Periodic motions of a class of forced infinite lattices with nearest neighbor interaction

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1 J. Mah. Anal. Appl Peroc oons of a class of force nfne laces wh neares neghbor neracon Chao Wang a,b,, Dngban Qan a a School of Maheacal Scence, Suzhou Unversy, Suzhou 2156, PR Chna b Deparen of Maheacs, Yancheng Teacher s College, Yancheng 2242, PR Chna Receve 26 February 27 Avalable onlne 19 Augus 27 Sube by Goong Chen Absrac In hs paper, by usng a pror bouns, opologcal egree an lng arguens, we suy he exsence of peroc soluons of a class of one-ensonal chan of parcles perocally perurbe an wh neares neghbor neracon beween parcles. We suy he necessary an suffcen conons for he exsence of peroc soluons n wo cases when he nuber of parcles s fne an nfne an oban fferen resuls. 27 Elsever Inc. All rghs reserve. Keywors: Peroc soluons; Lace; Neares neghbor neracon; Sub-lnear growh; A pror bouns; Topologcal egree 1. Inroucon In hs paper, we are concerne wh he exsence of T -peroc soluons of soe syses of classcal parcles perocally perurbe wh neares neghbor couplng. Ths proble whch have been sue n [1,2] for s apple backgrouns. In [1], soe resuls of exsence of an nfne of peroc soluons of a Toa ype an a sngular ype are obane, where, because of he lnear frcon er, classc varaonal echnques canno be use as n [3 6] for he auonoous conservave case. In [2], he auhor generalzes he resuls of [1] an presens he necessary an suffcen conons over he ean values of he exernal forces for he exsence of peroc soluons. Soe rearkably fferen an even oppose resuls abou he fne syse an he nfne syse are gven ou. A naural queson s ha weaher here s an analogous resuls n he suaon when he resorng forces are sub-lnear abou he sance beween parcles,.e. we assue g l x + g x sgnx =+, g g 1 l x x + x =, Z. Ths work s suppore by NNSF of Chna an NSF of Jangsu Provnce, Chna BK2646. * Corresponng auhor. E-al aresses: wangchaosuaah@163.co C. Wang, bqan@sua.eu.cn D. Qan X/$ see fron aer 27 Elsever Inc. All rghs reserve. o:1.116/j.jaa

2 C. Wang, D. Qan / J. Mah. Anal. Appl As n [2], we hope o presen an analogous necessary an suffcen conons for he exsence of peroc soluons n boh cases of fne syse an nfne syse. Bu, n our suaon, s neresng o fn here s no necessary conons over he ean values of he exernal forces for exsence of he peroc soluons o he nfne lace an a he sae e here are necessary conons n he case of fne syses. The eho of proof s ha of [1]: frs, we suy he fne syse hrough a change of varables an hen reuce syse o a spler sub-syse ha allows he obanng of a pror bouns of soe hooopc equaon. By usng he classc ools on opologcal egree, we ge resuls on fne syses. Secon, we are able o use he sae a pror bouns obane before o pass o he l an ge T -peroc soluons for he nfne lace. Bu, n our suaon, we canno ge he pror bouns recly as n [1]. Inspre by he eal eploye n [7], an usng recurrence eho, we solve he proble. 2. Fne syses Le us conser he fne syse of n equaons x 1 + cx 1 = g 1x 2 x 1 + h 1, x + cx = g 1x x 1 g x +1 x + h, = 2,...,n 1, x n + cx n = g n 1x n x n 1 + h n. We are gong o look for T -peroc soluons of 2.1 on he confguraon space { H n = x = x 1,...,x n C 2 R n : T } x =. By a T -peroc soluon we unersan a soluon x H n such ha x = x T, x = x T for each Z. Noe ha f x s a soluon of 2.1, hen also s x + C for every consan C, soweassue T x = asa noralzaon conon. By akng he change of varables { y = x1, 2.2 = x +1 x, = 1,...,n 1, we have he equvalen syse y + cy = g h 1, 1 + c 1 = 2g 1 1 g ĥ 1, + c = 2g g 1 1 g ĥ, = 2,...,n 2, n 1 + c n 1 = 2g n 1 n 1 g n 2 n 2 + ĥ n 1, where ĥ = h +1 h. Frs, we are gong o suy he sub-syse 2.4. Le us conser he followng hooopc syse: 1 + c 1 = 2g 1 1 g λĥ 1, + c = 2g g 1 1 g λĥ, = 2,...,n 2, 2.5 n 1 + c n 1 = 2g n 1 n 1 g n 2 n 2 + λĥ n 1, where λ [, 1] Lea 2.1. Assue ha n h j =. 2.6

