86
Lp-StabilityintheLargeofNon~Linear
今 卜Differential‑DifferenceEquations
SEINo KuDo ASO Shoichi
Miki Masamichi
(Receivedon310ctober,1974)
1. Introduction
WhatfurtherrestrictionsarenecessaryandsufficientfOreverysolutionoftheordi‑
narydifferentialequationstobeintegrableonthehalf‑linetotherightoftheinitial
point？
ThisquestionwasaskedbyL.Cesari,inl963andansweredbyhimforsecondorder lineardifferentialequations[1].LevinandNohel[2], [3]alsohaveobtainedsome resultsinthisdirectionusingLiapunov'ssecondmethod.Toprovideamorecomplete answer tothisquestion,AaronStraussfirst introducedandstudiedanewkindof stability,i.e､ ,Lp‑stabilityofordinarydifferentialequations, inhisdoctoraldissertation [4]. Inl967, D． RamakrishnaRaoextendedAaronStrauss'sresults tonon‑linear differential‑differenceequations
(1) x'$(0)=f(xt,t)
②y'0(0)=g(y$,t).
Inthatpaper[5],hestudiestneLp‑stabilityofthesystems.
ThepurposeofthispaperistoextendhisresultstoLp‑stabilityinthelargeofthe systems(1)and(2).Moreover,weextendthecontinuabilityofthesolutionsofordinary differentialequationtothecaseofdifferential‑differenceequation.
2． DefinitionsandNotations
Letldenotetheinterval O≦t<ooandRnisthespaceofnvectorsandforxERn,
￨￨xll isanyvectornorm.Givenanumberh>0,Cdenotesthespaceofcontinuousfunc‑
tionsmappingtheinterval [‑h, 0] intoRnandforpEC, ￨￨?￨￨‑SUpll?(6)ll.Ccowill
−h二9≦O
denotethesetof IDECforwhichlkpll<ooandD｡｡=I×Coo×Coo.Foranycontinuous functiony(u)definedonh≦u≦A,A≧0, anyfixedt, 0≦t≦A, thesymbolyt willdenotethefunctiony(t+0),‑h≦0≦0, i. e. , ytECand is thesegmentof the functiony(u)bylettingurangeintheintervalt‑h≦u≦t.
Letf(9，t）andg(少,t）benon‑linearinPand4jrespectivelyandbecontinuousint，
IDandint,'forallt≧Oand9， 少ECH,H>0.
Letx'､(0)denotetheright‑handderivativeofthefunctionx(u)atu=t.Letf(?,t) andg(少,t）beLipschitzianinPand少withLipschitzconstantL・ Let to≧0andlet
秋田高専研究紀要第10号
Lp-StabilityintheLargeofNon-LinearDifferential-DifferenceEquations 87
IDECHbeanygivenfunction.Afunctionxl(to") issaidtobeasolutionof(1)with initialfunctionipattimeto, ifthereexistsanA>Osuchthat
(i) x'(tO,のisdefinedforeacht, to≦t≦to+Aandx$(to,のECI,,
（ii）x ，(to,の＝I｡，
(iii)x'l(0)=f(xt,t), to≦t≦to+A.
Similardefinitioncanbegivenforthesolutionyi(to")oftheequation(2).
LetV(t,9,Wb)beacontinuousint, WDandf"fort≧0, ID, 'EC｡｡、
ThederivativeofValongthesolutionsof(1)and(2)willbedenotedby
●
V(,),e) andisdefinedas
十{V(t+h,x;"(t,,),y"｡(t,,))̲V(t",,))
V(,),⑨（t,？,妙）＝ lim
h→0＋
wherelpand'arethesolutionsof(1)and(2)respectivelyattimeto Wegivethefollowingdefinitions.
[Definitionl]
System(1) issaidtobestablewithrespectto(2), ifforeverye>0andto≧0, there exists6(to,e)>0, thatiscontinuousintoforeache>0andsuchthat ll ID− ￨￨<6(to,e) implies llx6(to,fD)‑y､(to,fj)￨￨<e forall t≧t｡.
