Galerkin's Procedure for Nonlinear Periodic Differential Difference Equations 利用統計を見る

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Original Report

Galerkin's Procedure for Nonlinear Periodic

Differential Difference Equations

MitsunobuKURIHARA ShinichiCHINO

（Received August 29，1978） Abstract We study a system of nonlinear perlodic differential difference equations and discuss the questions of existence， uniqueness and error estimation on approximate periodic solutions of the system obtained by the Galerkin，s method based on trigonometric polynomials． At that time we introduce certain isolation conditions on periodic solutions of the system． Further・ more we prove an existence theorem for an exact periodic solution of the system with the isolation conditions in some neighborhood of a given approximate solution and give an example for the existence theorem． These results we obtained are analoguous to those by M．Urabe for nonlinear periodic differential equations． 0． Introduction We consider a system of nonlinear periodic dif・ ferential difference equations （・・1）乞（t）−X［t，−x（t），・・（t一ω・），…， x（t一ω、）］．

Here and hereafter the following notations are

used． RN is the」V−dimensional Euclidean space and for x in RN lxl is the Euclidean norm ofx． L is the real line， that is L＝｛t；−oo＜t＜oo｝． The following hypotheses are imposed． ω1，．．．，ω々are given real constants． Let 1） be a bounded open domain in the space RN． X［t，x， y、，＿， Yk］is

the function mapping from the domain L×Dk＋1

into the space RN， where Z）is the closure of D and Z）n denotes the n times product set of the set ∠）．The function X［t， x， Yl，＿， Yk］is twice con’ tinuously differentiable of （t， x， Yl，．．．， Yk） on the domain L×Dk＋1 and periodic in t of period 2π， In this paper we discuss the questions of ex・やコ 1stence・unlqueness and error estimation on ap・ proximate solutions of the forms： m （0．2） Xm（t）＝ao十㎡百Σ（ancosnt十bnsinnt）．カ＝1 1t is reasonable that the coefHcients αo， al， bi，．．．， am， bm should be determined so that （・・3）6…ω一A・・ml（α）＋V万Σ｛An〔m）（α）cosnt＋B。（an）（α）sinnt｝ n ：1 ， Or eqUiValently （0．4） Fo㈲（α）＝0，、Fn（m）（α）＝0， Gn（m）（α）＝0， for n＝1，．．．，〃z． Here we denote F。（m）（α）＝A。（m）（α），F，、（m）（α）＝A。｛M）（α）−nb。， G。㈲（α）＝B。｛M｝（α）＋〃α。，

and

A・㈲（α）−il．，f：xx［…m（・），・・（・一ω1）・…・ Xm（s一ω盈）］ds，蝋α）一（1／vi・・）∫：πX［・，・m（・）・Xm（・一ω・）・＿，Xm（S−COIe）］COS・nsds， Bn・m・（α）一（1／vi；・）∫：πX［・，・m（・）・・m（ ω1）・＿，Xm（s−COIe）コsinnsds， for n＝1，＿， m and α＝ao， al， b，，＿， amt bm｝． The process for finding the approximate solutions （0．2） by the relation （0．3） or （0．4） is exactly the Galerkin’s method based on trigonometric poly・ nomials． Thus the finite series（0．2）satisfying the relation（0．3）or（0．4）is called Galerkin ap・ proximation of order m and the system（0．4）is called determining equation of Galerkin approxima・

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tions． After introducing certain su伍cient conditions on exact periodic solutions of the equation （0．1） which imply their isolution， we prove three main theorems on Galerkin approximations of she equa・ tion（0．1）． The first theorem， Theorem 1， says that for any periodic solution of the equation（0．1） 6f period 2π1ying in 1）for t in L with the isola− tion conditions there exists an Galerkin approxima・ tion accurately as it is desired by computing the

finite trigonometric polynomial． The second

theorem， Theorem 2， says that the obtained

Galerkin approximation corresponds one to one to the solution with the isolation conditions． The

third theorem， Theorem 3， says that one can

always assure the existence of an exact priodic solution of the equation（0．1）of priod 2πwith the isolation conditions by checking sevral con− ditions on the obtained Galerkin approximation and further it gives a method to obtain an error bound of the obtained approximate solution． The analoguous theorems were originally proved by M． Urabe［3］for a periodic system of ordinary differential equations． Our results are extension of those by M． Urabe［3］to a system of periodic

differential difference equations． Moreover our

results are also extenstion of those by S． Chino ［4］・1・th・，・el・ti・n t・Th…em 3・L・C・・a・i［1コ had proved existence of an exact periodic solution

in some neighborhood of a given Galerkin ap・

proximation． His process， however， requires a certain amount of discussions in terms in functional analysis which may not be easy． J． Mawhin［2］ studied existence of periodic solutions of functional

differential equations by using the method

analoguous to that of L． Cesari［1］． His argu・ ments， however， have no explicit relationship with Galerkin apProximations． In the present paper we introduce the isolation conditions on periodic solutions of the equation（0．1） in the section l and state three main theorems on Galerkin approximations in the section 2． In the section 3 we state some properties of trigonometric polynomials． In the section 4 we prove some pro・ perties of the determining equations in order to prove the main theorems． In the section 5−7 we give the proofs of the main theorems Theorem 1− 3respectively． In the last section 8 we give an example of our method 1． Isolation Conditions We denote by C the space of functions mapping Linto 1∼N continuous and periodic of period 2πin ton the line L． For any x＝x（りin C we define llxl［o＝sup｛lx（t）1；t in L｝