3 46 C. Wang, D. Qan / J. Mah. Anal. Appl Then, for every R 1,..., R n 1 R + n 1, here exs K 1,...,K n 1 R + n 1, such ha C 1 K for any T -peroc soluons = of 2.5 ha sasfy ax R, = 1,...,n 1 or n R, = 1,...,n 1. [,T ] [,T ] Proof. We only conser T -peroc soluons = o 2.5 ha sasfy ax R, = 1,...,n 1, [,T ] he oher suaon can be scusse n a slar way. By an negraon of 2.5 over a pero an a sple copuaon, where T g are seen lke unknowns of a lnear syse of n 1 equaons, an conserng 2.6, we can verfy ha T g = λ h j T, = 1,...,n By g, we see ha for each Z, >, nepenen of n, such ha g x sgnx > 2 h j, x. 2.8 So < g + g = λ h j T,an hen, we have Because T we have T < g = g = g 2 < g λ g + g + h j T g, h j T 2 ax [, R ] g + g x + h j T. h j T =: M. 2.9 Now we ulply each equaon of 2.5 by e c. Takng no accoun ha ec + c ec = ec an by usng 2.9, we ge 1 ec L 1 2 M 1 + M 2 + ĥ 1 L 1 e ct =: M 1, ec L 1 2 M + M 1 + M +1 + ĥ L 1 e ct =: M, = 2,...,n 2, n 1 ec L 1 2 M n 1 + M n 2 + ĥ n 1 L 1 e ct =: M n 1 afer an negraon over [,T] when c an over [ T,] when c<. Takng = n [,T ]. Then, ec = e c ec L 1 M

4 C. Wang, D. Qan / J. Mah. Anal. Appl for all [,T] when c an for all [ T,] when c<. In consequence, M. Fro 2.7 an 2.8, s possble o choose [,T], such ha >. Then we have = s s T TM for all [,T], so TM + TM + ax{, R }=:N, an so C 1 M + N =: K. Thus, Lea 2.1 s prove. In he followng, we wll seek for he a pror bouns for he T -peroc soluons o 2.5. Lea 2.2. Assue 2.6. Then, here exs R 1,...,R n 1 R + n 1, such ha C 1 R, = 1,...,n 1, for each T -peroc soluon 1,..., n 1 of 2.5. Proof. By conracon, we assue ha, for a ceran k {1,...,n 1}, here s a sequence {λ } [, 1 an a sequence of T -peroc soluons of 2.5, 1,..., n 1, whch correspons o he paraeer λ, such ha k C By Lea 2.1, clearly, here are, j {1,...,n 1} such ha Q q j := ax [,T ] + +, 2.1 := n [,T ] j Whou loss of generaly, we can suppose here s a subsequence, sll noe { }, such ha ax = Q. [,T ] Now, we ake, + [,Tfor every, an we efne +, = n{ [,T: = Q }. By 2.7 an 2.8, we know ha here us be ˆ [,T], such ha ˆ. Le α = ax{ <, + : = }, β = n{ >, + : = }, where s efne n Lea 2.1. Then, we have α <β <α + T, where α T,T,an α = β =, >, α,β, Q = ax { : [ α ]},β. Now we conser he h equaon c = 2g g 1 1 g λĥ. In he followng, we argue n wo cases n orer o ge a conracon.

5 48 C. Wang, D. Qan / J. Mah. Anal. Appl Case one, here s Z +, such ha Q > ax [α,β ] +1, Q > ax [α,β ] for every, hen we ulply he h equaon by. An we ge β β 2 + 2g g 1 1 g+1 +1 = λ ĥ β α α α afer an negraon over [α,β ]. An hen β β 2 + 2g g 1 1 g+1 +1 Q ĥ L 1. α α For suffcenly large, by usng he ean value heore an akng no accoun g, we ge β 2g g 1 1 g+1 +1 > g g+1 +1 β α α > ax g 1s + ax g +1s TQ [ 1,Q ] [ +1,Q ], where α,β an j j = 1,...,n 1 are efne as n Lea 2.1. An n hs case, fro g 1 an 2.15, we ge Snce β α 2 Q 2 Q ĥ 2T L 1. Q =, + +, = + α an by usng Cauchy Schwarz nequaly, we ge Q 2 β α β β 2 T 2. α α Then we have Q 2 Q ĥ 2T L 1, whch conracs 2.1. Case wo, here s a sub-sequence of funcon sequence {+1 } or { 1 }, sll noe { +1 } or { 1 } for convnce, such ha ax [α,β ] +1 Q or ax [α,β ] 1 Q for every. In hs case, we only scuss he forer suaon, he laer suaon can be scusse slarly. Obvously, Q +1 := ax [,T ] +1 Q

