATIONS 5 Taking s vu du csp2 s rs zs rsz s rs ds = s vu du csp2 s rs zs rsz s rs 1 r 1 s 1 r s ds and inegraing by pars he righ-hand side of 2.3, we obain s vu du csp2 s rs zs rsz s rs ds = p2 r 2p vs ds 1 2 c 1 r z2 r p2 r c 2p 1 r φ2 r where ks is given by 1.7. Inegraing by pars, we obain vs as p s = e R s vu du d s vu du ksz 2 s rs ds s vu du zs ds vu au p u pu zu du s vu du vs rs as rs p s rs ps rs 1 r szs rs ds = vs as p s r e R s vu du vs zs ds vu au p u pu zu du ds s vu du vs rs as rs p s rs ps rs 1 r szs rs ds Combining 2.2, 2.4 and 2.5, we obain ha a soluion of 2.1 has he form z = [ φ r p2 r 2p r vs as p s c 1 r φ2 r φs ds ] vs ds p2 r c 2p 1 r z2 r vs as p s zs ds

6 6 G. CHEN,D. LI, O. VAN GAANS, S. VERDUYN LUNEL EJDE-217/118 e R s vu du vs vu au p u pu zu du ds s vu du vs rs as rs p s rs ps rs 1 r szs rs ds 1 2 s vu du ks 2b 1 s z 2 s rs ds s vu du bsps rsγ gzs rs ds := where ks and b 1 s are defined in 1.7 and 1.8, respecively. 7 I i, Lemma 2.1. Le z Sφ l and define an operaor by P z = φ for [r, ] and for, P z = 7 i=1 I i. If condiions i ii and 1.9 in Theorem 1.3 are saisfied, hen here exiss δ > such ha for any iniial funcion φ : [r, ] δ, δ, we have ha P : Sφ l Sl φ and P is a conracion wih respec o he meric ρ defined on Sφ l. Proof. Se J = sup { vs ds }, by iii, J is well defined. Suppose ha 1.9 holds. We choose δ > such ha [ δ δ r p s vs as p 2 r ds 2p i=1 c 1 r δ2] J 1 αl. Le φ be a given small bounded iniial funcion wih φ < δ, and le ϕ Sφ l, hen ϕ l. Since g saisfies a locally Lipschiz condiion, from 1.6 in Theorem 1.3, we have P ϕ [ δ δ r vs as p s ds p2 r 2p c 1 r δ2] J αl l. Thus, P ϕ l. Nex, we show ha P ϕ as. I is clear ha I i for i = 1, 2, 3, 4, 5, 7, since e R vs ds, r and ϕ as. Now, we prove ha I 6 as. For r and ϕ, we obain ha for any ɛ >, here is a posiive number T 1 > such ha for T 1, ϕ r < ɛ, so we have I 6 = e R T 1 vu du s vu du ks 2b 1 s ϕ 2 s rs ds T 1 e sup ϕ r 1 2 ɛ2 T 1 e T1 e R T 1 vu s du ks 2b 1 s ϕ 2 s rs ds R s vu du ks 2b1 s ϕ 2 s rs ds 2e R T1 vu du T 1 R s vu du ks 2b 1 s ds e R T 1 vu s du ks 2b 1 s ds

7 EJDE-217/118 STABILITY OF NEUTRAL DELAY DIFFERENTIAL EQUATIONS 7 α 2 l2 T 1 vu du αɛ. By 1.9, here exiss T 2 > T 1 such ha > T 2 implies α 2 l2 vu T du 1 < ɛ, which implies I 6 as. Hence, we have P ϕ as. Finally, we show ha P is a conracion mapping. In fac, for ϕ, η Sφ l, using condiion 1.6 in Theorem 1.3, we obain P ϕ P η 2l p 2 r c 2p 1 r ϕ η r p s vs as ϕ η ds s vu du vs vu au p u ϕ η du ds pu s vu du vs rs as rs p s rs ps rs 1 r s ϕ η ds l L s vu du ks 2b 1 s ϕ η ds s α ϕ η. vu du bs ps rsγ ϕ η ds Therefore, P : Sφ l Sl φ is a conracion mapping. Proof of Theorem 1.3. Le P be defined as in Lemma 2.1. By he conracion mapping principle, P has a unique fixed poin z in Sφ l which is a soluion of 2.1 wih z = φ on [r, ] and z as. To prove sabiliy, le ɛ > be given, hen we choose m > so ha m < min{l, ɛ}. Replacing l wih m in Sφ l, we obain ha here is a δ > such ha φ < δ implies ha he unique soluion of 2.1 wih z = φ on [r, ] saisfies z m < ɛ for all r. This shows ha he zero soluion of 2.1 is asympoically sable if 1.9 holds. Conversely, we suppose ha 1.9 fails. Then by iii, here exiss a sequence { n }, n n as n such ha lim n vs ds = v for some v R. We may choose a posiive consan M such ha M n for all n 1. To simplify our expressions, we define ws = l ks 2b 1 s vs vs ds M 2.6 p u vu au du pu L bs ps rsγ vs rs as rs p s rs 1 r s ps rs

