Existence and exponential stability of periodic solutions for a class of Hamiltonian systems on time scales

Transcription

1 Yang e al. Advances in Difference Equaions 23, 23:8 R E S E A R C H Open Access Exisence exponenial sabiliy of periodic soluions for a class of Hamilonian sysems on ime scales Li Yang,YongzhiLiao 2 Yongkun Li * * Correspondence: yklie@ynu.edu.cn Deparmen of Mahemaics, Yunnan Universiy, Kunming, Yunnan 659, People s Republic of China Full lis of auhor informaion is available a he end of he aricle Absrac In his paper, by using a fixed poin heorem he heory of calculus on ime scales, we obain some sufficien condiions for he exisence exponenial sabiliy of periodic soluions for a class of Hamilonian sysems on ime scales. We also presen numerical examples o show he feasibiliy of our resuls. The resuls of his paper are compleely new complemenary o he previously known resuls even if he ime scale T = R or Z. Keywords: periodic soluion; Hamilonian sysem; exponenial sabiliy; ime scale Inroducion Hamilonian sysem, which was inroduced by he Irish mahemaician SWR Hamilon, is widely used in mahemaical sciences, life sciences so on. Many models in celesial mechanics, plasma physics, space science bio-engineering are in he form of a Hamilon sysem. Therefore, he sudy of a Hamilon sysem is useful meaningful in heory pracice. Recenly, various Hamilonian sysems have been exensively sudied see [ 5] references cied herein. Mos exising resuls on he sudy of Hamilonian sysems are for coninuous sysems. However, boh discree coninuous sysems are imporan in applicaions, i is roublesome o sudy he dynamical properies for coninuous discree sysems, respecively. Therefore, i is meaningful o sudy dynamical sysems on ime scales see [6 3] references cied herein, which helps avoid proving resuls wice, once for differenial equaions once for difference equaions. There have been some resuls devoed o Hamilonian sysems on ime scales [9,, 4 7]. For example, auhors in [4] inroduced Hamilonian sysems on ime scales auhors in [5, 6] sudied he following Hamiloniansysemonimescales: { x =αxσ + βy, y = γ xσ αy,. where T.In[5], auhors obained inequaliies of Lyapunov for., auhors in [6] sudied he sabiliy of. by using Floque heory. However, o he bes of our knowledge, up o now, here have been no papers published on he exisence exponenial sabiliy of a periodic soluion o.. 23 Yang e al.; licensee Springer. This is an Open Access aricle disribued under he erms of he Creaive Commons Aribuion License hp://creaivecommons.org/licenses/by/2., which permis unresriced use, disribuion, reproducion in any medium, provided he original work is properly cied.

2 Yang e al. Advances in Difference Equaions 23, 23:8 Page 2 of Moivaed by he above menioned works, in his paper, we sudy he exisence exponenial sabiliy of periodic soluions o., in which T is a periodic ime scale. The main aim of his paper is o sudy he exisence of periodic soluions o. byusinga fixed poin heorem. Moreover, we also sudy he exponenial sabiliy of he periodic soluion o.. Our resuls are new complemenary o he previously known resuls even if he ime scale T = R or Z. Remark. I is obvious ha when T = R,. reduces o he following coninuous ime Hamilonian sysem: x =αx+βy, y = γ x αy, R. When T = Z,. reduces o he following discree ime Hamilonian sysem: xn=αnxn ++βnyn, yn= γ nxn + αnyn, n Z. For convenience, we denoe [a, b] T = { [a, b] T} ϑ = sup T μ. For an ω- periodic funcion f : T R, wedenoef + = max [,ω]t f, f = min [,ω]t f f = max [,ω]t f. Throughou his paper, we assume ha H β, γ C rd T, R, α C rd T,,+ are all ω-periodic funcions e α ω,, α R +,wherer + = R + T, R={r :+μr>, T}. 2 Preliminaries In his secion, we inroduce some definiions sae some preliminary resuls. Definiion 2. [6] LeT be a nonempy closed subse ime scale of R. Theforward backward jump operaors σ, ρ : T T he graininess μ : T R + are defined, respecively, by σ =inf{s T : s > }, ρ=sup{s T : s < } μ=σ. Definiion 2.2 [6] Apoin T is called lef-dense if > inf T ρ=,lef-scaered if ρ <, righ-dense if < sup T σ =, righ-scaeredifσ >. IfT has a lef-scaered maximum m,hent k = T \{m};oherwiset k = T.IfThas a righ-scaered minimum m,hent k = T \{m};oherwiset k = T. Definiion 2.3 [6] A funcion r : T R is called regressive if +μr for all T k.ifr is a regressive funcion, hen he generalized exponenial funcion e r is defined by { } e r, s=exp ξ μτ rτ τ s for s, T,

