Luận văn Ứng dụng phương pháp bậc TôPô trong nghiệm tuần hoàn của phương trình vi phân

ỨNG DỤNG PHƯƠNG PHÁP BẬC TÔPÔ TRONG NGHIỆM TUẦN HOÀN CỦA PHƯƠNG TRÌNH VI PHÂN VÕ HOÀNG TRỤ Trang nhan đề Mở đầu Chương1: Bậc Tôpô của những toán tử. Chương2: Nghiệm tuần hoàn của phương trình vi phân thường. Kết luận Tài liệu tham khảo

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CHUaNG 2 NGHlltM TUAN HoAN CUA PHUONGTRINH VI PHAN TmJONG 2.1.Dfnhly chinh: La"y1=[0, 1],vaf: I x RD~ RD, thoanhii'ngdi~uki~ncaratheodory,vaky hi~uI x I la chu§:nEuclidecua x E RD,va(x,y) la tichvahtidngcuaxvay. Trongchtiangnay,chungtasechungminhst!t6ntC;iinghi~mchobai tmin. (2.1) { x1(t)=f(t,x(t)) , tEl x(o) = x(1) nhii'ngnghi~mnaysedtiqcgQila 1tu~nhoan. Chungtaky hi~uX lakhanggianconcuaC (1,RD)manhungph~n tll'cuanothoadi~uki~nthuhaitrong(2.1)vdichu§:nd~uthtiongdungla Ixl =maxlx(t)\' 0 tEl Z =L 1(I, RD),vdichu§:nthtiongdungla IIxiiI =fixet)ldt I va domLla khanggianconcuaX, manhungph~ntii'cuano la lient\lC tuy~tdot Anh x~L vaN l~nltiqtdtiQCxacdinhtrendomLvaX, bdi (L.x)(t)=x'(t), (Nx) (t)=f(t, x(t)) Vdi tEl, thlL vaN la'ynhunggiatritrongZ,vabailoan(2.1)ttiang duangvdivi~cgiaiphuongtrlnhthugQn Lx =Nx 25 ngoaifa, tll slj matatrongchuang1, KerL ={x E damL : xCi)=x(o), Vt E I} =ImP IrnL ={z EZ : fZ(I)dl=o}=KerQ d dayPx =x(o), Qx =fz(t)dt I VI the'L la anhx'].Fredholmvdi chisO'zerova do tinhchat1.5,taco N la L - hoanloanlien tl;lCtrongX. Ta c~nm(>tb6d€ d€ chungminhslj t6nt'].inghi~m. B6 d~2.1: Chor >0vaV E Cl(Rn,R),thoaman V' (x)* 0,vdi I x I =r ? d dayV' la gradientcuaV, vaIffy G :X ~ Z,duQcxacdinhbdi (Gx) (t)=- V' (x(t)), tEl va H =L - G, vdiX, Z vaL duQCxacdinhnhutren. ThlH E CL(B(r))va!DL(H,B(r))!=\Do(V',B(r))\ Chungminh Ta xetanhx']. H :XxI~Z (x,A) H Lx - AGX- (1- A)QGx Vdi Q duQcxacdtnhd ireD. Thl H la L - hoantoaDlientl;lcireDX x I. Ne'u(x, A) E X x I, saGcho H (x,A)=0,thlx lalienwctuy~td6i1- tu~n hoanva 26 (2.2)x'(t)=- AV'(X(t))-(l-A)fV'(X(S))ds, '\itEI I Do do x' lienwc trenI va liy tichvo huanghaiv~(2.2)vaix'(t), tichphantrenI vadungHnhchit 1- tu~nho~mcuax,taduQc. fix'(012dt=0 I Vi v~yx(t)=x(o), '\it E I, bdi vi x'(o) =-AV'(x(o))-(1- A)fV'(x(s))ds I Bdi (2.2),x(o)thoaphudngtrlnhV'(x(o))=0 f)i~unayguyrading \xlo =Ix(o)\"* r Theotinhchitbit bi~nd6ngluaucualythuy~tb~c,taco: DL(H,B(r)) =DL(H(.,l),B(r))=DL(H(.,O),B(r)) =DL(L - QG,B(r)) NhungQG:X ~ ImQ,vai z=ImL~ImQ Vi v~y, bdi tinhchit (1.13),taco \DL(L - QG,B(r))1=\Do(-QGKerL,B(r)nKerL)1 =IDo(V',B(r))! d dayB(r) duQcky hi~uquac~utam0, bankinhr. B6 d~duQc chungminh. Bay giC1tachungminhdinhly chinhcuachudng. Binh Iy 2.2. Gia sadingnhungdi~uki~nsaildayxayra 27 (i) comQtV E C1(Rn,R+),vdi : Vex) ~ +00,ne'u I x I ~ 00 va a ELI (I, R+),thoaman (2.3) (V'(x),f(t,x»)saCt) Vdi mQix ERn, mQitEl (ii) T6n t~ir >0 vaW E C1(Rn\ B(r), R), saocho (V' (x),w' (x» >0 VdimQix, Ixl ~r,va (2.4)f(W'(x(t),f(t,x(t)))dts0 I Vdi mQianhx~lien tl,lctuy~td6i 1- tugnhoan x : I ~ Rll,vdi minlx(t)1~rtel Thl baitoan(2.1)co it nha'tIDQtnghi~ID