Omega-limit sets and bounded solutions

Transcription

1 arxiv:3.369v [mah.gm] 3 May 6 Omega-limi ses and bounded soluions Dang Vu Giang Hanoi Insiue of Mahemaics Vienam Academy of Science and Technology 8 Hoang Quoc Vie, 37 Hanoi, Vienam dangvugiang@yahoo.com May 3, 6 Absrac. We prove among oher hings ha he omega-limi se of a bounded soluion of a Hamilon sysem ṗ = H q = H p is conaining a full-ime soluion so here are he limis of p(s)ds and q(s)dsas foranyboundedsoluion(p,q)ofhehamilonsysem. These limis are saionary poins of he Hamilon sysem so if a Hamilon sysem has no saionary poin hen every soluion of his sysem is unbounded. AMS Subjec Classificaion: 34K6 (47D6) Key Words: Beurling specrum of a bounded full-ime soluion, precompac orbi, unique ergodic

2 . INTRODUCTION k= In his paper (X, X ) denoes a complex Banach space. Le A : X X be a bounded linear operaor wih compac specrum σ(a) and posiive specral radiusr(a). In[]weprovedhaifσ(A) ir = {iξ,iξ,,iξ n }henevery bounded full-ime soluion of differenial equaion ẋ() = Ax() has he form u() = n e iξk v k,wherev,v,,v n arefixedvecorsofx.recallhafullime soluion is he soluion saisfying he differenial equaion for all R. For example, periodic soluions (if exis) are full-ime and bounded soluions. We used Beurling specrum [] and Fourier coefficiens of a bounded funcion (on he real line) in he proof. More exacly, we proved ha he Beurling specrum ofany boundedfull-imesoluion isasubse of{ξ,ξ,,ξ n }.For he delay equaion u() = u( τ)we proved ha every almos periodic soluion is periodic, so if here exiss an almos periodic soluion hen he delay τ mus be π/. Generally, he specrum of any bounded full-ime soluion of he delay equaion ẋ() = Ax( τ) is a compac subse of he inerval [ r(a), r(a)]. Now consider a bounded soluion x of { ẋ() = Ax() for > x() given in X. Assumehaheorbi {x() : }isrelaivelycompac. Thenheomegalimi se ω of x is a compac conneced subse of X [4]. Moreover, ω is invarian under he group T() = e A. Le v be a poin in his omegalimi se and u() = T()v. Then u is a bounded full-ime soluion of he differenial equaion ẋ = Ax. On he oher hand, Ω = ω {x() : } is a compac subse of X. Therefore, he semi-group {T ()} acs injecively on Ω. By an ergodic heorem [6] we have lim x(s)ds = lim u(s)ds. This limi is lying in he kernel of A. Specially, if σ(a) ir = hen is he only bounded full ime soluion. Thus, every bounded soluion ends o as. Now le (p,q) be a bounded soluion of he Hamilon sysem ṗ = H q = H p.

3 Then here is an injecive coninuous semi-flow T () : R n R n such ha (p(),q()) = T ()(p(),q()) Then he omega-limi se ω of (p,q) is a compac conneced subse of R n [4]. Moreover, ω is invarian under he group T(). The dynamical sysem ω,{t()} R is uniquely ergodic, since he only invarian (coninuous) funcion on ω,{t()} R is he consan funcion. Le v be a poin in his omega-limi se and u() = T()v. Then u is a bounded full-ime soluion of he differenial equaion ṗ = H q = H p. By an ergodic heorem [6] here are he limis of p(s)ds and q(s)ds as for any bounded soluion (p,q) of he Hamilon sysem. These limis are saionary poins of he Hamilon sysem. Therefore, we have Theorem A. If he gradien H of a smooh hamilonian H is nowhere hen every soluion of he Hamilon sysem ṗ = H q = H p is unbounded. For example, consider he sysem ẍ = sinx wih x() =. If ẋ() > hen x() is unbounded. If ẋ() = hen x() = arc sin e e + which is increasingly ending o π as. If ẋ() (,) hen x() is periodic and bounded by π in he ime and boh x(s)ds and end o as. Moreover, he period of his soluion is A dx, cosx +ẋ() 3 ẋ(s)ds

