FIXED POINTS AND STABILITY IN NEUTRAL DIFFERENTIAL EQUATIONS WITH VARIABLE DELAYS

Transcription

1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 136, Number 3, March 28, Page S (7)989-2 Aricle elecronically publihed on November 3, 27 FIXED POINTS AND STABILITY IN NEUTRAL DIFFERENTIAL EQUATIONS WITH VARIABLE DELAYS CHUHUA JIN AND JIAOWAN LUO (Communicaed by Carmen C. Chicone) Abrac. Inhipaperweconideralinearcalar neural delay differenial equaion wih variable delay and give ome new condiion o enure ha he zero oluion i aympoically able by mean of fixed poin heory. Thee condiion do no require he boundedne of delay, nor do hey ak for a fixed ign on he coefficien funcion. An aympoic abiliy heorem wih a neceary and ufficien condiion i proved. The reul of Buron, Raffoul, and Zhang are improved and generalized. 1. Inroducion Lyapunov direc mehod ha been very effecive in eablihing abiliy reul for a wide variey of differenial equaion. Ye, here i a large e of problem for which i ha been ineffecive. Recenly, Buron and oher applied fixed poin heory o udy abiliy [2 9]. I ha been hown ha many of hoe problem encounered in he udy of abiliy by mean of Lyapunov direc mehod can be olved by uing fixed poin heory. While Lyapunov direc mehod uually require poinwie condiion, he abiliy reul by fixed poin heory ak condiion of an averaging naure. In he preen paper we alo adop fixed poin heory o udy he aympoic abiliy of neural delay differenial equaion. A new echnique i ued, which make abiliy condiion more feaible and he reul in [3, 8, 9] are improved and generalized. The re of hi paper i organized a follow. In Secion 2, we ae ome known reul and our main heorem; he proof of our reul i alo given in hi ecion. In Secion 3, wo example how ha our abiliy reul, no only for delay differenial equaion bu alo for neural delay differenial equaion, i indeed beer han hoe in [3, 8, 9]. Received by he edior Ocober 1, Mahemaic Subjec Claificaion. Primary 34K2, 34K4. Key word and phrae. Fixed poin, abiliy, neural delay differenial equaion, variable delay. The econd auhor wa uppored in par by NNSF of China Gran # c 27 American Mahemaical Sociey Rever o public domain 28 year from publicaion Licene or copyrigh rericion may apply o rediribuion; ee hp://

2 91 CHUHUA JIN AND JIAOWAN LUO 2. Main reul Conider he following neural delay differenial equaion wih variable delay of he form (2.1) x () = a()x() b()x( τ()) c()x ( τ()), where a, b, c C(R,R)andτ C(R,R )wih τ() a. Equaion (2.1) and i pecial cae have been inveigaed by many auhor. For example, Buron in [3] and Zhang in [9] have udied he equaion (2.2) x () = b()x( τ()) and obained he following. Theorem A (Buron [3]). Suppoe ha τ() =r and here exi a conan α<1 uch ha (2.3) b( r) d b( r) b(ur)du b(u r) dud α r for all and b()d =. Then for every coninuou iniial funcion ψ :[ r, ] R, he oluion x() =x(,,ψ) of (2.2) i bounded and end o zero a. Theorem B (Zhang [9]). Suppoe ha τ i differeniable, he invere funcion g() of τ() exi, and here exi a conan α (, 1) uch ha for (2.4) (2.5) τ() b(g()) d lim inf b(g())d >, b(g(u))du b(g()) r b(g(u))du b() τ () d α. Then he zero oluion of (2.2) i aympoically able if and only if (2.6) b(g())d a. b(g(v)) dvd Obviouly, Theorem B improve Theorem A. On he oher hand, Raffoul in [8] ha inveigaed equaion (2.1) and obained Theorem C (Raffoul [8]). Le τ() be wice differeniable and τ () 1for all R. Suppoe ha here exi a conan α (, 1) uch ha for (2.7) and (2.8) c() 1 τ () a()d a, a(u)du b() [c()a()c ()](1 τ ()) c()τ () (1 τ ()) 2 d α. Then every oluion x() =x(,,ψ) of (2.1) wih a mall coninuou iniial funcion ψ() iboundedandendozeroa. Licene or copyrigh rericion may apply o rediribuion; ee hp://