ChUngmink . Ta mu6napdl,lngdinh19(1.17)vdi F =L - N vaH =L - G nhtttrongb6 d~(2.1)va tinhcha'tcQngtinh,clingvdi V' (x) "* 0, cho IDQi x E Rn vdi I x I ~r . H eCL(B(p»)vdimQip~r £)gutien chungta chI ra ding nhii'ngnghi~m1- tugnhoancuahQ nhii'ngphu'dngtrlnh. x'(t)=- (1 - A) V' (x(t» + Af (t, x(t», tEl, AE10,1[ la mQtti~nbi ch~n. Ne'udi~unaykh6ngKayfa, seco mQtday(An)nEN*vdi AnE ]O,l[ 28 vamQtday (xn)nEN*, vdi IXnlo2n vaxnlamQtnghi~m1- tu~nhoancua (2.5): x~(t)=-(l-An)V'(Xn(t))+Anf(t,xn(t)), tEl, n EN* Ngoaifa, vdim6itEl, n E N*, dungdi~uki~n(i) (d/dt)V (Xn(t))=(V'(Xn(t)),X~(t)) =-(1- An).IV'(xn(t))12+An(V'(Xn(t)),f(t,xn(t))):s;aCt) Md fQngXnva a fa R bdi 1 - tu~nllOan,tadu'Qcvdi m6iT E R va m6i t E [T, T + 1] v(xn(t)):S;V(xn(T))+ r a(s)dsJf Do tinh 1- tu~nhoancuaXn,di~unaysuyfading (2.6) maxV( Xn(t)) :s;mill V(xn(t))+Ilal!ltEl tEl Bay giO,n€u tnE I saocho \Xn(tn)1=maxlxn(t)1=\Xnl 2 n tEl 0 Taco maxV(Xn(t))2 V(xn(tn))tEl va bdi di~uki~n(i) V(Xn(tn))~oo n€u n ~oo. Ngoai fa bdi (2.6) millV(xn(t))~ 00n€u n ~ 00 tEl Hoan loan tu'dngtt,I'suyfa ding minlxn(t)1~ 00ne"un ~ 00 tEl 29 La'ynl E N* saocho,voi m6in ~nl minlxn(t)j~r tEl Bi~usailduQcsuyratti'(2.5) (d/ dt)W(xn(t))=-(1- An)(V'(xn(t)),W'(xn(t)))+An(f(t,xn(t)),W,(xn(t))) Va VIv~y,bditinh1- tuftnhoancuaXn,vavoin ~nl, ta suyra tti' (ii) 0=-(1- An)f(V'( xn(t)),W'(xn(t)))dt+AnJ( f(t,xn(t)),W'(xn(t)))dt<0 I I di~unaymallthu~n. VI v~ynhG'ngnghi~mcuahQnhG'ngphuongtrlnh1ati~nbi ch~n,vOi ti~nbi ch~nduQcky hi~ubdip.Chungtacoth€ chQnp saochop~r. Baygiodungb6d~(2.1),giadinh(i)vatinhcha't(1.15),chungtaco IDL(H,B(p))! =IDo(V',V(p)~=1 Vabdi dinh1y(1.17),dinh1yduQcchungminh. Binb Iy 2.2' : Gia sanhG'ngdi~uki~nsandayKayra (i') T6n t~iV cl (Rn,Rr), voi Vex) ~ +00ne'u I x I ~ 00 Vaa ELl (I, R+)saocho (2.3') (V'(x), f(t, x))~-aCt) voi mQix ERn, mQitEl (ii') T6n t~ir >0 vaW E Cl (Rn\B(r),R), saGcho . (V'(x), Wi(x)) > 0-,voimQix, Ix! ~rva 30 (2.4') f(W'(x(t)),f(t,x(t)))dt?::0 I Vai mQiX : 1~ Rnlient~ctuy~td6i 1- tuftnhO~lllvai minlx(t)l?::r. tEl ChUngminh D6i bi~n,bftngcachd~t =1- T Vai mQitEl coTEL Luc niiy(2.3') vii (2.4')trdthiinh (2.3') (- V'(x(1-T)),f(l- T,x(l- T)))?::-a(l- T) ~ (V'(x(1-T)),f(l- T,x(l- T)))~a(l- T) SI(-W'(x(1- T)), f(l- T,x(l- T))) d(1- T) ?::0(2.4) Bftngcachd6ibi~nmQtIftnnua1- T ='t Ta co (2.3')vii (2.4')trdthiinh (V'(x('t)),f('t,x('t)))~a('t) f(W'(X('t)),f('t,X('t))}t't~0 I Ap d~ngdinh19(2.2),dinh19duQcchungminh. 2.2.Ung d\lngcuadinhIy 2.2. Trongphftnniiy,chungtasechomQtviii ungd~ngthuvi cuadjnh19 2.2, vai sl;l'll;l'achQnd~cbi~t.Ung d~ngdftulien Iii chophuongtrinhvo huang(n=I). H~qua2.1: Giasan=1vii a.etEl, f(t,.)Iii khongtang.Thl biii loan(2.1)co 00 mQtnghi~mn~uvii chin~ut5nt~iy E L (I, R) saocho 31 (2.7) J f( t,y(t))dt=0 I Chungminh * Di~ukit%nc~n: Ta IffyY HimQtnghit%mcua(2.1),suyfa J f(t,yet)dt=J y'(t)dt=y(l) - yea)=0 I I * Di~ukit%ndu: VI f(t,.)la khongtangnentaco,voi mQix E R va a.etEl, xf(t,x) ~x f(t,O)~ \f(t,O)I.