8 8 G. CHEN,D. LI, O. VAN GAANS, S. VERDUYN LUNEL EJDE-217/118 for all s. By ii we have n n s vu du ws ds α. 2.7 Combining 2.6 and 2.7, we have n e R s vu du ws ds αe R n vu du αe M, which yields ha n e R s vu du ws ds is bounded. Hence, here exiss a convergen nk subsequence, we assume ha lim k e R s vu du ws ds = γ for some γ R. nk We choose a posiive ineger k 1 so large ha lim k nk1 e R s vu du ws ds δ 4J for all n k > n k1, where δ > saisfies 2δ Je M α < 1. Now, we consider he soluion z = z, nk1, φ of 2.1 wih φ nk1 = δ and φs δ for s nk1, and we may choose φ such ha z 1 for nk1 and φ nk1 nk1 nk1 r nk1 p2 nk1 r nk1 2p nk1 vs as p s φs ds c nk1 1 r nk1 φ2 nk1 r nk1 1 2 δ. So, i follows from 2.8 wih z = P z ha for k k 1, z nk p2 nk r nk c nk 2p nk 1 r nk z2 r nk nk [ vs as p s ] zs ds nk r nk 1 2 δ n k n k1 vu du nk nk1 e R n k s vu du ws ds = n k vu du [ n k1 1 2 δ e R n nk k1 vu du nk1 n k vu du [ n k1 1 nk 2 δ J e R ] s vu du ws ds nk1 e R ] s vu du ws ds δ n k vu du n k1 1 4 δ e 2M >. On he oher hand, suppose ha he soluion of 2.1 z = z, nk1, φ as. Since nk r nk as k, and ii holds, we have z nk p2 nk r nk c nk 2p nk 1 r nk z2 r nk nk [ vs as p s ] zs ds as k, nk r nk which conradics 2.9. Hence condiion 1.9 is necessary for he asympoic sabiliy of he zero soluion of 2.1. Since p is a posiive bounded funcion, from he above argumens we obain ha 1.9 is a necessary and sufficien condiion for he asympoic sabiliy of he zero soluion of 1.1. When gx = x 2 in 1.1, we have he following resul.

9 EJDE-217/118 STABILITY OF NEUTRAL DELAY DIFFERENTIAL EQUATIONS 9 Corollary 2.2. Suppose he following condiions are saisfied: i he delay r is wice differeniable wih r 1, and r as ; ii here exiss a bounded funcion p : [r,, wih p = 1 such ha p exiss on [r,, and here exiss a consan α, 1 and an arbirary coninuous funcions v : [r, R such ha { cp 2 r l p1 r ks 2bs e R } s vu du ds p s vs as ds r s vu du vs α, vu au p u du ds pu s vu du p s rs vs rs as rs 1 r s ds ps rs where ks is defined as in 1.7, 2.1 bs = bsp2 s rs csps rsp s rs, 2.11 and l > is defined as in 1.3; iii and such ha lim inf vs ds >. Then he zero soluion x, φ of 1.1 is asympoically sable if and only if vs ds as Proof of Theorem 1.4 We sar wih some preparaions. Define { Sφ l = ϕ C[r,, R : ϕ = sup ϕ l, ϕ = φ r } for [r, ], ϕ as. Then S l φ is complee meric space wih meric ρx, y = sup r { x y }. Le z = φ on [r, ], and le x = pz, for, from 1.2, we obain z = a p p cp rp r z z 2 r p cp2 r z rz r p p γ sk, s gzs ds. r p 3.1 Since p is bounded, we only need o prove ha he zero soluion of 3.1 is asympoically sable.