3 FIXED POINTS AND STABILITY IN NEUTRAL DIFFERENTIAL EQUATIONS 911 Le R =(, ), R =[, ), and R =(, ], repecively. C(S 1,S 2 ) denoe he e of all coninuou funcion φ : S 1 S 2.Foreach, define m( )= inf{ τ() : },andc( )=C([m( ), ],R) wih he upremum norm. For each (,φ) R C( ), a oluion of (2.1) hrough (,φ) i a coninuou funcion x :[m( ), α) R n for ome poiive conan α > uch ha x() aifie (2.1) on [, α) andx() =φ() for [m( ), ]. We denoe uch a oluion by x() =x(,,φ). For each (,φ) R C( ), here exi a unique oluion x() =x(,,φ) of (2.1) defined on [, ). For fixed,we define φ =max{ φ() : m( ) }. Sabiliy definiion may be found in [1], for example. Theorem 2.1. Le τ() be wice differeniable and τ () 1for all R. Suppoe ha here exi a conan α (, 1) andafuncionh C(R,R) uch ha for (i) (ii) (iii) c() 1 τ () τ() lim inf h() a() d h()d >, h(u)du b()[h( τ()) a( τ())](1 τ ()) [c()h()c ()](1 τ ()) c()τ () d (1 τ ()) 2 h(u)du h() h(v) a(v) dvd α. Then he zero oluion of (2.1) i aympoically able if and only if h()d a. Proof. Fir, uppoe ha (iii) hold. For each, we e (2.9) K =up{ h()d }. Le φ C( ) be fixed and define S = {x C([m( ), ),R):x() a,x() =φ() for [m( ), ]}. Then S i a complee meric pace wih meric ρ(x, y) =up { x() y() }. Muliply boh ide of (2.1) by e h()d andheninegraefrom o o obain x() =φ( ) h()d h(u)du [h() a()]x()d h(u)du b()x( τ())d h(u)du c()x ( τ())d. Licene or copyrigh rericion may apply o rediribuion; ee hp://

4 912 CHUHUA JIN AND JIAOWAN LUO Performing an inegraion by par, we have (2.1) x() =φ( ) h()d h(u)du d( ) [h(v) a(v)]x(v)dv h(u)du{ b()[h( τ()) a( τ())](1 τ ())} x( τ())d { = φ( ) c( ) 1 τ ( ) φ( τ( )) c() 1 τ () e h(u)du dx( τ()) τ( ) [h() a()]φ()d } h(u)du c() 1 τ x( τ()) [h() a()]x()d () τ() h(u)du{ b()[h() a( τ())](1 τ ()) [c()h()c ()](1 τ ()) c()τ () } (1 τ ()) 2 x( τ())d h(u)du h() [h(v) a(v)]x(v)dvd. Ue (2.1) o define he operaor P : S S by (Px)() =φ() for [m( ), ] and (2.11) { (Px)() = φ( ) c( ) 1 τ ( ) φ( τ( )) h(u)du τ( ) c() 1 τ x( τ()) () } [h() a()]φ()d τ() [h() a()]x()d h(u)du{ b()[h() a( τ())](1 τ ()) [c()h()c ()](1 τ ()) c()τ () } (1 τ ()) 2 x( τ())d h(u)du h() [h(v) a(v)]x(v)dvd for. I i clear ha (Px) C([m( ), ),R). We now how ha (Px)() a.sincex() and τ() a,foreachε>, here exi a T 1 > uch ha T 1 implie ha x( τ()) <ε.thu,for T 1,hela Licene or copyrigh rericion may apply o rediribuion; ee hp://

5 FIXED POINTS AND STABILITY IN NEUTRAL DIFFERENTIAL EQUATIONS 913 erm I 5 in (2.11) aifie I 5 = h(u)du h() T1 T 1 up σ m( ) T 1 ε h(u)du h() h(u)du h() [h(v) a(v)]x(v)dvd h(v) a(v) x(v) dvd h(v) a(v) x(v) dvd T1 x(σ) h(u)du h() h(v) a(v) dvd h(u)du h() h(v) a(v) dvd. By (iii), here exi T 2 >T 1 uch ha T 2 implie T1 up x(σ) h(u)du h() h(v) a(v) dvd σ m( ) = up x(σ) h(u)du T 1 σ m( ) T1 T1 h(u)du h() h(v) a(v) dvd < ε. Apply (ii) o obain I 5 ε αε < 2ε. Thu, I 5 a. Similarly, we can how ha he re of he erm in (2.11) approach zero a. Thi yield (Px)() a, and hence Px S. Alo, by (ii), P i a conracion mapping wih conracion conan α. By he Conracion Mapping Principle, P ha a unique fixed poin x in S which i a oluion of (2.1) wih x() =φ() on [m( ), ]andx() =x(,,φ) a. To obain aympoic abiliy, we need o how ha he zero oluion of (2.1) i able. Le ε> be given and chooe δ>(δ <ε) aifying 2δKe h(u)du αε < ε. If x() =x(,,φ) i a oluion of (2.1) wih φ <δ,henx() =(Px)() a defined in (2.11). We claim ha x() <εfor all. Noice ha x() <ε on [m( ), ]. If here exi > uch ha x( ) = ε and x() <εfor m( ) <, hen i follow from (2.11) ha ( x( c( ) ) φ 1 1 τ ( ) h() a() d ) h(u)du (2.12) c( ) ε 1 τ ( ) ε ε h(u)du τ( ) τ( ) h() a() d b()[h() a( τ())](1 τ ()) [c()h()c ()](1 τ ()) c()τ () d (1 τ ()) 2 ε h(u)du h() h(v) a(v) dvd 2δKe h(u)du αε < ε Licene or copyrigh rericion may apply o rediribuion; ee hp://