3 Yang e al. Advances in Difference Equaions 23, 23:8 Page 3 of wih he cylinder ransformaion ξ h z= { Log+hz h if h, z if h =. Le p, q : T R be wo regressive funcions, we define p q := p + q + μpq, p := p +μp, p q := p q. Then he generalized exponenial funcion has he following properies. Lemma 2. [6] Assume ha p, q : T R are wo regressive funcions, hen i e, s e p, ; ii e p σ, s=+μpe p, s; iii e p, s= e p s, = e ps, ; iv e p, se p s, r=e p, r. Lemma 2.2 [7] Assume ha f, g : T R are dela differeniable a T k, hen i ν f + ν 2 g = ν f + ν 2 g for any consans ν, ν 2 ; ii fg =f g+fσg =fg +f gσ. Lemma 2.3 [7] Assume ha p for s, hen e p, s. Definiion 2.4 [7] A funcion f : T R is posiively regressive if + μf > for all T. Lemma 2.4 [7] Suppose ha p R +, hen i e p, s>for all, s T; ii if p q hen e p, s e q, s for, s T. Lemma 2.5 [8] If p R + p<for all T, hen for all s T wih s, we have <e p, s exp pu u <. s Lemma 2.6 [8] Le x {x CT, R x + ω =x}. Then x σ exiss x σ = x, where x = max [,ω]t x. Lemma 2.7 [7] If p R a, b, c T, hen [ ep c, ] = p [ ep c, ] σ b a pe p c, σ = ep c, a e p c, b.

4 Yang e al. Advances in Difference Equaions 23, 23:8 Page 4 of Definiion 2.5 Le z =x, y T be a soluion of.z=x, y T be an arbirary soluion of.. If here exis posiive consans M >λwih λ R + such ha for T, z z M z z e λ,, T,, where z = max{ x x, y y }, z z = max{ z z *, z 2 z * 2 }, z i z * i = max s, ] T z i s z * i s, i =, 2, hen he soluion z is said o be exponenially sable. 3 Exisence uniqueness Lemma 3. Le H hold. Then every ω-periodic soluion x, y T of. is of he form: { x= +ω y= e α ω, e α ω, βsyse α s, s, e α s, γ sxσ s s. μsαs Proof Le x, y T be an ω-periodic soluion of.. We can rewrie he firs equaion of. as follows: x αx σ = βy. Muliplying boh sides of he above equaion by e α,,wehave e α,x = e α,βy. Inegraing boh sides of his equaion from o + ω noicing ha x + ω=x, we have x= e α ω, βsyse α s, s. On he oher h, we rewrie he second equaion of. as follows: αμ y +αy σ = γ x σ, which is equivalen o y + α y σ γ = αμ x σ. Muliplying boh sides of he above equaion by e α,,wehave e α,y = γ x σ e α σ,. Inegraing boh sides of his equaion from o + ω noicing ha y + ω= y, we have y= +ω e α s, e α ω, μsαs γ sx σ s s, which complees he proof.

5 Yang e al. Advances in Difference Equaions 23, 23:8 Page 5 of Theorem 3. Assume ha H H 2 min s [,ω]t { μsαs } { β + ωe ω ξ μτ ατ τ γ + ωe ω ξ μτ ατ τ } θ = max, < e α ω, e α ω, min s [,ω]t { μsαs } hold. Then. has a unique periodic soluion. Proof Le X = {z=x, y T CT, R 2 x + ω=x, y + ω=y} wih he norm z = max{ x, y },where x = max [,ω]t x y = max [,ω]t y.thenx is a Banach space. For z X, define he following operaor: : X X, z =x, y T z = z, 2 z T, where z= 2 z= e α ω, βsyse α s, s +ω e α s, e α ω, μsαs γ sx σ s s. We will show ha is a conracion. Firs we show ha for any z X, wehave z X. Noe ha z + ω= = = 2 z + ω= = = e α ω, e α ω, e α ω, +2ω +ω βsyse α s, + ω s βs + ωys + ωe α s + ω, + ω s βsyse α s, s +2ω e α s, + ω e α ω, +ω μsαs γ sx σ s s e α ω, e α ω, e α s + ω, + ω μs + ωαs + ω γ s + ωx σ s + ω s e α s, μsαs γ sx σ s s, which means ha z X. Nex, we prove ha is a conracion mapping. Togeher wih e α s, =e s ξ μτ ατ τ e ξ μτ ατ τ = e ω ξ μτ ατ τ, s + ω, s T,