(lxl+1) Do dodi~ukit%n(i) cuadinhly (2.2)thoamanvoi ( 1 1-1 V(x) =G)( ul +1) du aCt)=I f(t,0) I. Bdi VI Vex) ~ +00 ne'uI xI ~ 00 Vaa ELI (I, R+),saGcho (V' (x),f(t, x)) =x(\xl+1)-I.f(t, x) =(Ix!+lr1 x f(t,x) ~(lx\+lr1\f(t,0)1.(lx\+1) ~\f(t,o)1=aCt) Ta Iffyx E domLlaph~ntii'tilyY saGcho: minlx(t)1~Ilyll =f tel <X) 32 Dungtinhdondi~ukhongtangcuaf(t,.),tacoa.etEl, x(t) . f(t,x(t))sx(t). f(t,yet)) Suy fa fX(t).f(t,x(t»)dts fx(t) .f(t,yet))dt I x(t)1 Ix(t)!I I S:I: ff(t,y(t))dt=O I 2 11 x -- Nhu'ngla'yw(x) =2 12 u 2du x W'(x) =~ Suy fa 1 WEe (R \B(r), R) V'(x). W'(x) =(Ixl+It 1:(°,vdi .Ix I ~r Va Jw'(x(t) );f(t,x(t))dt= J1:~:~l"f(t,x(tJ)dt,;; 0I I Vdi x : I ~ R lien tl;lctuy~td6i 1- tu~nho~m.VI v~ydi~uki~n(ii) cuadinhly (2.2)xayfa,theodinhly nay,h~quadu'Qchangminh. H~qua2.1' : Giasan=1vaa.etEl, f (t,.)lakhonggiam.Thlbaitmln(2.1)co 00 mQtnghi~mne'uvachIt6nt~iy E L (1,R) saocho (2.7') ff(t,y(t»)dt=O - I 33 Chungminh * Di6uki~ncftn: Ta la'yy la mQtnghi~mcua(2.1),suyfa Jf(t,y(t»)dt=Jy'(t)dt=y(1)-y(O)=O I I * Di6uki~ndu: Vi f(t,.)la khonggiam,nentaco,vdim6ix E R va a.et E I, x.f(t,x)~x f(t, 0)~- \f(t,O)I(lx!+1) Dododi6uki~n(i') cuadinhly (2.2')thoaman,vdi l J -1 1 ~2 ! Vex)=(2)r u2+1 du. aCt)=!f(t,O)1 BdiviVex)~ +00 ne'uI x I ~ 00 vaa ELI (I, R+) Saocho: (V'(x),f(t,x»)= x(lxl+lr1f(t,x) =(Ix!+1)-lXf(t,x)~(lxl+lr1[-\f(t,O)I(lxl+1)] ~-\f(t,O)\=-aCt) Ta la'yx E domLlaphftntii'tilyY saocho Minlx(t)1~Ilylloo=f Ta dungtinhdondi~ukhonggiamcuaf(t,.),a.et E I, xCi). f(t, xCi»~ xCi)f(t, yet) 34 Suyra J X(t).f(t,X(t))dt;:::JX(t) f(t,y(t))dt IX(t)1 I IX(t)1I I ;::::tJf(t,y(t))dt=O I 1 x2 _! Nhu'ngla'yW(x)=2£2 u 2du W' (x)=x(t) IX(t)1 1 SuyraWEe (R\B(r),R) V' (X) .W' (X)=x(lxl+1r11~>0, vdi I x I ;?: r Vii JW'(x(t)).f(t,x(t))dt= J1:i:~I.f(t'X(t))dtz0,I I - Vdi x : I ~ R lien t~ctuy<%td6i 1- tuftnhoan.Do dodi~uki<%n(ii') cuadinhly (2.2') thoaman,apd~ngdinhly nay,h<%quadu'Qchungminh, H~qua2.2: Gia sttt6nt(;l.ir >0vaa E Ll (1,R+),saocho (2,8) (x, f(t, x)) ~ aCt)(lx12+1),a.et EI Vdi mQix E Rll,va (2.9) J(X(t),f(t,x(t)))dt~O, vdimQix E domL I minlx(t)l;:::r teI Thl bai loan (2.1)co it nha'tmQtnghi<%m 35 Chungminh 2 11 2 ,.:' 1 xl - x Lay Vex)= -~: (u+ 1) 1du, W(x) =- 2t 2 Suy fa ( 2 ) -1 V'(x) =x Ixl +1 , W'(x) =x Ta ki~mtradi8uki~n(i) va (ii) cuadinh 1y (2.2). Ta c6 V E C1 (Rn,R;.), vdi Vex) ~ +00 ne'u I x I ~ 00 (V'(x), f(t, x)) =(x(lxl2 +It,f(I,X)) ~a(t{lxI2+1)(lxl2 +1)-1 =aCt) Vdi mQix E Rn, a.etEL Do d6 di8u ki~n(i) cua dinh1y(2.2)thoaman: M~tkhac W E C1(Rn\B(r), R) (V'(x),W'(x))= (x(JxI2+It,~ >0 Vdi mQix, I x 12f , va f(W'( x(t)),f(t,x(t)))dt= J(X(t),f(t,x(t)))dt~0 I I Vdi mQix E domL minlx(t)12r tel Do d6 di8u ki~n(ii) cuadinh 1y (2.2)thoaman. 