6 914 CHUHUA JIN AND JIAOWAN LUO which conradic he definiion of. Thu x() <εfor all, and he zero oluion of (2.1) i able. Thi how ha he zero oluion of (2.1) i aympoically able if (iii) hold. Converely, uppoe (iii) fail. Then by (i) here exi a equence { n }, n n a n uch ha lim n h(u)du = l for ome l R. We may alo chooe a poiive conan J aifying J n h()d J for all n 1. To implify our expreion, we define ω() = b()[h() a( τ())](1 τ ()) [c()h()c ()](1 τ ()) c()τ () h() (1 τ ()) 2 for all. By (ii), we have h(v) a(v) dv n n h(u)du ω()d α. Thi yield n e h(u)du ω()d αe n h(u)du e J. The equence { n h(u)du ω()d} i bounded, o here exi a convergen ubequence. For breviy in noaion, we may aume ha e n lim e h(u)du ω()d = γ n for ome γ R and chooe a poiive ineger k o large ha n k e h(u)du ω()d < δ /4K for all n k, whereδ > aifie2δ Ke J α<1. By (i), K in (2.9) i well defined. We now conider he oluion x() =x(, k,φ) of (2.1) wih φ( k) =δ and φ() δ for k. An argumen imilar o ha in (2.12) how x() 1for k. Wemaychooeφ o ha φ( k) c( k) k 1 τ ( k) φ( k τ( k)) [h() a()]φ()d 1 2 δ. k τ( k) Licene or copyrigh rericion may apply o rediribuion; ee hp://

7 FIXED POINTS AND STABILITY IN NEUTRAL DIFFERENTIAL EQUATIONS 915 I follow from (2.11) wih x() =(Px)() haforn k, x( n ) c( n n) 1 τ ( n ) x( n τ( n )) [h() a()]x()d n τ( n ) 1 2 δ n n h(u)du k k n h(u)du ω()d (2.13) = 1 2 δ n h(u)du k e n h(u)du = ( n h(u)du 1 k 2 δ k h(u)du ( n h(u)du 1 k 2 δ K n 1 4 δ n h(u)du 1 k 4 δ 2J >. k n k n k e h(u)du ω()d e ) h(u)du ω()d e ) h(u)du ω()d On he oher hand, if he zero oluion of (2.1) i aympoically able, hen x() = x(, k,φ) a.since n τ( n ) a n and (ii) hold, we have x( n ) c( n n) 1 τ ( n ) x( n τ( n )) [h() a()]x()d a n, n τ( n ) which conradic (2.13). Hence condiion (iii) i neceary for he aympoic abiliy of he zero oluion of (2.1). The proof i complee. Remark 2.2. I follow from he fir par of he proof of Theorem 2.1 ha he zero oluion of (2.1) i able under (i) and (ii). Moreover, Theorem 2.1 ill hold if (ii) i aified for σ for ome σ R. Remark 2.3. When a() c(), Theorem 2.1 wih h() b(g()) reduce o Theorem B. On he oher hand, we chooe h() a(), hen Theorem 2.1 reduce o Theorem C. Remark 2.4. The mehod in hi paper can be exended o he following general neural differenial equaion wih everal variable delay: N M x () = a()x() b i ()x( τ i ()) c j ()x ( δ j ()). i=1 j=1 3. Example Example 3.1. Conider he delay differenial equaion (3.1) x () = b()x( τ()), Licene or copyrigh rericion may apply o rediribuion; ee hp://

