Luận án Bài toán hai điểm biên kỳ dị

CHU0NGI " ,::> "" A PHU0NG PHAP DIEM BAT DQNG TRONG BAI TOA.NBIEN KY Df " ,? I. CAC D~NHLY ca BAN Dinhly 1.1(Brouwer). En la khonggian tuye"ntinhdinhchu§'nhUllh~nchi~u . c la t~p dang,bi ch~ntrongEn thi ta"tcacacanhx~ f: C ~ C lien tlfcd~uca di~mc6dinh Dinh nghia 1.2 . 1) Anh x~ F: C ~ C du'<;1cgQila compacne"uF(X) chuatrongmQtt~p compaccua Y . 2) Anhx~ F: C ~ C dlf<;1cgQila compaclien Wene"uanhcuam6i t~pbi ch~nchifa trongmQtt~pcompaccuaY . 3) Anh x~ F: C ~ C du'<;1cgQila compachUllh~nchi~une"uF(X) chua trongkhonggiancontuye"ntinhhUllh~nchi~ucuaY . GQiA ={aI, a2, ..., an }lamQtt~pconcuakhonggiantuye"ntinhdinhchu§'nE vdi chu§'nII. II . Vdi S>0 c6dinh d~t : n Ac =UB(aj,s) , B(aj,s)={xEE:llx-aj II<E} i=I ~j: Ac ~ R saocho ~j =max{O,E-11x-aj II} GQiCoCA)la t~pl6i benha"tchuaA . Ta dinhnghlaphepchie"uSchauderla anhx~: n I ~I(x)a; Pc : Ac ~Co (A) sao c110Pc = i=ln I ~I (x) i =I 0 ; XE A c Nh~nxet : Cachdinhnghla PE la hoantoanca nghlavi ne"u: X E Ac ' 3io : x E B(aio' E) ~ E - II x - a,o II > 0 n ~ ~lio= E - II x - a,o II -:f= 0 ~L: ~li(X)-:f=0,=! 2) Chu y PE(AE)c CoCA)dom6iPE(x) la t6helptuytn tinhcua aI, a2,..., an. Dinh Iy 1.3 . Cho A c C c E voi A ={aI, a2, ..., an}; C la t?P l6i trongkhonggiantuytn tinhdinhchu§'nE .Ntu PE(x) la phepchitu Schauderthi : 1) PE la anhx~compaclien wc tUAE vao CoCA)c C . 2) II x- PE II < E 'v'x E AE . Chungminhdinh Iy : 1) Slf lien t\lccuaPE duQctha"ytn;fctitp . Chungtachungto tinhcompaccuaPE . GQi {PE(xm)boola mQtdaytrong PE(AE) voi V?y thidotinhcompaccuan_l?p phltdngclIngvdi CoCA) la mQtt?P dongsuy ratinhcompaccuaanhx~PE. 2) Chu Y - 1 II II x - Pc (x) 11-~(x) i =1 1 n 1 n :::; L ~lJx) II x - a, II < L ~li (X)E= E ~(x) i=l ~l(X) i=l bdi vi ~ i (x) = 0 tnr khi II x - a i II< E (x) = II li (x)aj t;! i (x)=> p" (x m) = :t,=1 (x) Chli Y voi m6i m (J(X) 2(x) II(X)) E[O,l]n., (X II) (X n) ,..., l(X II) Dinh If 1.4 (xa'pXl Schauder) . C Ia t~p16itrongkhonggiantuySntinhdinhchu§'nE . Anh x~ F: C -+ C la compaclientvcthlvdim6i8>0 , A ={aI, a2, ..., an }c F(E) la mQtanhx~ lien Wchuuh~nchieu Ft;:E -+C saGcho: 1) II Ft; - F(x) II < 8 '\Ix E At; . 2) F(E) c CoCA) c C . Chung minhdinh If : Ta co F(A) chltatrongmQtt~pcompac K cuaC . Do K bi ch~nhoanroanlien t6nt~imQtt~pA ={al , a2, ..., an} c F(E) vdi F(E) cAt; . GQiPt;: At; -+CoCA)la phepchiSu Schauderva dinhnghTaanhx~ Fc: E -+C b~ngFE(x)=Pt;(F(x», x E E rheadinh191.3taco kStqua. . Cho B la mQtt~pcuakhonggiantuySntinhdinhchu§'nE F : B -+ E . Vdi m6i 8 > 0 , b la mQt di~m trong B saG cho II b - F(b) II < 8 thl b duQc gQi la di~m8_c6 dinhcuaF . DinhIf 1.5 . ChoB la t~pcondongcuakhonggiantuySntinhdinhchu§'nE . F :B -+E la anhx~compaclien tvc thlF co di~mc6dinhnSuva chInSu F co 8_c6dinh vdim6i8>0 . Chungminhdinh If : Gia sU'F co di~m 8_c6dinh vdi m6i 8>0 . 8~tbnla di~m lIn _c6 dinhcua F. Ta co IIbn- F(bn)II< lIn (1.2) . Do F la compacsuyra F(B) chl1'atrongmQtt~pcompac K trongE do do co mQtdayconcacs6tv nhienS va x thuQcK saGcho : F(xn)-+x E K khi n -+ IX)trongS . VI v~y(1.2 ) suy ra bn-+ x khi n -+ IX) trongS vadoB la t~pdong=>x E B . DoF lientvc trenB suyra F(bn)-+F(x) . V~y taco IIx - F(x) II=0 hayF codi~mc6dinh. DinhIf 1.6(Schauder). ChoC lamQt~pcon16icuakhonggiantuySntinhdinhchu§'nE thltatca cac anhx~compaclientvc F :C -+C delicoitnhatmQtdi~mc6dinh. ....""""""...". ...... :::i.::AEla,{}£lit]ff4tQ&Jf :hung minhdinh Iy : Jung dinh111.5voi B =E =C chungtachungminhF co di€m E_c6dinh voi m~). ~'>\) .'--~ ~\~\.\. ~'>\) \.\\ ~\~\.\."\':)\...~~)\).'3,"Q \.~'-\.~\w:.~\.~~ ~C:-\.\~'-\.~'- \\.\i~ h(;lnchieu FE: C ~ C vdi IIFE(x) - F(x) II< E vdi x thuQcE vi FE(Co(A)) c CoCA) c C . Do CoCA)dongbi ch~nvi FE(Co(A)) c CoCA) chungtaco th€ apdvngdinh1:91.1 suyra XE=FE (xE) , XEE CoCA) . VI v~yII XE - F (XE) II =II FE (XE) - F (xE) II < E . . Hai anhx(;llientvc f, g : X ~ E dlfQcgQiIi d6ngluaun€u co mQtanh x(;llientl,lc H : X x [0,1] ~ E vdi H(x,O)=rex) vi H(x,1)=g(x) . Anh X(;lH duQcgQi Ii d6ng luau lien tl,lc vi tavi€t H: f ==g . Voi m6i t E [0,1] anh X? x ~ H(x,t) duQcvi€t Ii HI: x ~ E . . Chungtad~ding ki€m tradlfQcquailh~d6ngluauIi mQtquailh~tlfdng dudng . Anh X(;ld6ngluaulien tl,lcH Ii compacn€u no Ii compac. . Anh X(;ld6ngluaulien tl,lcH gQiIi co di€m co'ajnhkhongphl1thuQctren A c X nC'uvdi m6i t E [0,1] anhx(;llien tvc HIAX{t}: A ~ E kh6ngco di€m c6dinh. . GQi KA(X, C) Ii t~phQpta'"tca cacanhx(;llientvccompac F: X ~ C saochothuhyp FIA: A ~ C codi€m co'ajnh khongphl1thuQc. . Hai anhx(;llientvc F, G E KA(X, C) duQcgQiIi d6ngluau( taviC't F =:G) trong KA(X, C) n€u co mQtd6ngluaulien tl,lcH: X x [0,1] ~ E vdi Ht(u) =Hlxx{t} : X ~ E , t E [0,1] co di€m co'ajnhkhongphl1thuQc tren X vi Ho(x)=F(x) ; Hl(X) =G(x) DinhIy1.7. ChoF , G E KA(X , C), gia sll' vdi m6i (a,t) E X X [0,1]chungtaco: tG(a) + (1-t) F (a) "* a thl F=:G trong KA(X, C) ChungminhdinhIy : B~tH(x,t)=t G(x) +(1- t)F(x) ; (x,t) E X X [0,1] . Trudch€t tachungminh H Ii mQtanbX(;lcompac . La'"ymQtday ba'"tky (xn, tn) E X X [0,1] d€ kh6ng m§tHnht6ngquattaco th€ gia sll' tn~ t E [0,1] khi n ~ O'J . Do F vi G compaclienco mQtdayconS cacs6tv nhien vi F(x) , G(x) thuQcC nJ sao cho : F(xn) ~ F(x) , G(xn) ~ G(x) khi n ~ 00 trong S hdn nua do C 1dilien taco H(xn , tn) =tn G(xn) + (1- tn) ~ H(x, t) khi n ~ 00 trong S V~y H(x,t) 1amQtphep ddng1uanlien tl,lCcompac . Do t G(a) + (1-t) F (a) * a vdi m6i (a,t) E X x [0,1] lien Ht 1a di€m cO' dtnhkh6ngphl1thuQc. Cu6iclingdoHo= F , HI = G lien F ~G trong KA(X, C ) . Anh x~ F E KA(X , C ) duQcgQi1ac6tySu n6uta'tCelcacanhx~ G E KA(X, C) saochoFIA=GIAcodi€m cO'dinh . . Anh x~ F E KA(X , C ) du'QcgQi1akh6ngc6tySu n6u tdnt~i anhx~ G E KA(X, C) sao cho FIA = GIA 1adi€m cd dtnh kh6ngphl:lthuQc. Dinh Iy 1.8 . rho U 1amQtt~pconma cuamQtt~p1di C c E thIba'tky liOE U anhx~ hAng F( U )= ul) 1a c6t ySu trong Kcu( U , C). ChungminhdinhIy : GQi G: U ~ C 1amQtanh x~ compaclien tl,lcvdi G Iou= F lau= UI) chungtachungminh G codi€m c6 dinhtIeDU . Dinhnghla: lex) = { G(X); nSu x E U Uo ; n6u x E C \ U D~dangchungminh duQc 1 1a mQt anh x~ compac lien tl,lC . Tli 1.6 suyre\.. 1 codi€m cO'dinh U E C . K6t hQp vdi lex)= ul) E U vdi x EC \ U chungtacou E U, VIv~yU=leu)=G(u) vado G lau= Uo douE U suyraGcodi€m c6dinhu v~y F lac6tySu . ...,)"E:' ", <:.'..:..,.,Xi ~I: Wiliw.lt'WEW.~* q~~18" . . J '..... DinhIf 1.9 . GiclSl( (X,A) la mQtc~ptrongC c E . C la mQtt~pl6i trongkhonggiantuyen tinhdinhchugnE .Tacocactinhcha'tcuaF saDdayla tt(dngdl(dng: 1) F la khongc6tyell . 2) Co mQtanhX<;id €m co'dinhkhongph\!thuQcG E KA(X, C) saochoF ==G trong KA(X, C ) Ghichu : C~ptrongC conghlala c~pg6mt~pcontuyyX trongC vat~pcon A dongtrongX . ChungminhdinhIf : . Ta chungminhill'1) suyfa 2) . G9iG E KA(X, C) saochoFIA=GIAladi€m c6dinhkhongph\!thuQc. Ta co: t G(a) + (l-t) F (a) * a . D€ nh~ntha'ydi~unay tagiit Sl( :J a EAt G(a) + (l-t) F (a) =a do FIA =GIA => G(a) =a mall thu~n . Dodinh1:91.6tasuyra F ==G trong KA(X, C ) . Ta chungminhtu 2) suyfa 1) . G9i H : X x [0,1] ~ C lamQtd6ngluaulient\lctuG vaoF saocho Hlxx{t} la mQtddi€m co'dinhkhongrangbuQc vdi m6it thuQc[0,1] . D~t B ={ x : x =H(x , t) , t E [0 ,I]} . Neu B = ~ thl vdi m6i t E [0,1] , Ht khongco di€m c6 dinhvi theriengF khongco di€m c6 dinh suy ra F khong c6tyell . NeuB * ~ tacoB n A =~ , B la t~pdong. D€ tha'ydl(QCdi~unay tala'y XnE B tlic la Xn=H(xn, tn) ~ E vaXn~ x t6n t<;iit E [0,1] va mQtday concacs6tVnhienS saochotn~ t khi n ~ 00 trong S . Do S\(lien t\lccua H tacox =H(x , t) suyra x E B v~yB la t~pdong. G9i 'A : X ~ [0,1]la hamUrysohn lien l\!Cvdi 'A(A) =1, 'A(B)=0 . Binhnghlaham1t(x)=H(x, 'A(x)t); (x,t) E X x [0,1] , tac6 1,la mQtanbX<;i compaclien l\!c .Tadi chungminh1tladi€m c6dlnhkhongph\!thuQCva 1tlA=HtlA . Th~tv~ytachuy 1t(x)=x nghlala H(x , 'A(x)t)=x suyrax thuQc B vi v~y'A(x)=0 vaH(x,O)=x suyra mall thu~nvi Ho=G la di€m 0]'djnh khongph¥thuQc. Vi v~y1tladi€m co'djnhkhongph¥thuQc.Taco lieU x E A thi 'A(x)=1 va1t(x)=H(x , 'A(x)t) =H(x , t) =Ht(x)vi v~y1tlA=HtlA. I£)~t =1 suy ra Jt 1aanh Xc;tcompaclien tvc va la di~mco'ajnh khongphl;tlhuQC Do F =HI Den F kh6ng co'tye'u . DjnhIy 1.10 . Gia su(X,A) Ia mQtc~ptrongC c E . C la mQtt~pl6i trongkh6nggiantuye'n tinhdinhchu§'nE . Gia slf F va G la haianhXc;ttrong KA(X, C) saGcho F ~G trong KA(X, C) thl F co'tye'u ne'uva chi ne'uG c6tye'u . ChungminhdjnhIy : D~dangsuyratUdinh191.9. DinhIy1.11( Leray- Schauder). Giaslf C lamQt~pl6itrongkh6nggiantuye'ntinhdinhchu§'nE . U lamQt t?PconmacuaC , p* E U thl ta'tca cacanhXc;tcompaclien tvc F :U -7 C d6uco it nha'"tmQttronghai tinh cha'"tsail : 1) F codi~mco'dinh . 2) 3xEau vdi x =AF(x) + (l -A) p' , A E (0,1) ChungminhdjnhIy : Chungtacoth~giasu Flau ladi~mc6dinhkhongphVthuQctrongtru'onghQp: khongx§'yra . f)~t G: U -7 C la anhXc;th~ngu-7 p* . X6t ph6pd6ngluan compaclientvc Ht: U -7C lienketgifi'aGvaF la H(x,t)=tF(x)+(1-t)P* . X6t haitru'ongh9P: 1) H(x,t) ladi~mc6dinhkh6ngphv thuQctrenau . 2) H(x,t) kh6ngladi~mc6dinhkh6ngphVthuQctrenau . Neu tru'onghQp1) xayrathltUdinh191.6va 1.8suyraF ph.hcodi~mco'dinh Ngu'Qcl ;tineutru'onghQp2)xayfathl: 3XE au x= AF(x) + (l-A)P* vdiAE[O,l] TacoA:;t:0 vI P*~au va A:;t:1VI F lauladi~mc6dinhkhongphv thuQc. ~"'i "',::'' ::::'",:m,;',;I:;;&:1}::::I $ ~1I'1{« iM, 18 4i:~:,' i"j",]t,:ij'::'..:':' m@1 m" Dinh Iy 1.12 . Giii sU'A la t~pcondongcuakhonggiancacham C ([a,b],R) . Ntu A bi ch~n d~uva lien t\lcd6ngb~cthlA la compact.. DinhIy 1.13(Arzela-Ascoli) . GiiisU'A la t~pcondongcuakhonggiancachamC([a,b], R) . Ntu A bich?nd~uvalientl)cd6ngb~cthl A la compact. :~,-~'!.1f~lgrft{~m§~ ... 2. UNGD{)NG BaygiOtaapdvng1:9thuyttv~digmba'tdQngdS chungminhsll t6nt~inghi~m cuaphu'dngtrlnhvi phanco d~ng: 1 - (py,)'=q(t)f (t, Y, py,) pet) Voicacdi~uki~nbien I. XET sTJTONTAl NGHIEM CUA PHUONGTRINH: (1.1) 1 - (py I)' = q(t)f (1,Y,pyI ) pet) - exy(0) +~lim p(t)y,(t) = C1-70+ ay(1) +b lim p(t)Y1(t) =d1-71- Voi f: [0,1]x R2~ R lien tvc q E C(O,1) ; P E C[O,l]n C\O,l) p , q >0 tren(0,1) (1.2) t E (0,1) t E (0,1) a>O,f3:20 a~O,b~O a2+b2>0 Chungtac~nchungminht6nt~ihamYE C[O,l]n C2(0,1), py'E C[O,l] thoamanh~(1.1) Ta xeth~bai loanco d~ng: 1 - (py ')' = f~q(t)f (t, y, py ') P (1) - exy(0) + f3lim p(1)Y,(t) = C1-70+ ay(1) +b lim p(t)y ,(t) =d1-71- t,A E (0,1) (1.3) Dinhly 1.14. GiaSltp, q, f thoamandiSuki~n1.2va: fl ds < 00P (s )0 I f P (s)q (s)ds < 00 0 (1 .4) Gia sU'themt6nt(;lih~ngs6M >0 dQcl~pvdi A saacha : max{sup I yet) I,sup I p(t)y'(t) I} =max{1y 10,1py'lo}=Iyll~M te[O,I] te(O,!) vdi m6inghi~my cuaphu'dngtrinh(1.3)va m6i "A E (0,1) . Thl phltdngtrinh(1.1)sec6nghi~mYE C[O,1]nC2(0,1), py'E C[O,1]. Chung minhdinh ly: Giai bai tm'ln(1.3)tlfdngdu'dngtim 1hamYE C[O,1]vdi py'E C[O,1]thoaman: t ds t 1 1 yet) =B +Af -- Af -f q(x)p(x)f(x, y(x), py')dxds 0 pes) 0 pes)s (1 .5) Trang d6 : t d I I (ae+da) +A(af~ fpqf(x, y,py')dxds+Bfpqf(x, y,py')dx A= op(s)s 0 I d a(a f~ +b)+a~ 0 p(s) B =~A - C - ~A f q(x) p(x) f (x, y ,py ')dx a a 0 Chungtavie'tl';li1.5 t ds t 1 I yet)=(1- A)CO+A(CO+BI + Alf - -f -fpqf(x,y,py')dxds (1.6) 0 pes) 0 p(s)s t ds co= B2 + A2I peS) ac+dex A2 = 1 ds ex(af- +b)+exp 0 p(s) J3A2-C 0.7) B2= ex Vdi : BI = B Al - ~fq(x)p(x)f(x, y, py' )dx ex exo (1.8) 1 1 1 1 a(exf- fpqf(x,y,py')dxds +J3fpqf(x,y,py')dx op(s)o 0 Al = 1 ds ex(af -- +b) +aJ3 0 p(s) (1,9) D~tKI[O,1]={UEC[O,1]:pH'E C[O,1] vdi IU Id la 1khonggianBanachva Kko[0,1]= tuE K 1[0,1]:-exu(O)+ ~~~~p(t)u' (t) = au(1)+ b ~~~p(t)u' (t) =0J Dinh nghIa toaD t:ltM : KIBo[O,l] ~ K1Bo[0,1] sao cho : t ds t 1 I My (t) = B 1 +A 1 f- - f - fq (x )p (x ) f (x, y , py ') dxds 0 pes) 0 pes) s Slf t6n t~inghi~m tu'dngdu'dngvdi va"nde di~mc6 dinh: (1.10) y=(l-A)W+A[My+w] ==(1-A)W+)~Ny. r '' 1:JI-J"H -";r "'J. E ' N,'1. y', ,'... ,',r,I . ! TUtf \'IEN ! [ :- n03~3J. Ta chungminhM lien tQc: - Do f lien tlfctren[O,1]xR2~ R => f lien tlfcdeli tren[0,1]x[-M,Mf (vdiM la h~ngs6saocho Iy 11::;M) - \18>0,388>0:II (t,s,U)-(1',s',u') II If(t,s,u)- f(t' ,s',u') I <8. V?y vdi m6i t E [0,1] t ds t 1 1 IMy(t) - Mz(t)I::; /BI-B;I+IAI -A;lf -+ f -fpqlf(t,y,py') - f(t,z,pz')dzds O?(s) °p(s)s 1'.:.::'::::::::::::':::',].::':':':::':::::::::::,,'::::"'::::":':;~:,ili:..:.::::'::'::::.:.:::::':"":":::::,,,::::,::::::,:,':':""":.::...;:::.::"':::":@,B,lllil.liI-%I§@ a I 1I IAI -A;!::; d [as -fpqlf(t, y,py') - f(t,z,pz')ldxds +1 sop a(af-+b)+a~ s 0 p(s) 1 +~fpqlf(t,y,py') - f(t,z,pz')ldxds] ::;E 0 (do 1.4) IBI - B;I~ I~ !IAI -A;j+ I~ IIpq If(t,y,py ') - f(t,z,pz ')dx < 8 Suyra I My(t) - Mz(t) I <E (do 1.4). Vz,y E K1Bo[0,1] V~yM lien tl}c Ti€p rheatadungdinhly Arzela- Ascoli dS chungminhM la lien tl}co6ngb~c. E>Sthty du<jcdi€u nay,tad~tQ c K1Bo[0,1]Ia mQtt~pbi ch~nnghlala t6nt~i ffiQts6 Mo > 0 saGcho Iyll ::;Mo .Vdi m6i y E Q do f(t,y,py')lien tl,lco~utren [O,l]x[-M,M] nent6nt~iK> 0: If(t,y,py')I::;K . K€th<jpvdi (1.4)taco : vy E Q 3A*, B* la cach~ngs6 saGcho IE*I::;B va IA*I ::;A (A*,B*co thS phl) thuQcVaGM ) .V~ys1jbi ch~n cuaMQ Ol«jCsuyratli (1.10) Ti€p rheavdi YE Q ; t, s E [0,1]: 1 P(t)(My)' (t) =Al - J P(x) q(x) f (x, y(x ), py,)dx t => s Ip(t)(My)' (t) - p(s)(My)' (s)l::; I f p(x)q(x)f(x, y(x),py')dxlt Va t dx I I t 1 1 IMY(S)-My(t)lsIA"I I J- + J-jp(z)q(z)f(z,y,py')dzdxl s p(x) s p(x), V~yV t,s E [0,1]:It-sl<Edo rinhlien tl}ccuafva do (1.4)tasuyra: I p(t)(My)' (t) - p(s)(My)' (s)1<E IMy(s) - My(t) 1<E V~yMQ la lien tl}co6ngb~c. [I Ke'tlu<%nM la mQtloantiTlien tvchoanloan(haycongQila compactlien tvc) £)~t: K~[O,l]={uEK][0,1]:-au(O)+~limp(t)u'(t)=c,au(1)+blimp(t)u'(t)=d } t~O- t~l- U=~uEK1B[0,1]: luI1<M+l}; C=K1B[0,1] ;E=K1[0,1] Ap dvngdinh191.11vdi p*=conhu'ngvdi slj hfa chQnU nhlf trentinhchtt 2) cua 1.11kh6ngtheKayra v<%yN co diemc6 dinhtlic la phlfdngtrInh(1.1)co nghi~mYE C[O,I], py'E C[O,I] , y E c2 (0,1)la do 1.5vdi A =1. II. XET SUTONTAl NGHIEMCUA PHU0NGTRINH: (2.1) 1 - -Cry ')' = q(t)f(t, y,py ') P(t) lim p(t)Y ,(t) =C I~ o' ay(1) + b Jim- p(t)Y ,( t) = d 1 1 t E (0,1) a > O,b;::: 0 Trangrtt nhi€u lingdvngthljctt di€u giasiT J ds < 00 -kh6ngthoaman vi dV:pet)=to-1 (n ~2). 0 P (s ) V<%ytase thaygia thitt 1 ds J < 00 0 p (s ) B~nggiathitt : 1 1 s f fp(x)q(x)dxds<00 op(s)o X6t ffiQthQbai loan: (2.2) 1 -Cry ')'= Aq(t)f(t,y,py ') P (t) Jim p (t ) Y ,( t) = C I o' ay(l) +b Jim p(t)Y,(t) =d t 1- t E (0,1); A E (0,1) a>O,b~O Dinhly 1.15. Gia Sltf, p, q thoamandi@uki~n(1.2)ya 1 f P(x )q(x )dxds < CfJ 0 lIs , f -Jp(x)q(x)dxds < CfJ op(s)o (2.3) HdnmlagiaSlt:J h~ngs6M dQcl~pvdiA saGcho I y11 s:;M ydi m6inghi~my cua phu'dngtrlnh (2.2)va vdi m6i A E (0,1).Khi do phttdngtrlnh (2.1)se co nghi~mYE C [0,1] n C2 (0,1) , py' E C [0,1] . Chungminhdinhly Giai phttdngtrlnh(2.2)tu'dngdu'dngydi s\(fimhamYE C [0,1]ydi PY'E C [0,1] thoa man: d tIs b 1 yet) =- +A[ f -f p(x)q(x)f(x, y,py')dxds- - fpqf (x, y, py ')dx a op(s)o ao 1 I 1 I - - f-f p(x)q(x)f(x, y,py')dxds] aop(s)o d d d = (1- A)-+ A[-+ My(t)] ==(1- A)-+ ANy(t) a a a Trong do: M: K1Bo[0,1]-7 K1Bo[0,1]ydi: tIs b 1 My (t) =f - f p(x)q(x) f (x, y,py,)dxds- - f pqf (x, y,py,)dx op(s)o ao 1 I 1 1 - - f - f p(x) q(x) f (x, y, py,)dxds] aop(s)o Kko[0,1]={uEK1[0,1]: limp(t)y'(t)=ay(1)+blimp(t)y'(t)=O } t->O' t->f K1[0,1]={uE C[O,l] , pu'E C[O,l] } Lam tu'dng1\tnhu'trongdjnh191 trongph~nI taco M 1amQttmlntti'Compact lient\lc. f)~t: U =~uE K1B[O,1]:Iu 11<M+l ~ c =~K1B[0,1]~, E=K1[0,1] Kko[0,1]={uE KJ [0,1]:limp(t)y'(t)=0, ay(l)+blimp(t)y'(t)=d} (->0+ t->f Ta co toclntU'N co di~mc6dinhtucla phu'dngtrlnh(2.1)co nghiQm YE C [0,1] II C2 (0,1)vdi py' E C [0,1] . III. XET SUTONTAl NGHIEM CUA PHUONGTRINH: 1 -(py')'= q(t)f(t,y(t),py') pet) limp(t)y'(t)=a 1 0+ limp(t)y'(t) =b(-->1- t E (0,1) (3.1) X6t phltdngtrlnh: ~(PY')'= Aq(t)f(t,y(t),py') pet) Jimp(t)y'(t)=a(-->0+ limp(t)y'(t)=b1-->1- A , t E (0,1) (3.2) DinhIy 1.16. Gia sU'f(t,y,py'),pet), q(t)thoamandieukiQn(1A) va (1.5). N€u tdnt<;tih~ng s6M dQcl~pvdi A saochom6i nghiQmcuaphltdngtrlnh (3.2) thoamandieu kiQn: Iyll =max{sUPtE[o,l]ly(t)1, SUPtE(o,l)lp(t)y'(t)l} =max{Iylo,Ipy'10}~M thikhi a'yphudngtrinh3.1co nghiQm. ChungminhdinhIy . X6t phudngtrinh : 1 -(py')'= 0 pet) limp(t)y'(t) =0(-->0+ limp(tt."(t) =0 (-->]- t E (0,1) (3.3) phu'ongtrlnhnaychiconghi~mt§mthudng.V?y taco th€ bi€u di~nnghi~n: cuaphltongtrlnh3.2dudid~ng: yet)=A;31 (t) +B;.Y2(t) +SY\ (s)y 2(t) - YI (t)y 2(s) q(s)f(s, yes),py' )ds 0 w(s) tfongdoYl(t) , Y2(t)la hainghi~mdQCl?p tllyentinhcuaphltongtrlnh ( '"' 4'.). , _ ( 1 ) (py')'=0 ta co th€ gia SlYla : limp(t)y~(t) ;i: 0 di€u nay la hoanp t Ho' toaDco th€ xiy fa VI ne'ukh6ng lin~p(t)y~(t)= . lirqp(t)y;(t) =0khi do1->0 1->0 ta chQn U(X)=YI(X)lil1}p(t)y~(t)-Yix)lin!p(t)y~(t), d~dangki€m traduQc 1->1- I->C HeX)la mQtnghi~mcuaphuongtrlnh3.3 V?y u ==0 mall thu~nvdi s~tdQcl?p tuyentint cuaYl , Y2.Taco : A =aA2- bA] +A-As;. A3A] - A2A4 B =b-A3A;. ;, A I ] y](s)limp(t)y~(t) - Y2(s)limp(t)y;(t) As =AJ H]- () H]- q(s)f(s,y(s),py')ds0 w s AI =limp(t)y~(t) , A3 =limp(t)y;(t)1->0+ 1->0+ A: =lil~p(t)y~(t) , A4 =lil~p(t)y;(t)1->1 1->1 Trongdo w(s) la hamWronskian cuaYl,Y2t~is taco pw'(s)=0 SHYfa pw =C =h~ngs6 (taclingc§nchliy la AIA3 - A2A4;i: 0 bdi VIneukh6ngtachQn HeX)=A1Yl(X) -A3Y2(X) la mQtnghi~mcua phuongtrlnh 3.3 suyfa u ==0 mall thu~nvdi svdQcl?p tuyentint cuaYl , Y2). Ta viet l~icongthuc (3.4): yet) = 'A[CYI (t) +Dy2(t) +J Yl (s)y 2(t) - YI (t)y 2(s) q(s)f(s, yes),py')ds] + 0 w(s) + (1- 'A)[EYI(t) +Fy2(t)] a - EA 3 aAz - bA 1 A 5 +aAz - bA I a- CA 3 F= , E= , C= , D= , A] A3Az - A]A4 A3Az - A]A4 Al f)~t : K~={YEC[O,l]}, PY'EAC[O,l] , l~p(t)y'(t)=a, lim,p(t)y'(t)=b}1--+] 1--+0 Va N K] ~ KJB B Ta sechungminhN la toantUcompac,lien Wc . Chung minhNy, p(Ny)' E Kin: +Xet t~p[0,1]x[-M,M]2 (trongd6M la h~ngs6saDcho Iyll~ M) Do f :[0,1]xR2~R la lien t\1clien lien t\1cd~utren[0,l]x[-M,M]2 suyra : VE >0 ::188>0 : II (t,u,v) - (t' ,u' ,v')11 < 88 thl I f(t,u,v)-f(t' ,u' ,v') 1<E . V~y vdi t, tl E [0,1] : I t-tl I < 88 suy ra II (t,y,py') - (t',y,py') II< 88 do d6 If(t,y,py')-f(tl,y,Py')kE/2 . Do f lien t\1cd~u lien tacling c6 t6nt~i h~ngs6 K saDcho I f(t,y,py')I ~K do d6 t6nt~iC*, D* la cach~ngs6saDcho : IC I ~ C* , IDI~D* (C , D co the'ph\!thuQcvaoM ). V~y: I Ny(t)-Ny(z) I ~ C*ly](t)-yJz)I+D* I yz(t)-yz(z) I + + I Y2(t) II] yJs) q(s)f(s, y, py')ds I+I YI(t) I] Y2(S)q(s)f(s, y, py')ds I t w(s) 1 w(s) + IY2(t)-Y2(Z)II] yJs) q(s)f(s,y,py)ds1+ t w(s) + IYI(t)-YJ(z)II]Y2(s)q(s)f(s,y,py)dsl t w(s) ~ C'lyJt)-YI(z)I+D'IY2(t)-Y2(z)1 IYo(t)1 z + - supI yJs)11 Jp(s)q(s)f(s,y,py')dsl c SE[O,I] + I Y (t) I z , + 1 SUp I Y2(s)11 Jp(s)q(s)f(s,y,py )dsl C SE[O,J] + + I Y2 (t) - Y2 (z) I SUp I Yl (S) I KJ +I Y1 (t) - YJ (Z) I SUp I y2 (S) I KJ sE[O,l] sE[O,I] I (vdi KJ ~KJ p(s)q(s)ds)° Tu'dngt\( ta CO: I p(t)(Ny)'(t)-p(z)(Ny)'(z) I ~ ~ C'I p(t)y;(t)-p(z)y~(z) I +D'I p(t)y~(t)-p(z)y~(z) I + + I p(t)y~(t) II J yJs) q(s)f(s,y,py')dsI+Ip(t)y'Jt) I J Y2(S)q(s)f(s,Y,py')dsI- t w(s) t w(s) + I p(t)y~(t) - p(z)y~(z)111Yl (s) q(s)f(s, y, py' )ds I + t w(s) + I p(t)y;(t)-p(z)y;(z) II JY2(S) q(s)f(s,y,py')dsl t w(s) . Chung minh N lien ttJc : Do f :[0,1]xR2~R la lien tlJ.Cnenlien tl}cc1~utren[O,l]x[-M,Mf SHYra : \is >0 3be>0 : II (t,u,v)- (1',u',v')11<be thl I