4 ( where A = arccos ẋ() ) is he maximal value of x().. MAIN RESULTS Le T () : X X for denoe a semi-group wih (unbounded and close) generaor A. Le x() = T ()x() denoe a bounded soluion of he differenial equaion ẋ = Ax. Assume ha he orbi {x() : } is relaively compac. Then he omega-limi se ω of x is a compac conneced subse of X [4]. Moreover, ω is invarian under he semi-group {T()}. Clearly, T () : ω ω is bijecive. I is easy o prove ha he dynamical sysem ω,{t()} R is uniquely ergodic [6]. In fac, he only invarian (coninuous) funcion on ω,{t()} R is he consan funcion. Hence, here is a unique Borel probabiliy measure µ on ω [6] such ha lim ϕ(u(s))ds = ω ϕ(v)dµ(v). Here, ϕ denoes a coninuous funcion on ω and u(s) = T (s)v for some v ω. Therefore, here is he limi of u(s)ds as. Similarly, he limi of x(s)ds exiss as. Theorem B. LeAdenoehegeneraorofalinearsemigroupT () : X X for. Le x() = T ()x() denoe a bounded soluion of he differenial equaion ẋ = Ax. Assume ha he orbi {x() : } is pre-compac. Then he limi of x(s)ds exiss as. This limi is a vecor in he kernel of he operaor A. If σ(a) ir = {iξ,iξ,,iξ n } hen every bounded full-ime soluion of differenial equaion ẋ() = Ax() has he form u() = n e iξk v k, where v,v,,v n are fixed vecors of X. Specially, if k= σ(a) ir {} hen every bounded soluion of pre-compac orbi ends o a vecor in he kernel of A as. Proof: As we have menioned before, he dynamics on he omega limi se of x is uniquely ergodic. Moreover, his limi se conains a full ime bounded soluion. Le u denoe a bounded full-ime soluion of ẋ() = Ax(). Then (λ D) u() = (λ A) u() for any R and λ / ir σ(a). 4

5 Here D denoes he differenial operaor wih specrum ir. Therefore, for any poin ξ in he Beurling specrum of u we have iξ σ(a). Hence, if σ(a) ir = {iξ,iξ,,iξ n } hen he Beurling specrum of any bounded full-ime soluion is a subse of {ξ,ξ,,ξ n }. Thus, u() = n e iξk v k, where v,v,,v n are fixed vecors of X [], [5]. Now consider a bounded soluion x of pre-compac orbi. Then he omega-limi se of x should conain a bounded full ime soluion u() = n e iξk v k. Specially, if σ(a) ir {} k= hen he omega limi se of any bounded soluion wih pre-compac orbi has only one elemen. This elemen is a vecor of he kernel of A. The proof is now complee. Remark. The las saemen in our Theorem makes a significan exension of resuls in [], [3]. Indeed, he auhors have proved he exisence of he x(s)ds only. lim Acknowledgemen. Deepes appreciaion is exended owards he NAFOSTED (he Naional Foundaion for Science and Techology Developmen in Vienam) for he financial suppor. References [] Dang Vu Giang, Beurling specrum of funcions in Banach space, Aca Mah. Vienamica 39 (4) [] R. delaubenfels and Vu Quoc Phong. Sabiliy and almos periodiciy of soluions of ill-posed absrac Cauchy problems. Proc. Amer. Mah. Soc. 5 (997) [3] Vu Quoc Phong. On sabiliy of C -semigroups. Proc. Amer. Mah. Soc. 9 () [4] Hale, Jack K.; Verduyn Lunel, Sjoerd M. Inroducion o funcional-differenial equaions. Applied Mahemaical Sciences, 99. Springer-Verlag, New York, 993. x+447 pp. ISBN: [5] N.V. Minh, A new approach o he specral heory and Loomis-Arend-Bay-Vu heory, Journal of Differenial Equaions 47 (9) [6] Cornfeld, I. P.; Fomin, S. V.; Sina, Ya. G. Ergodic heory. Translaed from he Russian by A. B. Sosinski. Grundlehren der Mahemaischen Wissenschafen [Fundamenal Principles of Mahemaical Sciences], 45. Springer-Verlag, New York, 98. x+486 pp. k= 5