6 Yang e al. Advances in Difference Equaions 23, 23:8 Page 6 of for any z =x, y T, z 2 =x 2, y 2 T X,wehave z z 2 = max +ω [,ω] T βsy se α s, βsy 2 se α s, s e α ω, +ω max e α s, βs y s y 2 s s [,ω] T e α ω, ωβ + e ω ξ μτ ατ τ y y 2 e α ω, 2 z 2 z 2 = max [,ω] T e α ω, max [,ω] T e α ω, Hence, we have e α s, μsαs γ s x σ s x2 σ s s e α s, γ s x σ s x2 σ s s μsαs ωγ + e ω ξ μτ ατ τ e α ω, min s [,ω]t { μsαs } x x 2. z z 2 θ z z 2 < z z 2. I follows ha is a conracion. Therefore has a fixed poin in X, hais,. hasa unique periodic soluion in X. This compleeshe proof. 4 Exponenial sabiliy of periodic soluion In his secion, we sudy he exponenial sabiliy of he periodic soluion o.. Theorem 4. Assume ha H H 2 hold. Suppose furher ha α > γ +. Then he periodic soluion of. is exponenially sable. Proof By Theorem 3., onecanseeha. hasauniqueω-periodic soluion z = x, y T.Supposehaz =x, y T is an arbirary soluion of.. Denoe w =u, v T,whereu =x x, v =y y. Then i follows from. ha { u =αuσ + βv, 4. v = γuσ αv. For T, by Theorem 2.74 Theorem 2.77 in [8], we have u=e α, u + e α, sβsvs s 4.2 v=e α, v e α, σ s γ su σ s s. 4.3

7 Yang e al. Advances in Difference Equaions 23, 23:8 Page 7 of Take a consan λ >wih λ R + such ha λ > β + + ϑλα ɛ > γ +,whereɛ = min [,ω]t e λ,. Le { M > max λɛ 2 λ β + + ϑλ, α α ɛ γ + where ɛ 2 = max [,ω]t e λ,. I is easy o verify ha M >hence,wehave w w < Me λ, w,, ] T. }, We claim ha w < Me λ, w,,+ T, 4.4 which means ha u < Me λ, w,,+ T 4.5 v < Me λ, w,,+ T. 4.6 By way of conradicion, assume ha 4.4does no hold,hen we have he followinghree cases. Case one: 4.6isrue4.5 is no rue. Then here exiss,+ T such ha u Me λ, w, u < Me λ, w,, T. Hence, here mus be a consan p suchha u = pme λ, w, u < Me λ, w,, T. In view of 4.2, we have u = e α, u + e α, sβsvs s e α, u + e α, s βs vs s e α, w + β + pm w e α, se λ s, s = e α, w + β + pm w e λ, = pm w e λ, pm e α λ, +β + pm w e λ, M e λ, +β + e α λ s, s e α λ s, s e λ s, s

8 Yang e al. Advances in Difference Equaions 23, 23:8 Page 8 of = pm w e λ, M e λ, + β+ λ λe λ s, s pm w e λ, M e λ, + β+ λ + ϑλe λ, < pm w e λ,, 4.7 which is a conradicion. Case wo: 4.5isrue4.6 is no rue. Then here exiss 2,+ T such ha v2 Me λ 2, w, v < Me λ, w,, 2 T. In view of 4.3, we have v 2 = e α 2, v 2 e α 2, σ s γ su σ s s e α 2, v 2 + e α 2, σ s γ s u σ s s w + γ + M w 2 = w + γ + M w α e α 2 2, σ s s α e α 2, σ s s = w + γ + M w α e α 2, w + γ + M w α = M w e λ 2, Me λ 2, + γ + αe λ 2, < M w e λ 2,, which is also a conradicion. Case hree: Boh areunrue.bycaseonecasewo,wecanobaina conradicion. Therefore, 4.4 holds. Hence, wehaveha z z M z z e λ,, T,, which means ha he periodic soluion z of. is exponenially sable. This complees he proof. By Theorem 2. in [6], we have he following corollary. Corollary 4. Assume ha H H 2 hold. Suppose furher ha β, β, [, ω] T ω γ α2 >.If β ω ω ω β γ + <4exp ξ μ α