6 C. Wang, D. Qan / J. Mah. Anal. Appl for every, an Q Then we can efne +,+1,α+1 An hen, we conser he + 1h equaon,β+1 n he sae way as +,,α,β c +1 = 2g g g λĥ +1. We can repea argung n he sae way. Snce he syse 2.5 has fne equaons, so, eher here s k { + 1,...,n 2} an Z, such ha Q k > ax [α k,βk ] k+1, Q k > ax [α k,βk ] k for every an Q k + + ; or here s no. In he forer suaon, we conser he kh equaon k + c k = 2g k k g k 1 k 1 g k+1 k+1 + λĥ k ; n he laer suaon, we conser he fnal equaon n 1 + c n 1 = 2g n 1 n 1 g n 2 n 2 + λĥ n 1. Then, s obvous ha here s a subsequence of {n 1 }, sll noe by { n 1 } for convnce, such ha [α n 1 ax,β n 1 ] n 2 < Q n 1 = [α n 1 ax,β n 1 ] n 1. Now n boh above suaons, we can eploy he sae eho as n case one, an we wll also have a conracon. Ths coplee he proof. Reark 2.1. Obvously, R >, = n,..., n 1, epen no on n. In fac, f here s a subsequence of {n}, noe {n k }, an a sequence of soluons { n k n k,..., n k n k 1 } corresponng o {λ n k } [, 1], such ha 2.1 hols, hen we can argue as n Lea 2.2 an ge a conracon when n k suffcenly large. The followng lea s an eae consequence of he resuls n [8] see also Lea 2 n [1]. Lea 2.3. Le F : R n 1 R n 1 be a connuous funcon of coponens F = F 1,...,F n 1 efne by F 1 1,..., n 1 = 2g 1 1 g 2 2, F 1,..., n 1 = 2g g 1 1 g +1 +1, = 2,...,n 2, F n 1 1,..., n 1 = 2g n 1 n 1 g n 2 n 2 for all 1,..., n 1 R n 1. Assue ha here exss a copac se D R n 1 such ha 1,..., n 1 D, [,T], for any T -peroc soluon of 2.5, λ [, 1]. Then, f eg B F, Ω, for soe Ω open boune se conanng D, here exss a leas a T -peroc soluon of 2.4, where eg B enoe he Brouwer egree. Theore 2.1. Le conser syse 2.1 such ha g, g 1 are sasfe. Then, here exss a leas one T -peroc soluon x H n f an only f 2.6 hols. Proof. In orer o see ha conon 2.6 s necessary, we only have o a he n equaons an negrae over a pero. For he suffcency, we suy he equvalen syse Frs, we are gong o prove he exsence of a T -peroc soluon of 2.4. By Leas 2.2 an 2.3, we only have o prove ha eg B F, Ω,,