36 V~ybai tmin(2.1)co it nha'tmQtnghi~m H~qua2.2': Gia sut6ntq.ir >0 vaa ELI (I, R+),saocho (2.8') (Xf(t,X))2-a(t)(lxI2+1), a.etE1,vamQix ERn, va (2.9') f(X(t),f(t,x(0))dt20, vdimQix E domL I minjx(012r tel Thl baitmin(2.1)co it nha'tmQtnghi~m Chungminh 1 i 12 I ,2La'yVex)=- x (u+1)-ldu , W(x)=~2 2 SuyraV' (x)=x(!xj2+1)-1 , W'(x)=x Taseki~mtradi~uki~n(i') va(ii') cuadinhly (2.2'). TacoV E C1(Rn,R+),vdi Vex) ~ +00 , n6uI x I ~ 00 (V'Cx),f(t,x))=(x(lxl2+0-1 ,fCI,X))~-aCtJ(lxI2+1)(jxj2+r 2 - aCt) a.et E I vamQix ERn. Do dodi~uki~n(i') cuadinh1y(2.2')thoaman. M~t khac W E C1 (Rn\ B(r), R) (V'(x), W'(x» = (x(lxI2+It ,X)>0 37 Voi mQix, I x12r Va f(W'(x(t)),f(t,x(t))}it=f(X(t),f(t,X(t)))dt20 I I Voi mQix E domL,minlx(t)12r tel Do dodi~uki~n(ii') cuadinh1y(2.2')thoaman. V~ybaitmin(2.1)coit nha'tmQtnghi~m. H~qua2.3: Gia sacor >0 saocho (2.10) (x,f(t,x))~0, a.etEl vamQix E Rn Voi I x I =r Thl baitmin(2.1)co itnha'tmQtnghi~m ChUngminh Ta dinhnghIa g : 1x Rn ~ R0, bdi get,x)=f(t,x) neu I x I ~r, g(t, x) =+ -1:1)x+r(t'I:10,neu I x I ;:: r Bdi sl,txaydl,tngtren,g clingthoanhii'ngdi~uki~ncaratheodorynhtt f vatrungvoif trenI x I ~ r. Ngoaifa,neua ELI (I, R+)thoaman. I f(t,x) I ~aCt),a.etEl vamQix ERn, ydi I x I ~r, thl (x, g(t,x)) ~ aCt)(lxf +1), vdi mQix E Ro, a. et E I. Do dodi~uki~n(2.8)cuah~qua(2.2)thoamanchog 38 Neu x E domLthoamanminlx(t)1~r. tel Taco (2.11) (x(t),g( t,x(t»))=-( 1-lx~t)Jlx(tf+(x(t), f( 1,Ix~t)1X(t)] J ~- ( 1-~ J lx(t)12~OIx(t)! La'ytichphantrenI, di~uki~n(2.9)~uah~qua(2.2)thoamanchog vabailoan. (2.12) { x1(t)=g(t,x(t»),t EI x(o)=x(1) Co it nha'tmQtnghi~m.Ta chungminhnghi~mx naythoaman Ix(t)\~r ,tEl . NeuchungminhduQcdi~uki~nnay,thlnghi~mcua(2.12)clingla nghi~mcua(2.1). Neilx lamQtnghi~mcua(2.12),thlbai(2.11) Tacoa.etEl sacchoI x (t)I >r (2.13) G)(d/ dtJlx(tJl2=(x(tJ,g{t,x(t))) ~_ ( 1-~ )lx(t)12~0Ix(t)1 Dodonghi~mnaykhongth~la 1- tuftnbeanvathoamanI x(t) I > r vdi ffiQitEL SuyracomQtt'E I thoaman I x(t')I ~r . Neu ba'td£ngthuctrenkhongxayra vdi mQit'E I, co t" E I, t":;et' ma Ix (t") I>r. 39 Md fQngx vdi 1- tugnhO~lllfa mQtanhx~lien wc tfenR, tacoth€ giltsli' dingt" E [t', t'+l]. Bdi tinh lien tt;1c,o mQtkhoangmd ]ti, t2[c ] t', t' + 1 [ thoaman t" E ] ti, t2[, I x(t) I> f, vdi t E ] tl, t2[va Ix(tl) I =I X(t2)I =r nhung(2.13)suyra I X(tl) I > I X(t2)I mati thu~n V~y I x(t) I ::;r,mQit E I, h~quaduQcchungminh H~qua2.3': Gia sli't6nt~ir >0 saocho (2.10') (x, f(t,x));:::0 a.et E I vamQix E RDvdi I x I =r Thlbailoan(2.1)coitnhfftmQtnghi~m Chungminh Ta dinhnghla K: I x RD-+RD f(t,x) , ne'u Ix!::;r get,x) =~ ( r ) ( r JI-Ix! x+f t,!x( , ne'u Ixl;:::r Bdi s1;1'xay d1;1'ngnhu tren, g cling thoa man nhii'ngdi~uki~n Caratheodorynhufvatrungvdif trenI xI ::;r Ne'ua E Ll(I, 14)thoaman I f(t,x) I ;::: - aCt) a.et E I vamQix E RD~vdi I x I::;r, thl (x,g(t,x));:::-a(t{lxI2+1),vdimQix E RD,a.etEl 40 Do do di~u ki~n(2.8') cua h~ qua (2.3') thoa man cho g. Ne'u x E domL thoaman minlx(t)12rtel Ta co (2.11') (x(tJ,g(t,x(tJ))= (1-lx~t)I)X(tJI2+(X(tJ,r( t, Ix~tJI X(t)) J 2 ( 1-~ )lx(t)120x(t) La'ytichphantrenI, di~uki~n(2.9')cuah~qua(2.3')thoamancho gvabailoan. (2.12') { X'(t)=g(t,x(t)) ,tEl x(o)=x(1) co it nha'tmQtnghi~m. Ta chungminhnghi~mx naythoamanI x(t)I ~r , tEL Ne'uchung minhdl1qcdi~unay,thlnghi~mcua(2.12')clinglanghi~mcua(2.1),di~u nay sek€t thucdl1qchungminh. Ne'ux Ia mQtnghi~mcua(2.12'),thlbdi(2.11')tacoa.etEl sao chox(t)>r. (2.13') G)(d I dt~x(t)12=(x(t),g(t,x(t)))~(1-lx~t)I)X(t)12>0 Dodonghi~mtrenkhongth~Ia 1- tu§nhoanvathoamanI x(t)I>r vdimQitEL Tdidaychungminhhoanloantu'dngtvnhl1h~qua(2.3). H~quadadl1qchungminh. 