[Definition2]
Theequilibriumof(1) issaidtobeLp‑stableinthelargewithrespecttotheequi‑
libriumof(2), if(1) isstablewithrespectto(2), thesolutionsof (1)and(2)existon
!:llx&(t',,)̲y@(t,,')￨￨d[
〔to,CO）forevery（to,甲)， （to,｡)臣I×Coo, and the integral convergesforevery(to"),(to")EI×Coo．
[Definition3]
V(t,x,y) ismildlyunboundedonD｡｡ ifforeveryT>0,V(t,x,y)→ooas llxll→ooor
￨lyll‑→oouniformlyonthesetO≦t≦T、
3. PrelixninaryResults
Inl965,AaronStraussprovedthenexttheorem[6].
【TheoremA】
Leff(t,x)6eCO""""0"s""d/ocα"yZ,jpsc"鰯α"o"I×Coo.T"e〃肋gsO畑加〃F(t,to,xo) Qfｵ"eSyS花加
¥‑f(t,x)
(E)
Ca〃6ecO""""edro [tO, oo)んγg""y(tO,xO)EI×Cooがα"do"ﾉγが肋e"ee"Sおα "一
"eg"""e, 籾"〃y〃"加"""edsca〃γ〃"C伽〃V(t,x)d賊"g"o"I×Coo Sα"a/WzgMe
ん"0伽"gcO"成加"S;
ShoichiSEINo･MikiKuDo・MasamichiAso 88
(a)V(t,0)=0九γα〃t≧0,
(b) V(t,x)jscO""""0"so"I×Coo，
(c) V(t,x)isんcα"y〃，sc"zjα〃0〃I×Coo一{(t,0);t≧0},
●
(d) V(E)(t,x)≦00"I×Coo．
F"""γ柳O"e, オ〃s/""c加〃V(t,x)iS加s伽"ed賊"伽がα"do"/y〃肋e〃""め ""o/
(E) issm"9．
D.RamakrishnaRaoextendedAaronStrauss'sresults tonon‑lineardifferential‑
differenceequations.Werestatehismainresults[5].
【Lemma】
Leff(?,t)α"dg(',t)sα"sカオ舵α加"gCO"成加"s.Z,e〃舵so/"伽"Sx (to,？)α"my (to, ） or(1)""d(2) "es'ec""g/ysa抗カメ"eco"""o"
llx0(tO,の一y!(to,')￨￨≦K(to)e‑["(t)‑α(t@)] ￨IID− ￨￨,
""gγeMe/""c加犯α(t)isco""""o"s""一dec''easi"gα"d加SSeSSeS"CO"が""0"sde"@ﾉα伽g 九γt≧0α"d"ﾉ'"eK(t)isα加""α〃ん"c伽"．〃んγSOw@e9,
0<q<1, オ〃"ee"S応α〃""6"T>Os"c〃＃加＃，九γα〃t≧0, K(t)e‑q["(t+T)‑"(G)]≦1,
肋e〃肋e"ee"S応α／""C伽〃V(t",')co""""0"Sj〃t， ？α"。 んγα〃t≧0, iD,少巨CHo, HO>0, s"c〃オ伽r
ll甲一少￨￨≦V(t〃,少)≦K(tルー少ll,
●
v(,),(卸(t,？, )≦一(1‑q)α'(t)V(t",､),
IV(t,?,,',)‑V(t,?2,'2)￨≦eLTSUpe(x(t+f)一α(t)[￨￨?,‑?211+￨￨']− 211].
O≦で≦T
【TheoremB】
"V(t,xc(to,?),yL(to,4))sα"蛾esMeα加@ﾉeass"加加0"g"e""Le加加α，〃V〃s"c〃
j肋/〃saGMﾉα胸eαﾉ0"g#"esoﾉ"伽"sor(1)α"d(2) is
V(,),0(t,x&(to,の,y$(to,'))≦一CllX$(",?)‑y&(to,')￨￨p
んγα〃t≧tO, リリ, 'ECHo,HO>0α"dんγso"@ec>0,p>0．T舵〃＃加幻""めγ畑加"(1) js
●
Lp‑sm"ez0"んγes'ecオ加オ"ggq""めγ畑加"(2).
4. Results
【Theoreml】
〃肋e"ee"Sおα〃ﾉ〃y〃幼O""〃。〃"c伽"α1V(t〃, ）αg方"〃o〃，｡。sα"s""gオ"g ん"0伽"gco"d"0"s;
（i） V(t〃,少）isco""""0"so"D｡｡，
●
（ii）V(,),②(t,？,少）≦00"Dc｡,
ｵ加犯肋esOﾉ"伽"sQf(1)""d(2)c""beco"が""ed#o[tO,oo)んγe"el'y(to,P), (tO,')E I×Coo・
Proof・ Supposethatforsome（to,P）巳I×Coo，x (to,のcannotbecontinuedto〔to,CO)．
ThenthereisanumberT>tosuchthat llxC(to,ID)il→ooOr lly$(to,4)￨￨→ooaSt→T4.For
秋田高専研究紀要第10号
Lp-StabilityintheLargeofNOn-LinearDifferential-DiffercnceEquations
89instance,wesuppose llx0(t,")￨￨→"ast→T‑0． SinceV(t,ID,')ismildlyunbounded wemaychoosersocloSetoTthatV(s,xr(to,の,yr(t,,'))>V(t0",')uniformlyon thesetO三s≦Tandsothatto<r<T.Inparticular,then
V(で,x管（to〃),y雷（to, ))＞V(t0，1','),
isacontradictionbecauseV(t,xi(to"),yL(to,4))≦V(to",')forallto≦t≦Tbycondition (ii)．Weobtainthesameresultas lly$(to,4)￨￨→CO
●【Theorem2】
U"〃γｫ"edzss""@'"0〃伽T舵0γgw@1,"es""ose肋〃V(t, ID,少) sαがS伽s オルeα加"c ass""ゅ伽"sg"g""Z,e加加α.L"V(t",')"s"c〃ｫ肋ｫ伽伽""α物eα肋"g#"esoﾉ"伽"s Q/(1)""d(2) is
V(,),(2)(t,X (to,の,y (to,の)≦一CIIX&(to,の一yL(to,のli '' んγα〃t≧to",｡EC｡｡， α んγso"zec>0,p>0.
T加伽ォ"e〃"j"〃畑加Qf(1) isLp‑s加"g畑＃加加,gez""〃γespecfio"e〃""めγ畑加or(2).
Proof.WithassumptionsofV(t",'), theequilibriumof(1) isstablewithrespect totheequilibriumof(2),followedfromtheworkofHale[7].ByTheoreml,there existsthesolutionsof(1)and(2)on[to,oo).Nowlet
t
I)￨x､(t,,')̲y@(t@,')￨￨,d上
γ(t)＝V(t,x (to,？),y (to,少))＋c
to
forallt≧to.Lettbefixedin[to,oo),
厩十[1'(t+h)‑7(t)]
≦師十〔V(t+hox"．(t,,),y ･"(t,少))‑V(t,x (t"),y (t,'))〕+cllx (t ,,)一y (t．, )￨￨・
=V(,),(9(t,xI(t｡"),yL(to,'))+cIIx!(to")一y&(t,,')ll!'
≦0 ．
Hence7(t) isnon‑increasingin[to,oo).But 7(to)=V(to,jD,f"), therefore 7(t)≦V(to,ID,のforallt≧to.Andwehave
O≦V(t,x (to,？),y (to,少)）
≦‑!)…‑y$(to,')llpdt+V(…）
forallt≧to, sothat
恥叶yc(to,')￨剛≦÷v(崎｡州
Hencethetheoremfollows.
90
ShoichiSE,No・MikiKuDo･MasamichiAso References
(1] L.Cesari,AsyinPtoticbehaviorandstabilityproblemsinordinarydifferentialequations,
2nded.,AcademicPress,NewYork,1963.