f(t,u,v)-f(1',u',v') 1<s . V?y vdi t E [0,1]c6c1inhvalI(t,y,py')- (t,z,pz')11=lI(y,py')-(z,pz')1I< be ta co : INy(t)-Nz(t)1 ~ ly](t)IIC-C'I+IY2(t)IID-D'I+ + S tYI(S)Y2(t)-Yl(t)Y2(S) ( )If( ' ) - f( ' )Id Iq s s,Y,PY S,Z,pz s 0 w(s) I C - C'I = I As - aA2 - bA1 - A~- aA2 - bAI I ~ I As - A~ I A3A2 - A4AJ A3A2 - A4Aj A2A3 - A4A] IAs-A~1 ~ 1YI(s)limp(t)y~(t)-Y2(s)limp(t)y~(t) ~ IAI If Hl- () Hl- q(s)lf(s,y,py')-f(s,z,pz')0 w s I d I1 1 - I S - ~ KEfq(s)ds ~ KE(fp(S)q(s)ds)2(f_)l ~ EKKI ~ E. 0 0 opes) A => IC-C'I <E , ID-D'I = 1-2IIC-C'1 <E . Al V~yN lien tvc . f)~tU={uEKB1:IU/l<M+1};C=KBl ;E=K1={UE C[O,l] ;PU'E C(O,!)} Ap dvngdinh191.11trongbai 1ta suyraN co diSmc6 dinhtltc1ab~tiloan3. co nghi~m UE C[O,l] ; pu' E C(O,l) . IV.XET Su'TONTAl NGHIEMCUA PHUONGTRINH; 1 -(py')'= q(t)f(t,y(t),py') pet) limp(t)y'(t) = limp(t)y'(t)t-->O+ t-->l- y(O) = y(1) t E (0,1) (4.1) Tltdngtv nhlttrongph~nIII tad~t phltdngtrlnh: 1-(py')'= Aq(t)f(t,y(t),py') pet) limp(t)y'(t) = limp(t)y'(t)t~O+ t~l- y(O) = y(l) A, t E (0,1) (4.2) Bjnh If 1.17. Gia su' f(t,y,py'),pet), q(t)thoamanGi~uki~n(1.4) va (1.5).Ne'ut6nt~lih~ng s6M dQcl~pvoi A saGchom6inghi~mcuaphu'dngtrlnh (4.2) thoamandi~uki~n: Iyll =max{sUPtE[o.l]ly(t)1, SUPtE(o,1)lp(t)y'(t)l} =max {Iylo,Ipy' 10}::;M thlkhi§yphu'dngtrlnh4.1conghi~m. ChungminhdjnhIy . xetphu'dngtrlnh 1 -(py')' = 0 pet) Jimp(t)y'(t) =t~O+ y(O) = t E (0,1) Jimp(t)y'(t)t~l- y(l) (4.3) phltongtrlnhnaychiconghi~mt~mthu'ong.V~ytaco th€ bi€u diennghi~m cuaphu'dngtrlnh4.2du'oid(;lng: yet)=A"Yl(t)+B"Y2(t)+ JYl(S)Y2(t)~ ~1(t)Y2(S)q(s)f(s,y(s),py')ds0 w s (4.4) trongdoYl(t) , Y2(t)la hainghi~mGQcl~ptuye'ntinhcuaphu'dngtrlnh: 1 - (py')'=0 tacoth€ giasU'laY2(0)-Y2(1)7=0di~unaylahoantoancoth€ pet) x§yfa VIne'ukhong :12(0)-Y2(1)=Yl(O) -Yl(1)=0 khi do tachQn: HeX) =YI (x)[limP(t)Y'2(t)-lim p(t)y~(t)]- Y2(x)[limp(t)y;(t) -lim p(t)y~(t)] 1->0' 1->1+ 1->0+ 1->]' dSdangnh~ntha'yHeX)la mQtnghi~mcuaphu'dngtrinh4.3 lien u ==0 di€u nay matithu~nvdi stfdQcl~ptuySntinhcuaYI , Y2 Trangd6w(s)lahamWronskiancuaYI , Y2t~lis tac6 pw'(s)=0 SHYrapw =C =h~ngs6 taclingd~nchuy la [Y2(O)- Y2(l) ]10-[YI(O)- YI(l) ]II * 0 bdi VI n€u khongtachQn: u(x) =[Yl(1)- yJO) Y2(1)-Y2(O)]Y2(X)-Yl(X) . D~tha"yHeX)Iii nghi~mcua phu'dngtrinh 4.3 suy ra u ==amall thuc1nv6i s1/Q9C l~ptuySntinh cuaYI , Y2. Tac6: A = "A (12+IJ ). [Y2(0)-Y2(1)]Io-[yJO)-YI(1)]II II =t~~p(t)Y~(t)-~~}lp(t)y~(t) B =AJYI (0)- YI (1)]+"A 12 ). [Y2(0)- Y2(l)] 10=t~~p(t)Y~(t)=~~}lp(t)Y;(t) 13 = [y 2(0) - y , (1)]S Y 1(S) ~~}l p (t)Y ~ (t) - Y 2 (S) ~~~p ( t )Y; ( t ) - 0 w(s) q(s)f(s, yes),py')ds I =SYI(S)Y2(t)-Y2(S)YI(t) 2 0 w(s) q(s)f(s, yes),py')ds B~ngcach chungminhtu'dngttfnhu'trongph§n III ta SHYra S\tt6nt;~linghi~!J, cuaphu'dngtrInh4.1.