10 1 G. CHEN,D. LI, O. VAN GAANS, S. VERDUYN LUNEL EJDE-217/118 If we muliply boh sides of 3.1 by e R vs ds, inegrae from o, and perform an inegraion by pars, we obain z = { φ p2 r 2p r r vs as p s c 1 r φ2 r p2 r c 2p 1 r z2 r [ vs as p s s vu du vs φs ds } vs ds ] zs ds vu au p u pu zu du ds s vu du vs rs as rs p s rs ps rs 1 r szs rs ds 1 2 s vu du ks 2b 1 s z 2 s rs ds s vu du Ks, up γ u gzu du ds := where ks and b 1 s are defined as in 1.7 and 1.8 respecively. 7 I i Lemma 3.1. Le z Sφ l and define an operaor by P z = φ for [r, ] and for, P z = 7 i=1 I i. If condiions i-iii in Theorem 1.4 are saisfied, hen here exiss δ > such ha for any φ : [r, ] δ, δ, we have ha P : Sφ l Sl φ and P is a conracion mapping wih respec o he meric defined on Sφ l. Proof. Se J = sup { vs ds }, by iii, J is well defined. Suppose ha iii holds. Using he similar argumens as as he proof of Theorem 1.3, we obain ha P ϕ Sφ l for ϕ Sl φ. Now, we show ha P is a conracion mapping. In fac, for ϕ, η Sφ l, by using condiion 1.1 in Theorem 1.4, we obain ha P ϕ P η p2 r c 2l ϕ 2p 1 r η p s vs as ϕ η ds r s vu du vs i=1 vu au p u ϕ η du ds pu s vu du p s rs vs rs as rs ps rs 1 r s ϕ η ds

11 EJDE-217/118 STABILITY OF NEUTRAL DELAY DIFFERENTIAL EQUATIONS L α ϕ η. ks 2b 1 s 2l ϕ η s vu du Ks, u p γ u du ϕ η ds Hence, we obain ha P : Sφ l Sl φ is a conracion mapping. Proof of Theorem 1.4. Le P be defined as in Lemma 3.1. By he conracion mapping principle, P has a unique fixed poin z in Sφ l which is a soluion of 1.2 wih z = φ on [r, ] and z as. Le ɛ > be given, hen we choose m > so ha m < min{l, ɛ}. Replacing l wih m in Sφ l, we obain here is a δ > such ha φ < δ implies ha he unique soluion of 1.2 wih z = φ on [r, ] saisfies z m < ɛ for all r. This shows ha he zero soluion of 1.2 is asympoically sable if 1.11 holds. Following he similar argumens as he proof of Theorem 1.3, we obain ha 1.11 is necessary for he asympoic sabiliy of he zero soluion of 1.2. The proof is complee. When gx = x 2, we have he following corollary. Corollary 3.2. Suppose he following condiions are saisfied: i he delay r is wice differeniable, r 1, r as ; ii here exiss a bounded funcion p : [r,, wih p = 1 such ha p exiss on [r,, and here exiss a consan α, 1, a consan l > and a coninuous funcions v : [r, R such ha { cp 2 r l p1 r 2 Ks, up2 u r p s vs as [ ks 2b 1 s du ] ds s vu du ds s vu du vs rs as rs p s rs 1 r s ds ps rs e R s vu du vs p u vu au du ds α, pu where ks and b 1 s are defined as in 1.7 and 1.8; iii and such ha lim inf } vs ds >. 3.2 Then he zero soluion of 1.2 wih a small iniial funcion φ is asympoically sable if and only if vs ds as. 3.3

12 12 G. CHEN,D. LI, O. VAN GAANS, S. VERDUYN LUNEL EJDE-217/ Examples Example 4.1. Consider he nonlinear neural differenial equaion x cx rx r = ax bx 2 r 4.1 for, where a = 2 1, c =.95, r =.5, l = 1, b saisfies ks 2bs.3 s1, hen he zero soluion of 4.1 is asympoically sable. Proof. We check condiion 2.1 in Corollary 2.2, choosing v = and p = 1 1, we obain ha { cp 2 r l p1 r e R } s vu du ks 2bs ds vs as p s r ds e R s vu du vs vu au p u pu du ds e R s vu du vs p s rs rs as rs 1 r s ds ps rs < =.941 < 1, and since vs ds = 1.5 s1 ds = 1.5 ln 1 as, p 1, so he condiions of Corollary 2.2 are saisfied. Therefore, he zero soluion of 4.1 is asympoically sable. Noe ha c 1 r = 1; herefore he resul in [12] is no applicable. Example 4.2. Consider he nonlinear neural Volerra inegral equaion x cx rx r = ax r K, sx 2 s ds 4.2 for, where a = , c =.1, r =.5, l = 1, K, s = 1.1, hen he zero soluion of 4.2 is asympoically sable. Proof. We check he condiion 3.2 in Corollary 3.2, choosing v = 2.1 p =.1.1, we obain ha l cp 2 r l p1 r r [ ks 2b 1 s 2 vs as p s ds Ks, up2 u ] du s vu du ds s vu du vs p u vu au du ds pu s vu du vs rs as rs p s rs ps rs 1 r s ds < =.824 < 1, and