and

ll・ll・一 o±∫：π1・・（り1・dt｝1／2・ We begin with the follwing lemma， Lemma 1， proved by M。 Urabe［3］． Lemma 1． Let d x（り＝A（りx（り＋f（り（1．1） dt be a given linear periodic system， where／1（り is acontinuous periodic 2V×」V matrix of period 2π andプ（りin C． If the multipliers of the correspond・

ing homogeneous system

d y（の＝A（t）y（り（1．2） dt are all different from one， then the equation（1．1） has one and only one periodic solution of period 2π，which is given by （1・3）・（t）一轣FπH（ち・）∫（・）d・・ where H（t， s）is the piecewise continuous periodic ロ

matrlX

（1．4） H（t， s） 1∼（t）［E−R（2π）］−iR−1（s） for O≦s≦t≦2π R（り［E−R（2π）］−1R（2π）R−1（s） for O≦t＜s≦2， H（ちs）＝H（t＋2〃2π，s＋2κπ） for any integer〃t， n， is the unit matrix and R（り is the matrix of the equation （1．2）with

where E

fundamental

R（0）＝E． The formula（1．3）de丘nes a linear operator H from the space C into itself． Consequently the norms of the linear operator H；C→C are defined corresponding to the norms in the space C， that are ［IHI lo and llHl1Q． Then we call the linear

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December 1978

No．29

operator H；C−＞C defined by（1．3）and the matrix H（t，s）defind by（1．4）H−mapping and H・matrix corresponding to the equation （1．2） respectively・ We assure the existence of the H−mapping and H・ matrix corresponding to the equation（1．2）under the assumptions of Lemma 1． We denote byφ［t， x， Yl，＿， y，］andψゴ［ちx， Y1，＿，脇］，元＝1，＿，々，the Jacobian matrices of the function X［t， x，胡，＿， yle］with respect to the arguments x and yj，ノ＝1，＿， le respectively． For the sake of convenience we use the notations X［t；x］＝X［ちx（t），x（t一ω1），＿， x（t−tok）］， φ［t；x］＝φ［ちx（t）， x（t一ω1），…， x（t一ω、）］

and

ψゴ［t；x］＝ψゴ［ちx（t）・x（t一ω1）・・… x（t−tOk）コ・元＝1，＿，k， for any function x＝x（り lying in D as the occasion demands． Let x ＝＝ te（t） be a periodic solution of the epuation （0．1）of period 2π 1ying in 1） for、any t in L． The solution x＝￡（りis called to satisfy the isola・ tion conditions if the multipliers of the system of linear differential equations （1・5）晶（り一φ［ち・e（t）・al（t一ω1），…， 2（t−（Ole）］y（り A are all different frcm one and the H−mapping H corresponding to the equation （1．5） satisfies the relations ん A （1．6） lIHIloρ〈1 and llHII，／1＜1，

where

k （1．7） R＝max｛Σ1gb i［ちi（t）、 it（t−tO1），＿， 1＝1 1（t一ω々）］1；t inL｝．

The word “isolation’comes from the following

lemma， Lemma 2． Lamma 2． Let x＝元（t）be a periodic solution of the equation （0．1）of period 2πlying in 1） for any t in L and satisfying the isolation conditions． Then， besides the solution x＝元（t）， there is no other periodic solution of the equation （0．1） of period 2πin a su伍ciently small neighborhood of x＝di（t）． The proof of the above lemma， Lemma 2， can be given in the following way： Choose a positive constant ε such that ヘパ（1．8） 211H［1σε＜1−1［H‖oR． for such ε， by the uniform continuities of the functions φ［ち x・Yl・・… yle］ and ψプ［ち x， Yi，＿．， Yk］，元＝1，．．．，〃， there is a positive constant δsuch that 1）δ＝｛x；［x−x（り1≦δfor some tinL｝⊂・D

and

（1．9） 1φ［ちx，Yl，＿， y，］−95［t， x， y？、，＿，ψ、］1 ≦ε， k Σ1ψノ［ちx，Yl，・」＝1 9々］1≦ε ・Yk］一ψゴ［ち元， Oi，…， for any t in L and x・元・〃〆・ 9ゴ・ノ＝1， … ，〃， in Z） satisfying ． Ix一元1≦δ and lyゴー9ゴ1≦δ，元＝1，＿っ〃・ Suppose that， besides x＝元（t）， there is a periodic solution x＝x（t） of the equation （0．1） of period