41 2.3Dngd~ngchonhilnghamVectd. Ta sechoddaymQtungd\lngcuadinhly (2.3)de'nphudngphapcua nhunghamcohuang. Giasaf thoamannhungdi~ukit%nduQcmatadphgndinhly chinh Binh nghla2.1. V E Cl (Rn,R) duQcgQilamQthamhuang(ng~t)chophudngtrlnh (2.14) x' =f(t, x) ne'ut6nt~ir >0 saDchoa.et E I vamQi x E Rllvai I x I ~r taco (2.15) (V' (x), f(t,x)) ~0 « 0). Chungtasechungtos\!'t6nt~icuamQthamcohuangthoamanmQt vaidi~ukit%nb6sung,thlsuyras\!'t6nt~icuamQtnghit%m1- tugnhoan Tinh cha't2.1: Gia saphudngtrlnh(2.14)comQthamcohuangV thoaman V' (x) :f=0vai I x I ~r Va (2.16) Vex) ~ +00 ne'uI x I ~ 00 thlbai loan(2. 1)co it nha'"tmQtnghit%m Chungminh Bdi nhungdi~ukit%ncaratheodoryva tinh lien t\lCcua V' t6n t~i a E L 1(1,R+)saDchoa.et E I vamQix vai I x I ~r Taco I V' (x) 1.1f(t, x) ~aCt). ~~tkhac,dung(2.15) Ta coa.et E I vamQix E RD (V'(x),f(t,x» ~aCt). Do do di~ukit%n(i) cuadinhly (2.2)thoaman,vai nhunganhx~ V vaaviladuQcxacdinhnhutren. 42 N6u x E domLthoamanmill I x(t)12r tEl dung(2. 15),taco a.etEl (V'(x(t)), f(t,x(t)))~0, Suyra f(V'X(t)),f(t,X(t)}lt~O, I L1y w(x) =Vex),suyra f(W'(x(t)),f(t,x(t)))dt~Ova (V'(x),W'(x))>O, I nhuv~y,di~uki~n(ii) cuadinhly (2.2)xayfa, vai r duQcgiai thi~utrong dinhnghla.Do do theobai tmln(2.3),tinhch1tduQcchungminh. Tinh cha't2.1'. Gia sli'ding phudngtrlnh (2. 14) co mQthamco huangV thoaman. V' (x) :;to0 vai I x I 2 r (2.17)Vex)~ - 00 n6uI x I ~ 00 thlbaitmln(2.1)coitnha'tmQtnghi~m ChUngminh La'yW(x) =- Vex), X ERn, thl W'(X) =- V'(X) :;to0, cho mQix E Rn ~. I I >VOl X - r, W(x) ~ +00 n6u I x I ~ 00 (W(X), f (t,X))2 0,Va a.e tEl mQi x E Ril I x12r lamnhutrongchungminhcuatinh cha't2.1. Ta co (W'(x),f(t,x))2 - aCt), a.etEl mQix E Rilvaa ELI (I, R+),va f(W'(~(t))f(t,x(t)))dt20, I 43 Vai mQix E domL,mill I x (t)I ~r.Dungdinh192.2' Chungtaco,bai tmin(2.1)co it nhcltm(>tnghi~m fJtnh nghia2.2: V E C1(Rll,R) gQila m(>thamg~ncohuangchophuongtrlnh (2.18) x' =f(t, x), neut6nt~ir >0 saocho,a.et E I va mQix E Rfivai I x I ~r taco (2. 19) (V' (x),f(t,x))~O. Tinh cha't2.2: GiastYdingphuongtrlnh(2.18)com(>thamg6mco huangV,thoaman V' (x) :;t0 vai I x I ~r Va (2.20) Vex)~ + 00 neuI x I ~ 00 Thl bai tmin(2.1)coit nhcltm(>tnghi~m Chungminh Bdi nhungdi~uki~nCaratheodoryva tinhlien we cuaV' t6nt~i a ELI (I, R+)saocho,a.et E I, mQix, vai I x I :::;r chungta co, I V' (x) 1.1f (t,x) I ~ - aCt), ngoairadung(2.19)(V'(x), f(t,x) ~0 chungtacoa.et E I va mQix E Rll, (V' (x), f (t, x)) ~ - aCt) dododi~uki~n(i') cuadinh19(2.2')xayfa.Vai nhunganhx~V vaa.vila xacdinhdtren. Baygio,neux E domL,thoamanmillI x(t)I ~ r bdi(2.19),chungta co, a.e t E I, Suyra (V'(x(t, x(t))~0 J(V'( x(t)),f(t,x(t)))dt~0, I 44 La'y W(x)=Vex),suyfa (V'(x), W'(x) >0 f(W'(x(t)).f(t,x(t)))dt~0, I Vdi f duQcgiOi thi~utfong dinh nghla,di~uki~n(ii') cua dinh1:9 (2.2')xayfa.V~y(2.1)co it nha'tffiQtnghi~ffi Tinh cha't2.2':aia sadingphudngtrlnh(2.18)coffiQthamgftnco huangV, thoaman V' (x) :;t0 vdi I x I ~f Va (2.20') V (x) ~ - 00 ne'u I x I ~ 00 Thl bai tmin(2.1)coit nha'tffiQtnghi~m Chungminh f)~tW(x)=- Vex), X E Rll, thl W'(x) =- V' (x) :;t0 vdi ffiQix E Rll ~. I I >VOl X - f, W (x) ~ +00 ne'u I x I ~ 00 (W'(x), f(t,x))~0,Va a.etEl, va ffiQix E Rll vdi I xI ~r.TheoHnhcha't(2.1)tacotinh cha'tduQcchungminh. 