8 916 CHUHUA JIN AND JIAOWAN LUO 1 where τ() =.281, b() = Following he noaion in Theorem B, we have b(g()) = 1.Thu,a, 1 τ() b(g()) d =.719 b(g(u))du b(g()) 1 1 d =ln b(g(v)) dvd ln(.719), = 1 [ln( 1) ln(.719 1)]d 1 =ln( 1) 1/.719 ln(.719 1) ln(.719), 1 Thu, we have { lim up b(g(u))du b() τ () d = = (.281 ln(.719 1) ) τ() b(g()) d b(g(u))du b() τ () d d b(g(u))du b(g()) b(g(v)) dvd } = 2[ln(.719)] =1.58. In addiion, he lef-hand ide of he following inequaliy i increaing in >, hen here exi ome > uch ha for, τ() b(g()) d b(g(u))du b(g()) b(g(u))du b() τ () d > 1.5. b(g(v)) dvd Thi implie ha condiion (2.5) doe no hold. Thu, Theorem B canno be applied o equaion (3.1). However, chooing h() = 1.2 1,wehave τ() = < h() d = d =1.2ln <.396, h(u)du b()h( τ())(1 τ ()) d u1 du d u1 du 1.2 d <.1592, 1 and h(u)du h() h(v) dvd <.396. Le α := =.9512 < 1, hen he zero oluion of (3.1) i aympoically able by Theorem 2.1. Licene or copyrigh rericion may apply o rediribuion; ee hp://

9 FIXED POINTS AND STABILITY IN NEUTRAL DIFFERENTIAL EQUATIONS 917 Example 3.2. Conider he neural differenial equaion (3.2) x () = a()x()c()x ( τ()), where a() = 1 1, τ() =.5, c() =.48. Obviouly, c() 1 τ () a(u)du [c()a()c ()](1 τ ()) c()τ () d (1 τ ()) (3.3) 2.48(2 1) =.95( 1). Since he righ-hand ide of (3.3) i increaing in >and {.48(2 1) } lim up =1.15,.95( 1) hen here exi ome > uch ha, c() 1 τ () a(u)du [c()a()c ()](1 τ ()) c()τ () d > 1.1. (1 τ ()) 2 Thi implie ha condiion (2.8) doe no hold. Thu, Theorem C canno be applied o equaion (3.2). However, chooing h() = 2.2 1,wehave c() 1 τ () <.56, and τ() h() a() d = h(u)du h() d =1.2[ln( 1 )] <.62,.95 1 h(v) a(v) dvd <.62, h(u)du [h( τ()) a( τ())](1 τ ()) [c()h()c ()](1 τ ()) c()τ () d (1 τ ()) 2 = 2.2 du( ) u1 d < ( 1) =.41. Le α := =.671 < 1, hen he zero oluion of (3.2) i aympoically able by Theorem 2.1. Acknowledgmen The auhor hank very incerely he anonymou referee for heir valuable commen and helpful uggeion. Reference 1. T. A. Buron, Sabiliy and Periodic Soluion of Ordinary and Funcional Differenial Equaion, Academic Pre, New York, MR (87f:341) 2. T. A. Buron, Liapunov funcional, fixed poin, and abiliy by Kranoelkii heorem, Nonlinear Sudie, 9 (21), MR (23e:34133) Licene or copyrigh rericion may apply o rediribuion; ee hp://

10 918 CHUHUA JIN AND JIAOWAN LUO 3. T. A. Buron, Sabiliy by fixed poin heory or Liapunov heory: a comparion, Fixed Poin Theory, 4 (23), MR (24j:3411) 4. T. A. Buron, Fixed poin and abiliy of a nonconvoluion equaion, Proc. Amer. Mah. Soc., 132 (24), MR28491 (25g:34186) 5. T. A. Buron and T. Furumochi, A noe on abiliy by Schauder heorem, Funkcialaj Ekvacioj, 44 (21), MR (22d:3475) 6. T. A. Buron and T. Furumochi, Fixed poin and problem in abiliy heory, Dynamical Syem and Appl., 1 (21), MR (22c:3476) 7. T. A. Buron and T. Furumochi, Kranoelkii fixed poin heorem and abiliy, Nonlinear Analyi, 49 (22), MR (23e:3487) 8. Y. N. Raffoul, Sabiliy in neural nonlinear differenial equaion wih funcional delay uing fixed-poin heory, Mahemaical and Compuer Modelling, 4 (24), MR B. Zhang, Fixed poin and abiliy in differenial equaion wih variable delay, Nonlinear Analyi, 63 (25), e233-e242. Faculy of Applied Mahemaic, Guangdong Univeriy of Technology, Guangzhou, Guangdong 519, People Republic of China addre: jinchuhua@om.com Correponding auhor. School of Mahemaic and Informaion Science, Guangzhou Univeriy, Guangzhou, Guangdong 516, People Republic of China addre: mahluo@yahoo.com Licene or copyrigh rericion may apply o rediribuion; ee hp://