2πsatisfying

lx（t）一￡（t）1≦δfor any t in L． If we Iet y（t）＝x（t）一元（り， we have ly（t）1≦δfor any tin L and d mt y（り＝x［t；x］−x［’；司一∫1｛φ［t・i＋・・］y（り h 十Σψゴ［t；元十θy］y（t一ωゴ）｝dθ，」＝1 that is d y（り＝φ［t；元］y（t）＋∫（り，（1．10） dt

where

ほ（1．11）ア（の＝Σψゾ［オ；司y（’一ωブ）ノ＝1 ＋∫9｛｛φ［ぽθ・］一φ瞬コ｝・（・） k ＋Σ｛ψゴ［t；i＋刎一ψプ［t；司｝y（t一ωブ）｝dθ・ゴ＝1 Since y（’） is periodic solution of the equation （1．10）of period 2π， by Lemma 1， we have ん y（り＝（Hf）（t），パ where H is the」研一mapping corresponding to the equation（1．5）． Thus we obtain ハ｜ly［1σ≦‖丑llcl1∫11。・ On the other hand it follows from the relations （1．7） and （1．9） that the functionノてt） in （1．11） satisfies Hf［1。≦R‖yll。＋ε1回1σ＋εily［［a・

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This implies that A 1［yl1。≦‖酬c（刀一F2ε）［lyll。， that is パ A ｛（1−HHIIoR）−211Hllcε｝11yllo≦o・ Noting the relation（1．8）， we have llyllσ＝0， that is y（り＝O for any t in L． This proves Lemma 2． 2． Main Theorems We impose the hypotheses stated in the section Oon the equation（0．1）． Furthermore we use the definitions “Galerkin approximations”，“isolation conditions”and“Jacobian matricesφ，ψゴ，ノ＝1，＿，カ” stated in the sections O and 1． Our aim in the present paper is to prove the following main theorems： Theorem 1． Suppose that there exists a periodic solution of the equation（0．1）of period 2π Iying in 1）for any t in L and satisfying the isolation conditions． Then for some number mo there exists

aGalerkin approximation x＝輪（t）of any order

m≧m・su・h・h・t th・・e・i・・｛…（り｝・nd｛嘉（り｝ converge uniformly on the line L to the solution and its derivative respectively as〃z→Oo．

Theorem 2． The Galerkin apProximationりc＝

輪（りstated in Theorem l is determined uniquely in a suthciently small neighborhood of the solution provided that the order 〃z of the Galerkin ap・ proximation x＝輪（りis suMciently high． Theorem 3． Assume that the equation（0．1） has a periodic approximate solution x＝死（り of period 2π， for which there are a positive constant δ，anonnegative constant κ＜1 and a continuous periodic」V×」V matrix∠4（りof period 2πon the line L such that （1）the multipliers of the homogeneous equations d y＝・4（りy（2．1） dt are all different from one，（2） 1）δ＝｛x；lx一元（り≦δfor some t inL｝⊂Z），（3） 1φ［t， x， 2／1，．．．， Yk］−A（t）1≦κ／1レI for any tin L and any x， Yl，．．．， Yk in 1）δ，（4） κ÷ノ匪μ＜1，（5） Mr／（1一κ一ルfμ）＜δ， where H is the H・mapping corresponding to the equaiton （2．1） satisfying the relations 1［Hlle≦ノしf and （2．2）

and

（2．3） for any llH11。≦ハイ， k μ＝max｛Σ1ψゴ［ちX・Y1・…，」；1 x，Yl，＿， y、 in D、｝ y，］1；tinL， period ic solution x＝x（t） of period 2π the solution x＝元（t）satis丘es the isolation conditlons and the relation （2．4） llx一元llc≦Mr／（1一κ一ルfμ）・ The proofs of these theorems are given later in the present paper． 3． Properties of Trigonometric polynomials For any function f＝ノてt） in C we have the Fourier series expansion ＿（3．1）ゾ（t）∼Co十∼／2Σ（cηcosnt十dπsinnt）， lt＝＝1

where

・・一訂：π∫ぴ）dちCn−（㎡2π）−1∫：πf（り・・…dt

and

d・・＝＝（⑫π）−1轣Fπプ（り・inn・dt・ The Parsevars equality for the expansion（3．1） implies that （3・2） ICo12十Σ（ICnl2十ldnl2）＝・llfl］Q2・ n＝1

We de丘ne the operator Pm which express the

truncation of the Fourier series expansion（3．1） of the operand discarding the terms of the order higher than m， that is for any function f＝ノてり in C expanded in the form（3．1） m （3．3）（Pmf）（t）＝・ Co十∼／2Σ（cπcosnt十dπsinnt）． n＝1 1t is clear that Pm is a linear operator from the space C into itself． Later we use the following lemmata proved by M．Urabe［3］． 1・emma 3． Let f＝ノ（の be a continuously dif・ ferentiable function in C． Then dfi （3．4）［1（1−Pm）fllo≦σ（m） dt Q

d

f（t）−X［ち元ぴ），te（t一ω、），…・

dt

x（t−（Ok）コ≦r

tin L． Then there exists uniquely a

＾ the equation （0．1） of lying in Dδfor any t in L． Moreover ・

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and

（3・5）‖（・一馴・≦・（m）都

whereσ（m）andρ（m）are monotone decreasing

functions of〃2 for〃t≧1 satisfying （3．6）㎡百／（〃2十1）＜σ（〃2）＜㎡1「／∨／万 and ρ（m）＝1／（〃2十1） and I is the identity operator in the space C． Lemma 4． The finite Fourier series m fm（り＝Co十Σ（らCOS〃t十dπsin nリカ＝1 satisfies the relations （3．7） ‖fmilc≦㎡2〃2十11γl and lILnliQ＝1γ1， whereγ＝（co，01，41，．．．， cm， dm）and m lγ1＝｛ICo12十Σ（｜c？じ12十1dnl2）｝1／2・ n＝1 By the relations（3．2）and（3．7）it follows that （3．8） 11PmfHe＝1［fmHe≦11fll， for any function∫in C， where Pmf＝塩． 4． Properties of Determining Equations Let x＝￡（t）be a periodic solution of the equation （0．1）of period 2πlying in a bounded open domain Z）in RN for any t in L and satisfying the isola・ tion conditions． There exists a positive constant δ such that （4．1） Dδ＝｛x；lx一元（り1≦δfor some t in L｝ ⊂D． Denote that 2rn＝P誘． It follows from the relations （3．4） and （3．5） that