2.4. Ung d1}ngsf!t6n t~icuanhiingdinhly thuQclo~i Landesman - Lazer's Trangphftnnay,chungta se tha'ynhfi'ngdi~uki~nloC;\i(2.4)trong dinh1:92.2bdi ffiQtdi~uki~n,matfenffi6iquailh~xli'l:9nhugifi'aF vaV, va quailsatmQtvai tru'onghQpd~cbi~t. Tinh cha't2.3: aia sadingt6ntC;\iV E C1(Rn,R+),vdi V (x) ~ 00+khi Ix I ~ 00, Va 1 a E L (1,R+)saocho 45 (2.21) (V'(x), f(t,x))~ a (t), a.etEl, vamQi XE Rn,va Ilimsup(V'(x),f(t,x)),dt<O. I Ixl~oo Thl baitmin(2 .1)co itnha'tmQtnghi~m (2.22) ChUngmink Sad\lng(2.22)vab6d~Fatou'schungtaco limsupJ(V'(x),f(t,x))dt<0 , Ix! ~ 00 I Va VIv~ycor>0saochoV' (x)*0,mQix ERn, vdi I x I ~f dodo,di~uki~n(i) cuadinhly 2.2Kayfa vdi f ,V va a nhu'(J tren.BaygiG, chungtachungtodi~uki~n(ii) cuadinhly 2.2Kayfa vdi f nayvaW =v. Ne"udi~unaykhongKayfa,set6nt~imQtday(xn)neN*, Vdi Xn E domL vamill I Kit) I ~n,thoaman tel f(V'(xn(t)),f(1,xn(t)):fit>0 , n E N* I Dungb6d~Fatou's,di~unaysuyfa dug flimsup(V'(xn(t)),f(t,xn(t)))dt~0, r n~oo vavI v~y,khi I xn(t)I ~ 00vdi m6i tEl flimsup(V'(X),f(t,x))dt~0, I ixl~oo di€u nay,thlmallthuftnvdi(2.22),dododi~uki~n(ii) cuadinhly 2.2Kay fa. V~y(2.1)co it nha'tffiQtnghi~m. Tinh cha't2.3' Gia sadingt6nt~iV E Cl (Rn,R+),vdi V (x)~ +00 khi Ix I ~ 00, 46 Vaa E L 1(1,R+) saocho (2.21') (V'(x), f (t,x) ~- aCt), a.etEl, va mQiXE Rn,va Sliminf(V'(x),f(t,x))dt>0 Ixl~ooI Thl bai tmln(2. 1)co it nhiltm9tnghi~m Chungminh Sadl,mg(2.22') vab6d~Fatou's,chungtaco liminf S(V'(x),f(t,x))dt>o. Ixl~oo .I (2.22') Va vl v~ycof >0 saochoV'(x) "*0,mQix ERn, voi I xI ~f, di~u ki~n(i') cuadinh192. 2' thoaman,voif, v va a nhu'd tfen.Bay gio,chung tachungtodi~uki~n(ii') cuadinhly (2.2') Kayfa. Lily W =V, BAngphanchung,giasadi~uki~n(ii') cuadinhly 2.2' khong Kay fa, se t6n t~i m9t day (xn) n E N*' voi Xn E domL va minlxn(t)1~n, thoaman. tEl S(V'(xn(t»),f(t,Xn(t»)}1t<O,~EN* I Dimgb6d~Fatou's, Slim inf(V'( xn(t)),f( t,xn(t)))dt:s;0, .n~oo I cho \xn(t)\~ 00 voim6itEl, Slim inf(V' (x),f(t,x) )dt:s;0, I Ixl~oo mallthuftnvoi(2.22').V~y(2.1)coitnhiltm9tnghi~m. Bay giC1,n~uchungta,lily n =1,thldi~uki~n(2.22)(lanlu'QtIf! (2.22'),coth~thaybdidi~uki~ny~uhdn(2.22") 47 max ( flimsupV'(x)f(t,x)dt,flimSUPV'(X)f(t,X)dt ] <0 I x~oo I x~-oo Th~tv~y,ne'u(Xn)neN* lamQtdayKayratrongchungrninhcuatinh cha't2. 