7 5 C. Wang, D. Qan / J. Mah. Anal. Appl where F s efne n Lea 2.3 an Ω s an open boune se of R 2n large enough. To hs purpose, we ake a convex hooopy beween F an F : R 2n R 2n efne by F n n,..., n 1 = 2 g n g n+1, F n,..., n 1 = 2 g g 1 g +1, = n + 1,...,n 2, F n 1 n,..., n 1 = 2 g n 1 g n 2, where g s a connuous funcon wh connuous an posve ervave, sasfyng gx >, x R, g 1 an such ha gx > g x, x R, = 1,...,n I s clear ha hs choce s possble. Then, he respecve Brouwer egrees conce aybe wh a large Ω fwe fn a pror esaes for he soluons of λf 1,..., n λ F 1,..., n 1 =, λ [, 1], 2.18 ha s, 2 λg λ g 1 λg λ g 2 =, 2 λg + 1 λ g λg λ g +1 λg λ g 1 =, = 2,...,n 2, 2 λg n 1 n λ g n 1 λg n 2 n λ g n 2 = wh λ [, 1]. I s easy o see ha λg + 1 λ g can be consere as unknowns of a lnear syse of equaons wh a unque soluon, naely, λg + 1 λ g =, = 1,...,n 1. Fro here, by usng 2.17, g >, = 1,...,n 1, an as g s srcly ncreasng, here exss he nverse g 1 an > g 1, = 1,...,n 1. On he oher han, g <, = 1,...,n 1, an by g follows ha here s ψ > such ha <ψ for all. In concluson, we have foun a pror bouns for he soluons of he convex hooopy an hence s prove ha eg B F, Ω, = eg B F,Ω, for Ω large enough. Fnally, we copue hs las egree. If A s efne as he coeffcen arx of lnear syse F n,..., n 1 =, s easy o prove ha e A. Takng no accoun ha g s srcly ncreasng, hen he vecor fel has he unque zero ξ,...,ξ wh ξ = g 1. If F s he Jacoban arx of he vecor fel F, by he efnon of Brouwer egree an soe easy copuaons we ge eg B F,Ω, = sgn e F ξ,...,ξ= sgn e g ξa. Therefore, he exsence of a T -peroc soluon of sub-syse 2.4 s prove. Fnally, he exsence of a T -peroc soluon of Eq. 2.3 s rval because usng 2.3 he rgh-han eber has ean value zero. Now, we conclue he proof.

8 C. Wang, D. Qan / J. Mah. Anal. Appl Infne syses In hs secon, we are concerne wh he exsence of T -peroc soluons of he nfne syse of nonauonoous fferenal equaons x + cx = g 1x x 1 g x +1 x + h, Z. 3.1 We are gong o look for T -peroc soluons of 3.1 on he confguraon space { H = x ={x } Z C 2 R Z : T } x =. Theore 3.1. Le conser syse 3.1 such ha g, g 1 are sasfe. Then, for any K R here exss a T - peroc soluon x H of 3.1 such ha T g x1 x = KT. 3.2 Proof. The ea of he proof s o pass o he l fro a fne syse. Takng K a fxe real nuber, we conser he fne syse of 2n + 1 equaons x n + cx n = g n nx n+1 x n + h n + K h, = x + cx = g 1x x 1 g x +1 x + h, = n + 1,...,n 1, 3.3 x n + cx n = g n 1x n x n 1 + h n K n h. =1 Ths syse has been sue n Secon 2, an s easy o check ha he suffcen conons of Theore 2.1 s sasfe an so, has a T -peroc soluon {x n } = n,...,n. Le x be chosen such ha T x =. Fnally, he bouns euce n he proof of Lea 2.2 can be use as n [1] o prove he convergence of hs sequence o a soluon of 3.1. Beses, by 2.7, conon 3.2 s obane by ang up he equaons for = n,..., an hen negrang over a pero an passng o he l wh λ = 1. Reark 3.1. For any n N, f x s a soluon of hooopc equaons of 3.3, whch efne as n 2.5, hen T T g +1 = KT λ g = KT + λ j= h j T, n 1, 3.4 h j T, 1 n So, Reark 2.1 ples ha he bouns euce n he proof of Lea 2.2 o no epen on n bu, only epen on K, an c. We reark ha conon 3.2 assures he exsence of an nfne nuber of essenally fferen T -peroc soluons.

9 52 C. Wang, D. Qan / J. Mah. Anal. Appl References [1] P.J. Torres, Peroc oons of force nfne laces wh neares neghbor neracon, Z. Angew. Mah. Phys [2] P.J. Torres, Necessary an suffcen conons for exsence of peroc oons of force syses of parcles, Z. Angew. Mah. Phys [3] G. Arol, J. Chabrowsk, Peroc oons of a ynacal syse conssng of an nfne lace of parcles, Dyna. Syses Appl [4] G. Arol, F. Gazzola, Peroc oons of an nfne lace of parcles wh neares neghbor neracon, Nonlnear Anal [5] G. Arol, F. Gazzola, Exsence an approxaon of peroc oons of an nfne lace of parcles, Z. Angew. Mah. Phys [6] G. Arol, F. Gazzola, S. Terracn, Mulbup peroc oons of an nfne lace of parcles, Mah. Z [7] R. Iannacc, M.N. Nkashaa, P. Oar, F. Zanoln, Peroc soluons of force Lénar equaons wh jupng nonlneares uner nonunfor conons, Proc. Roy. Soc. Enburgh Sec. A [8] A. Capeo, J. Mawhn, F. Zanoln, Connuaon heores for peroc perurbaons of auonoous syses, Trans. Aer. Mah. Soc