（4・・）暗一釧・≦噺）篭，≦κ1姻）

and

（4・3魂一dill・≦・（〆膓手、≦K，ρ（〆・

where KI is a constant depending only on the

structure of the equation （0．1）． Hereafter we denote by K’s constants depending only on the structure of the equation （0．1）． In order to determine a domain where the func・ tions Fo㈲（α）， Fπω（α）， Gη㈲（α），％＝1，…・〃1・in the determining equations（0．4）are well defined， we choose a number勿l such that （4．4） 1（1σ（m）≦δfor any 〃1≧〃21 by the relation（3．6）． It follows from the relations （4．2） and （4．4） that the function xm（’） 1ies in

thedomainDδforany t in L and any m≧Ml．

Let us put ＿ m べ（4．5） dim（り＝2，十∼／2Σ＠cosητ十力ηsinηリカ＝1

and

ベム（4．6）〈を＝（∂，，∂1，b，，．．．，∂m， bm）． Define the domain （4．7）・s2η、＝｛α；1α一∂1≦［δ一K1σ（m）コ／∨／2〃t十1｝． This is the desired domain where the functions Fo（m｝（α）， F。（M）（α）， Gn（m）（α）， n＝1，＿， m， are well defined． In fact， if we choose any vector α＝（ao， al， b，，．．．， am， bm）in the domain J2m and let Xm（’）＝a。＋∼／2Σ（a。cos nt＋b。sin nt）， n＝1 then we obtain llx．−Xll。≦｜ixバtemi1σ＋1｜dim−il1。

≦V2m＋1｜α一al＋Klσ（⇒

≦㎡2〃2十1［δ一K，o（m）］／∼／2m十1−FKia（m）＝δ for any m≧mユ， which implies the function x肌（t） 1ies in 1）δfor any t in L and any〃z≧〃zl． It fol− 10ws from the expressions（0．4）that Fo（m）（α）， 1㌦（m）（α），Gπ㈲（d）， va＝1，＿，〃￠， are continuously differentiable functions ofαin the domain、2 η↓° Denote that （4．8） F（m）（α）＝（Fo㈲（α）， F，（m）（α）， G 1（m）（α），＿，Fm（m）（α）， Gm〔m）（α））． Letム（α）be the Jacobian matrix of the function F（刎（α）with respect to the argumentα． To in・ vestigate the properties of the matrix／n、（α）， we consider a system of linear equations of the form；（4．9）ノm（α）ξ十γ＝0， whereα＝（ao，α1， b，，＿， am， bm）in・S2．，γ＝（co， cl， d，，＿，Cm， dm）andξ＝（Uo， Ul， Vl，．＿， Um， Vm）． Let us put that Xm（t）＝a。＋V2Σ（a。C・S・nt＋b。sin nt）， n＝1 ゾ（t）＝Co十∼／百Σ（CπCOS nt十dπsin nt） n＝1

and

y（t）＝uo十∀2Σ（u．cos nt十vπsin nt）． n＝1 1t can be proved that the system（4．9）is equiva・ 1ent to the periodic system of the form： d （4・10）万y（り’＝P・｛φ［t；x・］y（t）

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k ＋Σψゴ［t；Xm］Y（t一ωプ）｝＋f（t）・ゴ＝1 where the functionsφandψゴ，元＝1，＿， k， are the Jacobian matrices defined in the section l and Pm is defined in （3．3）． Substituting 元m（り into xm（り in the equation （4．10），equivalently doing 4 intoαin． the system （4．9），we obtain （4・11）老τ・（り一φ［t・勾・（り k ＋Σψ」［t；‘i）］y（τ一ωゴ）＋f（り＋9（t）・」＝1

where

（4．12） g（t）＝一（1−Pm）φ［t；元］y（t）＋Pm｛φ［t；妬一φ［τ；‘i）］｝y（り k −（1−P肌）Σψ元［t；司〃（‥ω」）」＝1 k 十P．Σ｛傷［t；礼］一ψゴ［t；勾｝y（t一ω」）・」；1 Since y（りis periodic of period 2π， we have by

Lemma 1

（4．13），（t）一（2π．i」（t，・）｛左ψ、［・、X］y（・一ω、）」＝1 」0 ＋f（s）＋9（s）｝ds， where H（t， s）is the H−matrix corresponding to the equation（1．5）． From the expression（4．13） it follows that パ_{（4．14） 11yllQ≦liH‖Q｛，allyl1Q一トilfll9十llgllQ｝’} On the other hand， using the relations（3．5），（3．8），（4．2）and（4．10）， we obtain for the functiorl g（t） in （4．12） 11・ll・≦・（m）｛Kll・ll・＋K馴