3, thl,bdiVI chungla nhii'nghamtht!clien t~c(coth€ gia sula day con,ne'ucftnthie't)hdnnii'aminxn(t)~n ho~cmaxxn(t)~-n. . . t~ t~ R6i sail do suyra tll (2.22"), di~unaymallthu~n,b~ngcachlam tu'dngtt!trongphftncu6icuachungrninhcuatinhcha't2.3. Bay giG,chungta xernxet mQtvai tru'GnghQpd~cbi~t,tu'dngung vdi st!chQn. ( J -1 12 (x+d) V(x) =G)t u-2- +c du, Vdi c >0 va d E [0,1],VI the'di~uki~n(2.21),(lftnIu'Qt(2.21) trd thanh (2.23) (x,f(t,x»)~a(t)(lxI1+d+c), bdiVI: [ 1+d J -1 V'(x) =x Ix! +c lfin Iu'Qt. (2.23') (x,f(t,x»)~-a(t>(lxI1+d+c) Tu'dngtt!:di~uki~n(2.22),(lftnIu'Qt)2.22'» tu'dngdu'dngvdi f (x,f(t,x»)(2.24) limsup 1+d dt<0 I Ixl~oo Ixl (lftnlu'Qt (2.24') Jlim inf(x,f(t,x)) ~ I Ixl~oo Ixl1+d dt>°) 48 bdi VI (V' (x),fer,x)) = ( 1+: 'f(t, X)J , Ixl +c Va, khi n=1,di~uki~n(2.22") (1~nlu'Qt(2.22"')) tu'dngdu'dng Vai (2.24") flimsuplxl-df(t,X)dt<0<fliminf!xl-df(t,x)dt I x-++oo I x-+oo ( Iftnlu'Qt (2.24"') flimsuplx\-df(t,X)dt<0< fliminflx!-df(t,X)dt )I x-+-oo I x-++oo *) Baygio,chungtaxemxettnionghQpvohuang(n=1)vad=0 vad~t L(t) =liminff(t,x) , f+(t)=liminff(t,x) x-+-00 x-++00 F_(t)=limsupf(t,x), F+(t)=limsupf(t,x) X-+-oo X-++OO (Hill lu(:1t f)i~uki~n(2.23)(l~nlu'Qt(2.23') tu'dngdudngvai t6nt~i 1 bEL (I, R+),saocho,a.e. tEl, fer,x) ~bet)khi x >0 vaf (t,x) ;?:- bet)khix <0l~nl11Qt fer,x) ;?:- bet)khix >0vafer,x)~bet)khix <0),vadi~uki~n (2.24"),(1~nlu'Qt(2. 24"') tudngdudngvai fF+(t)dt<0<fL(t)dt I I JF_(t)dt<O< Jf+(t)dt J . I I Chungtagiclsar~ng,vaimQix E Rnvaa.e tEl, Iftnluqt F+(t)~fer,x) ~ L(t)). F_(t)~f(t,x) ~ f+(t). 49 Bay giOgia sa r~ngf(t, x) =bet,x) + get),vdi gEL 1(1,Ril) va h : I x Ril ~ Ril thoanhii'ngdi6ukit%ncaratheodoryva saGcho,vdi mQt dE] 0, 1],a.etEl va mQix E Rntacobet,sx)=Sdh(t, x) vdi mQis ~O. Trong truonghQpnay,nhii'ngdi6ukit%nva (2. 23') ca hai d6uthoaman. rvl~tkhac dungd thuftnnha'tduongcuah, di6ukit%n(2. 24), (2.24')trd thanh (2.25') fsup(y,h(t,y))dt<0, rIY\=l f inf (y,h(t,y))dt>o r\yl=l (2.25) Tinh cha't2.4: Xet phuongtrlnhtuye'nHnh,kh6ngthuftnnha't (2.26) x' =A(t) x+get), va phuongtrlnhthuftnnha'tuonglingcua(2.26) (2.27) x' =A(t) x, vdid=1vas6 sup(y,A(t)y) \yl=l la chu§:nlogaritI-L(A(t))cuaA(t) : duQcxacdinhbdi I-L(A(t))=lirnh-1(11+hA(t)I-1) h--+O+ Ky hit%uyet)la matr~ncobanchinhcuaphuongtrlnh(2.27),chung t6i co IY(l)!~exp J I-L(A(t) )dt, r vaVIv~y,di6ukit%n(2.25)suyradug IYO)\<1,dodophuongtrInh,(2.27) kh6ngco nghit%rn1- tuftnhoankh6ngtftmthu'ong,bdiVI \Y(l)!<1tuong dliclng ydi 11:1<1, hay 1<1 voly. 50 Vi v~y,chungta chI xet phudngtrlnh(2.26)trongtruonghQpd~c bi<$tvai n=1,(phudngtrlnhvohuang). Chungtoi co ke'tqml:di~uki<$ncftnva dud~phudngtrlnh(2.26)co mQtnghi<$m1tuftnhoan,vai m6igEL 1(I, R),la fA(t)dt *0 I ChUngmink Gia sa(2.26)co mQtnghi<$m1tuftnhoan,vai m6igEL 1(1,R), thl theotren,taco di~uki<$n(2.25)ho~c(2.25')xayfa, nghlala fA(t)dt* 0, I dffchungminhduQcdi~uki<$ncftn.Bay giG,chungminhdi~uki<$nduo Gia sa fA(t)dt*0 thl ta co di~uki<$n(2. 25) ho~c(2.25') xay fa, I nghlala phudngtrlnh(2.26)co mQtnghi<$m1tuftnhoan,dinh 19dffduQc chungminh Dinh nghza2.3 : MQtt~pmabi ch~nG c RDduQcla mQt~pbienng~tchophudng trlnh(2.1)ne'um6iu E aG, comQtVuc1 (RD,R) saochonhii'ngdi~uki<$n ? sailxayra (i) G C {vERll:VU(V)<o} (ii) Vu(u)=0 (iii)Vaim6itEl (Vu'(u),f(t,u»)* o. Bay giC1giasaG c RDla ffiQtt~pconIDa,bi ch~n,va 16isaocho0 E G, thl vai m6i u E a G, co it nha'tmQtn(u) E RD\ {a}saocho (n(u), u) * 0 va (2.28): G c {vE RD: (sign(n(u),u)(n(u),v =u)< O}.n(u)nay duQcgQilamQtphaptuye'ncuaaGt~idi~mu,vad~cbi<$thdnlamQtphap tuye'ntrang,phaptuye'ngoaitudngung(n(u),u)o. 51 Tinh cha't 2. 