9 Yang e al. Advances in Difference Equaions 23, 23:8 Page 9 of or ω α ω + Then. has a sable ω-periodic soluion. /2 ω /2 β γ + <2, γ + =max {, γ }. 5 Examples In his secion, we presen wo examples o illusrae he feasibiliy of our resuls obained in previous secions. Example 5. Consider he following Hamilonian sysem on T = R: { x =αx+βy, y = γ x αy, R, 5. in which we ake he coefficiens as follows: α=2+sin, β =.3 sin, γ =.5cos. Since T = R,hen μ=.by calculaing,we have α + = ᾱ =3, α =, β + =.3, γ + =.5, { ω } exp αs ds = e 4π θ.34 <. All he condiions in Theorem 3. Theorem 4. are saisfied. Hence, 5. hasanexponeniallysable 2π-periodic soluion. Example 5.2 Consider he following Hamilonian sysem on T = Z: { xn=αnxn ++βnyn, yn= γ nxn + αnyn, n Z, 5.2 in which we ake he coefficiens as follows: αn = sin nπ 3, βn =.2 cos nπ 3, γ =.4sin nπ 3. Since T = Z,henμ =. By calculaing, we have α + = ᾱ =.9,α =.5,β + =.2,γ + =.4 ω k= αk exp{ ω log αk } ω k= αk.47, θ.634 <. All he condiions in Theorem 3. Theorem 4. are saisfied. Hence, 5.2 hasanexponenially sable 6-periodic soluion.

10 Yang e al. Advances in Difference Equaions 23, 23:8 Page of Remark 5. Since in , β orβn may be negaive, Theorem 2. in [6] is no suiable for our examples. Bu from our resuls, we can obain ha boh have exponenially sable ω-periodic soluions. Compeing ineress The auhors declare ha hey have no compeing ineress. Auhors conribuions All auhors conribued equally o he manuscrip yped, read approved he final manuscrip. Auhor deails Deparmen of Mahemaics, Yunnan Universiy, Kunming, Yunnan 659, People s Republic of China. 2 School of Mahemaics Compuer Science, Panzhihua Universiy, Panzhihua, Sichuan 67, People s Republic of China. Acknowledgemens This sudy was suppored by he Naional Naural Sciences Foundaion of People s Republic of China under Gran Received: 6 January 23 Acceped: 7 June 23 Published: 25 June 23 References. Ahlbr, C, Peerson, A: Discree Hamilonian Sysems: Difference Equaions, Coninued Fracions, Riccai Equaions. Kluwer Academic, Boson Tang, XH, Zhang, MR: Lyapunov inequaliies sabiliy for linear Hamilonian sysems. J. Differ. Equ. 252, Liu, ZL, Su, JB, Wang, ZQ: A wis condiion periodic soluions of Hamilonian sysems. Adv. Mah. 28, Han, ZQ: Compuaions of cohomology groups nonrivial periodic soluions of Hamilonian sysems. J. Mah. Anal. Appl. 33, Sun, JT, Chen, HB, Nieo, JJ: Homoclinic soluions for a class of subquadraic second-order Hamilonian sysems. J. Mah. Anal. Appl. 373, Hilger, S: Analysis on measure chains-a unified approach o coninuous discree calculus. Resuls Mah. 8, Bohner, M, Peerson, A: Dynamic Equaions on Time Scales, an Inroducion wih Applicaions. Birkhäuser, Boson 2 8. Kaufmann, ER, Raffoul, YN: Periodic soluions for a neural nonlinear dynamical equaions on a ime scale. J. Mah. Anal. Appl. 39, Zhou, JW, Li, YK: Sobolev s spaces on ime scales is applicaions o a class of second order Hamilonian sysems on ime scales. Nonlinear Anal. 73, Zhou, JW, Li, YK: Variaional approach o a class of second order Hamilonian sysems on ime scales. Aca Appl. Mah. 7, Zhang, HT, Li, YK: Exisence of posiive periodic soluions for funcional differenial equaions wih impulse effecs on ime scales. Commun. Nonlinear Sci. Numer. Simul. 4, Li, YK, Yang, L, Wu, WQ: Ani-periodic soluions for a class of Cohen-Grossberg neural neworks wih ime-varying delays on ime scales. In. J. Sys. Sci. 42, Li, YK, Wang, C: Uniformly almos periodic funcions almos periodic soluions o dynamic equaions on ime scales.absr.appl.anal.2, Aricle ID Ahlbr, CD, Bohner, M: Hamilonian sysems on ime scales. J. Mah. Anal. Appl. 25, He, XF, Zhang, QM, Tang, XH: On inequaliies of Lyapunov for linear Hamilonian sysems on ime scales. J. Mah. Anal. Appl. 38, Zafer, A: The sabiliy of linear periodic Hamilonian sysems on ime scales. Appl. Mah. Le. 26, Bohner, M, Zafer, A: Lyapunov ype inequaliies for planar linear dynamic Hamilonian sysems. Appl. Anal. Discree Mah. 23. doi:.2298/aadm324b 8. Adivar, M, Raffoul, YN: Exisence of periodic soluions in oally nonlinear delay dynamic equaions. Elecron. J. Qual. Theory Differ. Equ., doi:.86/ Cie his aricle as: Yang e al.: Exisence exponenial sabiliy of periodic soluions for a class of Hamilonian sysems on ime scales. Advances in Difference Equaions 23 23:8.