＋嚇一釧・M・＋・（m）｛Kll・11畔糺｝

＋K！ltl．−al11。llYllQ

and

凱≦Kl1酬服

It readily follows from（4．14）that パ｛（1−1fiHllQ刀）一［ρ（m）＋σ（m）コκ｝llyliQ パ ≦llHllQ［1十ρ（n）Kコ1レell《∼・ Noting the relation（1．6）， we choose a number M2 ≧MI such that for any m≧M2 A A （4．15） 11yllQ≦1レ¶《2‖∬ll（2［1十ρ（m）Kコ／｛（1−llHllρfi）一［ρ（m）一｛一σ（〃∂）］K≦ノlfllfll，

for some constant M≧O independent of m． The

relation（4．11）implies ｜ξ1≦Mlγ1＝Ml1−（∂）ξl for the vectors∂，γandξin the system（4．9）． It follows that （4．16） det Jm（∂）キO and lJl’肌一1（a・）1≦M． This relations（4．16）will play an impotant role in the proof of Theorem 1． Letα’＝（αo’，・al t， b1’，．．．， am’， bm’），α”＝（αo”， a、”，bt”，＿， am』”， bm’） be arbitrary vectors in the domain S2m． For any vectorξ＝（〃o， ul， v1，＿， ec。、， vm）we cosider the systems of linear equations （4．17） Jm（α’）ξ＝γ’and Jm（α”）ξ＝γ”， whereα’＝（Co’・c1’， dl，．．・， Cmt， dm’），α”＝（Co”， Cl”， d、”，＿，Cm”， dm”）． Let us put m Xm’（t）＝αo’一トA／2 Σ：］（an’cos nt−Fbn’sin nt）， n＝1 Xm「’（t）＝ao”十∀2Σ（an”cos nt十bn”sin nt）， n＝1 m y（t）ニU。＋⑫Σ（U。C・・nt＋V。・in nt）， n＝1 〃（り＝Co’H−∼／2Σ（c．’cos nt十dn’sin〃り n＝1

and

〃’（t）＝c。”＋㎡2Σ（Cn”c・s〃Z＋d。”sin〃り． n＝1 Corresponding to th．e system（4．17）， we have d 4zy（り＝P・｛φ［t；x・’］y（t） k ＋Σψゴ［t；Xmtコy（t一ω」）｝＋h’（り」＝1

and

6・（t）−Pm｛φ［t・・m〃］y（t） k ＋Σψゴ［t；Xm’t］y（t一ω」）｝＋h”（り．，i＝1 1t readily follows that −［h’（t）−h”（t）］＝Pm｛｛φ［t；Xm’コーφ［t；Xm”］｝y（り k ＋妥、｛ψ・［t・ x・’］一・b・［t・ x・”］｝・（t一ω・）｝・

Then we have

ll〃一〃’II，≦κ、｜1x．’−x。11・lly［1，，

which implies

lγ’一γ”1≦K2∼／2−t十11α’一α”川ξl by the relation（3．7）． It is concluded from the 、

system（4．17）that

（4・18） 1／m（α’）一ノm（α”）1≦K2∨／2勿十11evt一αtt1

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No．29

for any 〃2≧〃22 andα’，α”in J2m． The relation （4．18）will also play an impotant role in the proof of Theorem 1． 5． Pmof of Theorem l In order to prove Theorem 1， we use the follow・

ing lemma， Lemma 5 proved by M． Urabe［3］

based on Newton−Raphson，s procedure for nonlinear algebraic equations． Lemma 5． Let （5．1） F（α）＝O be a real system of equations， whereαand F（α） are vectors of the same dimension and F（α）is a continuously differentiable function ofαdefined in

some region、20f theα・space． Assume that the

system（5．1）has an approximate solutionα＝∂

for which the determinant of Jacobian matrixノ（α） of F（α） with respect toαdoes not vanish and that there are a positive constant δand a non・ negative constant κ＜1 such that （1）・9δ＝｛α；1α一∂1≦δ｝⊂」2 （5．2）（2） 1ノ『（α）一ノ（己）1≦κ／M for anyαin 52δ （3） M7／（1一κ）≦δ， where r and M are constants such as （5．3） IF（∂）1≦r and lノー1（∂）1≦ル1． Then the system（5．1）has one and only one solu・ tionα＝a in，S2δand forα＝五it holds that （5．4） det∫（元）・≒Oand lα一al≦Mr／（1一κ）． The proof of Theorem l is given in the follow− ing way． Suppose that there existS a periodic solution of the equation （0．1） of period 2π 1ying in the domain l）for any t in L and satisfying the isolation conditions． In the previous section 4， the following conclusions were obtained． There exists apositive constantδsatisfying the relation（4．1）．

Moreover there exists a number M2 su缶ciently

large such that for any m≧〃z2 the relation （4．4） holds and such that the function F（m）（α）in（4．8） is continuously differentiable ofαin the domain 2m in（4．7）． Its Jacobian matrix九（α）has the inverse ．1’m−1（α） atα＝∂ satisfying the relation （4．16） and satisfies the relation （4・18） for any m≧〃z2． Here X，m（り and 4 are defined in（4．5） and （4．6） respectively． Let us put h（t）一￡元。（り一P。X［t・ i’m］＿＝Pm｛X［t、；司一X［オ；dim］｝．