13 EJDE-217/118 STABILITY OF NEUTRAL DELAY DIFFERENTIAL EQUATIONS 13 and since vs ds = 2 s.1 ds = 2 ln.1 as, p 1, so he condiions of Corollary 3.2 are saisfied. Therefore, he zero soluion of 4.2 is asympoically sable. c Noe ha 1 r = no applicable. as ; herefore he resul in [12] is References [1] Abdelouaheb Ardjouni, Ahcene Djoudi; Fixed poins and sabiliy in linear neural differenial equaions wih variable delays, Nonlinear Analysis , [2] A. Ardjouni, A. Djoudi; Fixed poins and sabiliy in neural nonlinear differenial equaions wih variable delays, Opuscula Mahemaica , No. 1. [3] A. Ardjouni, A. Djoudi; Fixed poins and sabiliy in nonlinear neural Volerra inegraldifferenial equaions wih variable delays, Elecronic Journal of Qualiaive Theory of Differenial Equaions 213, No. 28, [4] N. V. Azbelev, V. P. Maksimov, L. F. Rakhmaullina; Inroducion o he heory of funcional differenial equaions: Mehods and applicaions. English Conemporary Mahemaics and Is Applicaions 3. New York, NY: Hindawi Publishing Corporaion, 27. [5] L. C. Becker, T. A. Buron; Sabiliy, fixed poins and inverse of delays, Proceedings of he Royal Sociey of Edinburgh, 136A 26, [6] T. A. Buron; Sabiliy by fixed poin mehods for highly nonlinear delay equaions, Fixed Poin Theory 5 24, No.1, 3-2. [7] T. A. Buron; Sabiliy by fixed poin heory for funcional differenial equaions, Dover Publicaion, New York, 26. [8] G. Chen, O. van Gaans, S. M. Verduyn Lunel; Asympoic behavior and sabiliy of second order neural delay differenial equaions, Indagaiones Mahemaicae-new series, , [9] G. Chen, O. van Gaans, S. M. Verduyn Lunel; Fixed poins and ph momen exponenial sabiliy of sochasic delayed recurren neural neworks wih impulses, Applied Mahemaics Leers, 27214, [1] C. Corduneanu; Inegral equaions and sabiliy of feedback sysems, Mahemaics in Science and Engineering, Vol. 14, Academic Press, Orlando, FL, [11] L. M. Ding, X. Li, Z. X. Li; Fixed poins and sabiliy in nonlinear equaions wih variable delays. Fixed Poin Theory Appl. 21, Ar. ID , 14 pages. [12] A. Djoudi, R. Khemis; Fixed poins echniques and sabiliy for neural nonlinear differenial equaions wih unbounded delays, Georgian Mahemaical Journal 13 26, No. 1, [13] J. Hale, S. M. Verduyn Lunel; Inroducion o funcional differenial equaions, Springer Verlag, New York, [14] M. B. Mesmouli, A. Ardjouni, A. Djoudi; Sudy of he sabiliy in nonlinear neural differenial equaions wih funcional delay using Krasnoselskii-Buron s fixed-poin, Appl. Mah. Compu., , [15] Y. N. Raffoul; Sabiliy in neural nonlinear differenial equaions wih funcional delays using fixed-poin heory, Mahemaical and compuer Modelling 4 24, [16] D. Zhao; New crieria for sabiliy of neural differenial equaions wih variable delays by fixed poins mehod, Advances in Difference Equaions, 211, 211: 48. Guiling Chen Deparmen of Mahemaics, Souhwes Jiaoong Universiy, Chengdu 6131, China address: guiling@home.swju.edu.cn Dingshi Li corresponding auhor Deparmen of Mahemaics, Souhwes Jiaoong Universiy, Chengdu 6131, China address: lidingshi26@163.com

14 14 G. CHEN,D. LI, O. VAN GAANS, S. VERDUYN LUNEL EJDE-217/118 Onno van Gaans Mahemaical Insiue, Leiden Universiy, P.O. Box 9512, 23 RA, Leiden, The Neherlands address: Sjoerd Verduyn Lunel Mahemaical Insiue, Urech Universiy, P.O. Box 81, 358 TA, Urech, The Neherlands address: