Chỉnh hóa một số bài phương trình tích chặp

o F: X ~ Y la mQtloan tli giua hatkhonggiancljnhchllfin.Phtiongtrlnh Fx ~y (finx) c1uQcgQila chlnhne'uthoabadi€u ki~nsauday: , I. Vy E Y,3x EX saochoFx =y, 2. VX1,X2EX, ne'uFx, =Fx2 thlx,=x2, 3. V {x11}C X, ne'u Fx 11~ Fx tIll XI1~ x khi n ~ co. NhungphtCongtrlnhFx =y khongthoamQttrangdc c1i~uki~ntrenc1tCQcgQi lakhongchlnh. Nilanxet2.1.2 I. Di~uki~nt6nt~inghi~mclla phuongtrlnhFx =y c6 th~c1~tc1tiQCkhi!ta md rQngkhonggiannghi~l11. ! 2.TnionghQpphtidngIrlnhFx =y c6quamOtnghi~m,ILi'cJa Lathie'uthongtin. VI v~yl11u6nd~ltduQcHnhduy nh5tnghi~mta din Omthemthongtin v~ nghi~m. 3. D~c tnCngchinhcita phuongtrlnhkhongchlnhla tlnhkhong6n c1inhct'ta nghi~m.PhuongtrlnhkhongchinhtrenthlfcIe'khangth~giili c1u'Qcbai vi khi do d~cta luan luon m~cphili satso'trenelaki~n.Do tinhkhong6n c1jnhclla nghi<';masatso'giUanghi~mtinhloan va nghi<';mchinhxac co th~rf{tIon, dLIchosais6Lreneluki~nnho. Chu'ung2: Bal loan!chongchlnhvaphuunglrinhtIchch('ip 4, Tinh 6n dinhcuanghit$mco lien quande'nchuflnCl'tacackhonggianX va Y. MQt phudngtrlnhco th~1achinhho~ckhongchinhkhi c1uQcta bie'nd6i trongcacchuflnkhacnhau.X6t vi d~ldlioi day: Dnh t6ngchu6iFouriervoi cach~so'gftndungtrongkhonggian12, Gia sll' 00 fJt) = 2:>11cos(nt) II~O TI b?' £ ~' 1 ' ilay all 01 CII =all +-VO1 n;::: va Co=ao'taOUQc: , n (1) f2(t)=IclI cos(nt) 11=0 Khi d6 saiso'trendCi'ki~nlEt: I I { } - ( J - 00 2 0012 £1 = I(clI - aJ2 =E I2 11=0 11=1n VI chu6iso'f -\- hQit\,lneB £1 -+ 0 khi £ -+ 0, II~In Ta co 00 1 fl (t)- f2(t)=£I -cos(nt) 1I~1n Danhgiasaiso'clh t6ngchu6itrongcackhonggiankhacnhau,taco: . Ne'usaiso'cuat6ngchu6idliQCd(lnhgia trongkhonggiancaeh~lJnlien l~le C till t6ngchu6iFourierkh6ng6ndjnh, VI: { OOI } 001 IIf2 - fille =sup £I -- cos(nt) =£I - =00, ~ I 11=1n 11=1II ( I ", '" ~1 h" k' )ClUO1so 1-.J- p an y, 1I~1n . Ne'usaiso'clla t6ngchu6idudcdanhgia trongkh6nggianl} [ 0;n] lhl nha. ! dtnhly Parseva1taduQc: { n 2 } ~ { n "0, 2 \ ~ H 00 1 11[2-f1112= flfl(t)-f2(t~dr . = - I(cn -aJ = -.8I2~O 0 2 n=1 2 n=1 n khi £ -+O.Do do t6ngchu6iFourierla 6nc1jnh. * TruanghQploan tli F kh6ngcompact,phlidngtrlnhFx =y la kh6ngchinh ne'utathemgiathie'tv~RangeF. Chuang2: Bih loankh6ngchlnhvaphuangtrlnhtlchch(Lp Menh c1~2.1.3 Cho F: X ~ Y la loan ta tuye'ntinh lien t~lCva I-I giua hai kh6ng gian Banach.Gia saRangeF:;t:Y va RangeP=Y. Khi doFI kh6nglien t~lC. ClllJ'ngminh: Tli gia thie'tvi: RangeF, SHYfa: 3yEY \ RangeF, 3{yII}c RangeF saochoyn~ Y kllin~ 00 D~ty"=Px,, Vn, taco: Fxn ~ Y khi n ~ 00 Ne'uFI lien t~ICtIll 3M> 0 sao cho: Ilxnll=llp-'YnllsMIIYnll n=1,2,3... Do {Yn}laday hQi t\1nen {Yn}laday Cauchy.TiXb5t dfll1gtllLi'ctrensuy ra {xn}la day Cauchytrongkh6nggianBanachX, nghlala t6nt~ix E X va Xn~ x khi n ~ 00. M~tkhac:F lien t~lC=?Fxn"~Fx khi n -j- 00 VI gidi fwnclla day {Fxn}la duynh5tDensuyfa: Fx =Y, nghlafaYE:Ral1geF. Dii:ll naymallthu:invdi y E Y \ RangeF.V~yFI kh6nglien t~IC. [J , * PhlfdngtdnhFx =y co th~d~tC11fQctinh6n c1jnhne'llta thuhypkhonggian nghi~m. Menh c1~2.1.4 Cho X va Y fa hai kh6nggian(11nhchufln.Gia stiXI c X la mQtkh6nggian cancompactvaF: XI ~ Y Ia loanttilientl,lCva 1-1.Khi doFI lient~lC. Chungminh: X / / ? F-lettoantlf -: Range F ~ Xl Giclslfy E RangeF vaday {yn}C RangeF thc')a:Yn--t Y khi n "~ 00 Ta phaich(fngminh: p-'Yn-tP-'y khin~oo D~t F --I xn = --<yII Vn, x =F-1y Ne'uXn~ x thlt6nt~idaycan {Xnk}cuaday {x,,}saochovdim9ik taco: Ilx", -xI12E>O (2.1) Chuang2: Bdi toclnkh6ngchlnhvdphuongtrinhtfchCh(7P Mal khac,doXI la t~pcompactneBtOnt~lidaycon tXl1klfclla oay {Xl1k} Ve)dJ X()E X I sao cho: xn --+x() khi 1~ 00k, VI F lien tl,lC,SHYra YI1 = FIx" )--? F(x,,) khi I-> 00k1 \ k I ~ x =x" (00 Y11~ y) (eloF 1aI - I) ~ F(xJ =Y Tt'((2.1), ta co: Ilxl1kl-xll~£>o V~ythl: o=llx()-xll=llx-xll~£>o v0Iy. Nht(v~yphaicoXn~ x khi n --.t00. [I Kh£ng uinhcua m~nhde (2.1.4)v~nc1ungtrongtruc')nghQpX I C X la khtJllg . ? X ' X x, X" 1/ X ' I'~ " X" j ' II A.gu:lI1con eua va I = + trongco a t~lpcompactva a '-long glall conhi:'(uh~lnchieuclJaX.(Xem [41). 2.2 Phu'dngtrinh tich chgp l;!iflJu1gh I~J. 2. 1 Pht(c1ngtrlnhtinhphancod<;tng: (K*vXx)= fK(x- t).v(t)dt=u(x),xE R",11~I (22 ) R" (K, u la cachamehotrude,v la h~llnffn) du(;cgQila phu'(ingtrlnhrichchap. Ta gc,JihamK Ia nhan,hamu Ia oil ki~nclla pht((ingtrlnhtichch~p(2.2). 2.3Vi duv~tinhkhongchinhcu~hu'dngtrinf:ll!~JL~b~1p Cho loan lu' A: e(R)-~ L2(R) ut(Qc xac ujnh bc.~i c{Jng! lh(i'c .HXJ Av(x)= fK(x-t).v(t)ut,vdi KEL'(R)nL}(R). -Cf) Ta Omnghi~mv E L2(R)cuapht(c1nglrlnh : A vex)=u(x) Phu'cingtrlnh(2.3)c1U\1cvie'tot((iicl<;tngtfehch~p: (2..) ) +Cf) (K*vXx)= fK(x - t).v(t}lt=u(x) (2.cj) -Cf) Chuang2: Bai loankhongchlnhvaphuangtrinhtIchch(ip . Ta ch((ngminhr~ngb?1itoclnd;;lI1g(2.4)noi chungla kb6ngcbInb. Menh dif 2.3.1 ToantuA la tuye'ntinhlient\lC. CluIngminh: , TU' dinhly 1.3.1suyraA hoanloanX;lcdinbtrenL2(R). Vdi mQivj, V2thuOcl}(R) vas,ria dc s6tilyy, taco: +00 A[s.vJx)+r.vz(x)]= fK(x-t)[S.Vl(t)+r.Vz(t)]dt = -00 +00 +00 =s fK(x-t).vJt)dt+r fK(x-t).vz(t)dt=s.Av,(x)+r.Av2(X) -00 --00 ::::> A tuye'ntinb. TheodinhIy 1.3.1,taco: IIAvllz = IlK * vl12~ IIKIIlllvl12 ::::> A lien t~IC. , * Phlidngtrlnhtichch~p(2.4)thu'C5ngco nbi€u hdnmOtngbi~m.Xet m~nhd€ saudily: Menhdif2.3.2 , Pht(dngtrlnh (2.4)co nhi€u nhfltmOtnghi~mkhi va chI khi t~phQp E ={t ER \ K(t)=o} codOdob~ngkh6ng. ClllCllK-11lill h: Ki hi~u:mla dOdotrenR, XM la hamd~ctn(ngtrent~pM. I 1.::::»Gi;l su phlidngtrlnh(2.4)co nhi€u nhflt:mOtnghi~m,taph;lich((ngminh m(E)=O. Th~tv~y,tacobiSudi~n E=~{E(1[-n,n]},0=1 Ne'um(E) *"0 thl t6n t~inoE N sac cho 0 <m(E(1[- no,nJ) <+00 A A TaIflyVo(t)=XEn[-no,no](t)thl Vo(t)* 0 ::::> vo(x)*"0 Chuang2: Bai todnkhi5ngchlnhvaphuangtrinhtlchch~lp -------------- 1\ M~Hkhacvo =XEn[-no.n..JEL2(R):::>voEL2(R) +c.o Cho u(x)==0,phu'ongtrlnh(2.4)trdthanh fK(x -- t).v(t)dt=0 (2.5) --co Khi do phuongtrlnh (2.5)co nhi~uhon I11Qtnghi~l11.£)i~unay tnii vdi gi<l thiC't.V~ym(E)=O. 1\ 2, <=)Gia Sl(m(E)=0,nghIala K(t)1=0 h.k.ntrenR BiC'ndc5iFourierhaivC'cila phuongtr1nh(2.4)tadl(QC: 1\ 1\ 1\ K (t). vet)= oCt)=> 1\ 1\ vCt)= oCt)1\ K (t) " NC'u ~ ~e(R) th1v ~L2(R) :phl((jngtr111h(2.4)vo I1ghi~l11. 1\ . NC'uv EL2(R) vaVEL 2(R)th1phl((ingtr1nh(2.4)co nghi~mcluyI1h51. [] Menh c1~2.3.3 Phuong trlnh tich ch{lP d<;lng(2.4)n6i chunglil khong chlnh. CluIng minh: 1. roantif A JiJ 1-1 Th~tv~y,gillsuA vex)=0 => (K*v) (x)=0 => 1\ 1\ K(t).v(t)=0 (1«(t)1=O Ker A ={a} h.k.ntrenR ) => 1\ v(t)==O h.k.ntrenR => vex)=0 => 2. Stf tiin tqinghiQm 1\ Ne'uphuongtrlnh(2.4)co nghi~mvEr}(R) th1~=~ E I}(R). Nhu v~yK phuongtrlnh(2.4)chuach~cco I1ghi~111v E L\R). Ch~ngh<:tnlily u E L2(R) l11a 1\ 1\ 1\ u==2K.Theogia thie't,K E LI(R) nI}(R)=>KEI}(R) 1\ 1\ 1\ 2K - - 2 =>2K E e(R). Tuynhienv==-;:- =2~e(R) =>v ~L (R) K r-- j!:)H.kH TlrNl-tlF;;) ,Chuang2: Bai loankhongchlnhvaphuangldnh tichchcJp 3. Tfnhtindjnh .Tli chungminhd 2.,mN cacht6ngquattasoyraRangeA"* L\R) .B@apd~tngm~nhd~2.1.3tachungminhthemRangeA trom~trangL2(R). Ki hi~uCc(R)la kh6nggiancachamlien tl,lccogiacompactrenR. B~t G ={U E e(R)\ ~E Cc(R)} .TnI'och€t tachungtoG c RangeA. Th~tv~y,H(yutoyythuQcG till ~E Cc(R). VI K E L1(R) Hen J( E C(R),soyra f\ -~ECc(R) (HchmQthamlien t\lcco giacompactvoi mQthamlien t~IC). K MQi hamso'thuQcCc(R)c L2(R) d~uco bi€n d6i FouriernglI'Qc.Do c10t6nt~li f\ 2 f\ U f\ f\ f\ VEL (R) thoa v =- hay K. v=Uf\ K f\ f\ K"'V=ll K '"v = u.V~yt6n tUE RangeA => G c RangeA , Bay gio H(yu toyy thuQc L2(R), khi c10 ~IE em.). Do CcCR)trll m~ttrong f\ L\R) Hent6nt~idaytrongCc(R)hQit~lv~ u, tucla t6nt~tiday {un}c G thoa il~n- ~112~ 0 khi n ~ 00. IIUn -u112 ~ 0 khi n ~ 00 =>Un ~ U trang e(R) VI {Un}C G nen {un}c RangeA. V~y voi U E e(R), t6n t~i day {un}c RangeA tboa Un~ U khi n -+ 00 RangeA=L\R)Dodo Thea m~nhd~2.1.3,ta soyra A-I kh6nglien t\lC,nghIala nghi~mclla pl1lI'(jng trlnh(2.4)kh6ngph\1thuQclien t\lCrheadli ki~n. K€t quad 2.va 3.clIngvoim~nhc1~2.3.2,c15kh~ngc1inhphltdngtrInh(2.4) noichunglakh6ngchlnh. ._.