Then we have

1［hl1《∼≦Kl1元一din川（∼≦ρ（〃z）1（3， which implies （5・5） lF（m）（a’）1＝［lh【le≦ρ（m）K3．

In order to apply Lemma 5 to the determining

equation F（m）（α）＝0， we choose an arbitrary non− negative constantκ＜1 and put δ1＝min｛κ／K，M，δ一Kla（M2）｝， where the constants K2， M and Kt are defined in the relations （4・18），（4．16） and （4．2） or （4．3） respectively． There exists a number M3≧M2 such that ［M／（1一κ）］K，ρ（〃z）＜δ1／V2〃z十1 for any m≧M3． If we choose a constantδm such that ［M／（1一κ）］K，ρ（〃z）＜δm＜δ1／∨／2〃z十1， then we obtain （5・6） 2（δm）＝｛α；1ev−∂1≦δm｝⊂、s2m． In fact， for anyαin、2（δm）and any m≧M3 1cu−（ib i≦δm＜δ1／∨t2 M十1≦［δ一1（1σ（〃12）コ／㎡2〃2十1 ≦［δ一K，σ（m）］／A／2〃z十1， which impliesαin JS2m by the de丘nition（4．7）． Moreover it follows from the relation（4．18）that （5．7） 1／m（α）−1m（∂）1≦K2㎡2〃z十11α一（lrl ≦．κ2レ2〃2十1δm＜K2δ，≦κ／ノレT for anyαin 9（δm）and any〃z≧〃13． By the rela− tion（5．5）we have （5．8）ノレIIF（m）（4）【／（1一κ）≦［M／（1−rc）］K，ρ（〃1）＜δm． Thus the relations（5．6），（5．7）and（5．8）imply that the conditions（5．2）in Lemma 5 are fulfi11ed． By Lemma 5 we see that the determining equations F｛m）（α）＝0，that are the equations（0．4）， have one and only one solutionα＝厩in the domain 9（δm） satisfying det Jm（α）・≒O and ［a−al≦［M／（1一κ）コIF（m）（め1 ≦［M／（1一κ）］K3ρ（〃1）． If we put d＝（∂。，α、，b，，＿，ぱ融， bin）and I ・・の一α・＋鋲C・・…＋5・・i・…）・ then出r㌫（りis a Galerkin approximation of order 〃z．

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The Galerkin approximation妬（りsatisfies l鳳一釧。≦｜1輪一釧。＋1臨一訓。 ≦V27■＋lla−al＋K、o（m） ≦［M／（1一κ）］κ、ρ（m）V2−．＋1＋・K．1σ（m） for any m≧M3． This implies uniform convergence ・fth・G・1・・ki・・pP・・xim・ti・口・（り・t・th…1uti・n 元（りon the line L as M−〉・oo． Furthemore we can prove uniform convergence of the derivative of the Galerkin approximation臨（t）． In fact，

嘉（t）一芸元（t）−Pm｛x［t…］−x四｝

一（1−Pm）x［t；勾 It readily follows that ￡’・一劃c≦㎡2勿＋1Kl1・・m−abll・＋K・（m） ≦㎡2m十IK｛HX．一・e働1［Q十11di．−all［e｝十Kσ（m） ≦∨t2m一トIK｛［M／（1一κ）］K，ρ（m）十K，ρ（m）｝＋Kσ（m） for any m≧M3， which implies皿iform convergence of the derivatives dtem（t）／dt to 4元（り／dt on the line L as m→oo． This proves Theorem 1． 6． Proof of Theorem 2 Let りc＝元（t） be a periodic solution of the equa・ tion（0．1）of period 2πlying in D for any t in L and satisfying the isolation conditions． There exists apositive constantδsatisfying the relation（4．1）． We choose an arbitrary constantεsatisfying O＜ε ≦δ． Then there exists a number mo such that

for any m≧mo

（6．1） ρ（m）＜ε． Suppose that for any m≧mo there are two Galerkin ．エ

apProxlmatlons

x＝Xm’（t） and x＝Xmtf（t） satisfying （6．2） IIX肌’一元lic≦ε and l1X．”一元ll6≦ε． Denote that l）、＝｛X；lX一欽τ）≦εfor some t in L｝⊂D． Then for any m≧mo臨’（t）and Xm”（t）1ie in the domain 1）、 for any t in L． Let us put y（t）＝Xm’（t）一 di．”（t）． It follows that d y（t）＝P己X［t；顧］−X〔t；Xm”コ｝（6．3） dt

where

（6．4）＝φ［t；ゐ司y（り＋Σψゴ［t；メ＝1 2］y（t一ωプ）＋h（り Applying Lemma l to the equation y（り＝

which implies

A_{（6・5） i12川Q≦ll正∬‖（∼｛，6LllyHQ十［lh［IQ｝・} On the other hand， from the function （6．4）it follows that （6・6）Uhll・≦・（m）｛Kll・1｜・＋凱｝＋K∫li［（・mtt＋の一釧・｜1・il・d・・

Moreover we obtain

（6・・7）2／ll，≦Klfi・li・

and

（6．8） ll（Xm”十θ3ノ）−211c≦ε for anyθin ［0，1］ by the relations（6．3）and（6．2）respectively． From the relations（6．5）・（6．8）it follows that （1−llHllQPt−1（ε）制IQ≦o・ Noting the isolation conditions（1．6）， we have （6・9） llyllQ＝O since the constantεwas arbitrarily chosen such that O＜ε≦δ． The relation（6．9）implies that teml（t）＝元m”（t） for any t in L． This proves Theorem 2． 7． Proof of Theorem 3 For a given periodic apProximate solution x＝菰（の of the equation（0．1）of period 2πwe put d te（り＝x［t；司＋h（り＝A（t）ヱ（り＋ゾ（り，（7．1） dt

where

f（t）＝x［t；x］−A（t）x（t）＋h（り． Applying Lemma l to the equation（7．1）， we have （7・2）・（り一轣FπH（ち・）f（・）ds・ where H（t， s）is the H・matrix corresponding to h（t）＝一（1−Pm）｛φ［t；司y（t） k 十Σψゴ［t；x］y（τ一ω」）｝ ∫＝1 ＋P・∫：｛φ［t・・・”＋・・］一φ［t・di］｝・（’）dθ

＋P・∫遺｛ψ・臨’＋θy］

一ψゴ［t；勾｝y（t一ωゴ）dθ・（6．3），we have ∫：a（ちs）｛量ψ・［・・i］・（t一ω・）＋h（t）｝ds・

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No．29

the equation （2．1）． To seek an exact periodic solution of the equa− tion（0．1）of period 2π， we consider the iterative

process

（7．3） for n＝0， the process

（7．3）can be continued

the space C

and that the relations （7・4） 11Xn＋i−−Xnllo≦（κ十1げμ）nllXi−Xollo

and

（7・5） 11Xn＋1−Xollo≦δ hold for n＝0， 1，2，．．．． In fact， for n＝O the relation （7．4） is evident． Since ・1（t）−x・（り一一轣FrrH（ち・）h（・）ds by the relation（7．2）， it follows that ［［Xl−x。11。≦［IH［1。1［hlla≦Mr≦δ． This proves（7．5）for n＝0． To prove our state・ ment by induction，1et us suppose that the iterative process （7．3） has been continued up to n−1 and we have obtained（7．4）and（7．5）up to n−1． We can make xn＋1ぴ）by the relations（7．5）and（7．3）． It follows that （7．6） x。（t）・x（の， Xn．1（t）一「2πH（ち・）｛x［、、。nコ Jo −A（s）x。（s）｝ds

Hence by

sumptions of Theorem 3 we have （7・7） IIXn＋1−Xnllσ≦11Hllo｛（κ／ノレ1）［IXn−Xn＿illc 十μll工π一Xn＿111σ｝≦（κ十」1イμ）‖Xn−Xn＿1110． This implies the relation（7．4）by the assumptiom of the induction and 21 11Xn＋i−XO［IC≦Σ（κ十Mμ）illX1−XollC i＝0 ≦［1／（1一κ一ルfμ）］［［Xi−Xollo≦Mγ／（1一κ一ルfμ）＜δ． It is easily seen that the relation（7．4）implies tt_{unif°「m c°nve「ge} 垂モ?汲?@th・・e・i・・｛x・（り｝i・th・・・＋1（り一・n（t）一轣BH（ち・）｛x［s・・・］ −x［s；x。．1コーA（s）［Xn（s）一出。．、（s）］｝ds −∫：πH（ち・）∫1｛｛φ［…n−1＋θ（・。−Xn−1）コーA（s）｝［x。（s）一出。．1（s）］ k 十Σψゴ［s；Xn．1十θ（Xn−Xn．1）］元＝1 ×［Xn（s一ω」）−Xn＿1（s一ωゴ）］｝dθds． the relations （2．2） and （3） in the as・ space C obtained by the iterative process（7．3）to afunction元（りin C．’Then by the relations（7．3）

and（7．5）we have

l1元一司lc≦Mr／（1一κ一ルfμ）＜δ

and

（7・・8）2・（り一轣FπH（ち・）｛x［・・勾一噸・）｝d・・

which implies

￡元（り一A（脚）＋x［t・hi］−A（・）di（り＝X［t；勾 Then the function 元（t） is our desired periodic soluiton of the equation （0．1）of period 2π lying in Dδ for any t in L and satisfying the relation （2．4）． In order to prove the uniqueness of the periodic solution of the equation（0．1）， we consider another periodic solution x＝2ノ（’） of the equation （0．1） of period 2π． Then we have 4，元’（り＝A（り2’（り＋x［t；釣一A（り元’（り・ which is equivalently （7・9）abt（り一轣FπH（ち・）｛獅’］−A（・）‘ii’（・）｝d・・ Using the relations（7．8）and（7．9）we obtain l1元一元’ilo≦（κ十ルfμ）ll元一元¶σ by the same arguments as those in proceeding from （7．6）to（7．7）， This implies 1｜2一釧1σ＝o by the relation（4）in the assumptions of Theorem 3． This proves the uniqueness of the periodic solution of the equation （0．1）． Finally in order to prove the isolation conditions of the solution x＝2（り，1et us consider the system ハ吻／dt＝A（りy， A（り一φ［t；勾， whose multipliers are easily proved to be all different from one and the periodic systems d d − （7・10）認＝A（t）y＋∫（りand万・＝A（り・＋∫（り・

whereプ（りis in C． By Lemma l we have

A （7．11） y＝Hf and z＝Hf，ハ where H and、H are the corresponding、H・mappings with respect to the equations（7．10）respectively． From the equations（7．10）and the relation（3）in the assumptions of Theorem 3 it follows that d − （y−z）＝ A（t）（y一の一［A（t）−A（リコz， dt

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which implies

Y−2＝−H｛［A−A］z｝

and

‖y−zllc≦ll珂1。ll［A−・4］z｜1c≦M（κμ4）11zllσ＝κ1｜zll。 From the relations（7．11）it follows that A ハ ll（H−H）∫‖。＝11y−zlla≦rcl！zlla＝rcl1Hf［［e A ≦κ‖HII。llflla for any function fin C． Then we have ム｜匡1−Hllσ≦κ｜IH‖o

and moreover

ぶムム ll酬o≦HHII．＋IIH−Hlla≦ll珂b＋rc［IH［1σ，

which implies

ハ（7．12） IIH‖σ≦［1／（1一κ）］1HIIσ≦ノレ1／（1一κ）． On the other hand， from the definitions（1．7）and （2．2）and the relation（4）in the assumptions of Theorem 3， it follows that （7．13），EZ≦μ（1一κ）／ノ匪．

Therefore by（7．12）and（7．13）we have

A ll酬。P＜1． Similarly we can prove that パ 11Hl1Q〆1． This proves the isolation conditions of the solution x＝al（t）and then completes the proof of Theorem 3．

8．An Example

We give an example of our method using the differential difference equations （8．1）

which has

consider an apProximate （8．1）in the form oder 3

（8・2）｛1：12ゴ：：1；lll：；：＝慧．

Then we obtain the constantグin the relation

（2．3） such asグ＝0．001． In order to apply Theorem 3 to our equation （8．1），we consider the matrix A（り一

k∴，；］．

The fundamental matrix R（り with R（0）＝E’of

the homogeneous equation

d

Xl（り＝x2（t）

dt

d

ZITt X・（t）＝−1・52x・（t）・−O・005［Xl（t一π）十1．59941sin t−0．00004 sin 3t］2十2sin t・ been introduced by S． Chino［4］． We solution of the equation of a Galerkin approximation of for any x and y in Dδ， where Dδ1s defined in（2）in the assumptions of This implies that the constantμ in （2．2）may be chosen such that Lt ＝＝ O．01｛2×1．59945十δ｝． d y＝A（t）y（8．3） dt is given in the form： R（り一

klllSs：∵’5＝5「

Then the multipliers of the equation（8．3）are all different from one since det［R（2π）−E］＝（1−cos 3π）2キ0． It follows that there exists the H−mapping H cor・ responding to the equation（8．3）which satisfies （8．4） llH］1Q，1）Hl［c≦（13∼／2／12）π＜5． Then we choose the constant M such that M＝5． The Jacobian matrices of the right side of the equations（8．1）are derived in the forms： φ［ちx，y］＝A（り

and

ψ［t，x， y］一［一。．。1［，1＋1．59941：。t−。．。。。。4、i。3t］1］， where x＝（x．1， x2）and y＝（Y1， Y2）． It follows that 1ψ［：t， x， y］1≦o．01｛lYi l十1．59941十〇． oooo4｝ ≦0．01｛2×1．59945十δ｝

the domain

Theorem 3．

the relation In order for all assumptions of Theorem 3 to be ful丘11ed， it is sufncient that there exist the con− stants O≦κ＜1 andδ＞O satisfying the relations（4） and（5）in the assumptions of Theorem 3， that are，（8．5） κ十5×0．01｛2×1．59945」一δ｝＜1

and

（8．6） 5×0．001／（1一κ一5×0．01｛2×1．59945−Fδ｝） ≦δ． Actually we can choose the constantsκ＝Oandδ＝ 0．006so as to satisfy the relations（8．5）and（8．6）． Therefore it is concluded by Theorem 3 that there exists uniquely an exact periodic solution x＝元（t）＝（21（t），元2（り）of period 2π of the equ− ation （8．1） in the neighborhood of the function （8．2），that is ［lab1（t）−Xl（t）12十lX2（t）一元2（t）12］1／2≦δ＝0．006． Since the exact solution x＝元（t） satisfies the isola− tion conditions， if necessary， a Galerkin approxima一

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December 1978

tion can be obtained acculately as it is desired by computing the finite trigonometric polynomial by Theorem l and Theorem 2．

References

1） Cesari L；Functional Analysis and Galerkin’s Method， Michigan Math． J皿rn．，11，385−414 （1964＞

2）Mawhin J．；Perodic Solutions of Nonlinear

Functional Differential Equations， Journ． of Diff． Eqs．，10，240−261（1971）． 3）Urabe M．；Galerkin’s Procedure for Nonlinear Periodic Systems， Arch． Rat． Mech． Analy．，20， 120−152 （1965）． 4） Chino S．；On Periodic ApProximate Solutions of Differential Difference Equations， Master’s Thesis in Yamanashi Univ．（1978）（in Japanese）．