5 : Cho G c Rn la mQtt~pmd bi ch~n,16i sac cho 0 E G, giasadingvai m6iU E 8G, m6itEl, saccho(n(u),f(t,u)) *-0vai lieU)la phapVectdcua8Gt(}.iu.thlG la mQt~pbien ng~tcho(2.1) ChUngmink Vai m6iu E 8G,chungtadinhnghlaVu E C1(Rn,R), bdi Vu(v)=(sign(n(u),u))(n(u),v-u) Ta seki~mtraca3di~uki~ncuadinhnghla2.3 xayra (i) thee(2.28),taco G c {v E Rn : Vu (v) < a}, do do di~uki~n(I) cuadinhnghia 2.3xayfa. (ii) Vu(u)=Viv) =(sign(n(u),u)) (n(u),u-u) , di~uki~n(ii) xayra (iii) (v~(u),f(t,U))=0, vai tEl. V~ytinhchit dffdu<;1cchungminh. Tinh cha't2.6: GiasacomQthamcohuangng~tcho(2.1)laW saccho (2.29) I W(v) I ~ 00 ne'uv~ 00 thl t6nt(}.imQtt~pbienng~tcho(2.1) ChUngmink Liy y>0 saccho y >maxIW(u)1 lul~p vatadinhnghiaGeRn, bdi (2.30) G={vERn:lw(v)1<y} {vERn:-y<w(v)<y} Bdi st!dinhghiacuayva(2.29)suyraG lamQt~pconrod,bich~n cuaRn,sacchoB(p) c G,va 52 8G={vERn:IW(V)=YI} ={vERn:w(v) =Y}U{VERn:w(v) =-Y} Voi m6iU E8G Tadinhnghla n Vu: R ~R Vu(y) =- w(y)- Y Vu(y) =- w(y)- Y ne'uWell)Y ne'uWell)=- Y Thl VuE c1 (Rn, R) Ta ki~mtrahaidieuki~nd~ucuadinhnghla(2.3) Voi mQiv E G, guyra- y <Well)<yrheadinhnghlacuaVu, tacoVu(v)<O. Do dodieuki~n(i) duQcki~mtra Vu(u)=y-y=O Do dodieuki~n(ii) duQcki~mtra. Ne'utEl vau ERn, I U I ~PvoiS E I Taco Iw(x(s))1~y =Iw(u)\ Tli 2.15),guyra (w'(u),f(t,u))<0 BdidinhnghlacuaVu,guyfa (V~(u), fer,U))=I;0 Do d6dieuki~n(iii) dffdu<;Jcki~mtra V~y tfnhcha'tduQcchungminh *) Tli dayv~gall,tagiiislYf la ham1- tu~nhoanrheat f (1,.) f(O,.)= 53 Do dochungtaco th~marQngf Wi R x C bai 1- tu~nhO~lll,v~n1a anh x~lien t1;1cma bie'mnhii'ngt~pbi ch~nthanhnhii'ngt~pbi ch~n,va clingmarQngX de"nkhonggian. {x E C (R, Rn): x(t)=x(t+1),voit E R} vaclingcochuffn hutru'oc Tinh cha't2.7: Gia sa t6nt~iIDQtt~pbienng~tG chobai loan(2.1)dinhnghia [""={x E C (Ir,Rn): x (t) E G, tEl}, c X, thl[""labi ch~nvax Ea[""nC1 (R, Rn)voi, x la nghi~ID(ne'uco)cua { x1(t)=Af(t,x), tEl, A E]O,I],If =[-r,I] (*) x(o)=xCI) Chungminh La'yx E C (In Rn)n Cl(R, Rn)1amQtnghi~m(ne'uco)cua(*). Dung phanchung,giasax E a [""nCl(R, Rn),vdi A E ] 0,1]thl xCI)c rva co t' E I sao cho x(t') E aG, bait' * 0vat' * 1,guyra Vx(t')(x(t)sO =Vx(t')(x(t')),tEl, VI the'v x(t')(x(.))~omQtclfcd~it~idi~mtrongt' cuaI . Dodo0=(d/ dt)(Vx(t')(X(t)))t=t'=(V'X(t')(x(t')),x'(t')) =(V'X(t')(x(t')),f(t')) di~unay,mallthuffnvdidi~uki~n(iii) cuadinhnghia2.3. V ~Y tinhcha'tda:duQcchungminh. Binh ly 2.3: Giasa dingt6nt~imQt~pbienng~tG chobai loan(2.1)vadinh nghlaf: Rn~ Rn,bai (2.31) f(u)=jf(s,u)ds, I 54 v~phai cua(2. 31),u du'qcky hi<$ula ph~ntatu'dngungvdi anhXC;lhang, nh?ngiatriu.N~u (2.32) - Do ( f , G) *-0, thlbai toan(2.1)co itnh:1tmQtnghi<$mx, saGchoxCI)c G . E>~chungminhdinhly nay,tru'dctientachungminhcacb6d~sail. B(}d~2. 2 : E>?tF =L - N, vdiN la L - compacttren0 vagiasa nhungdi~uki~nsailxayfa: (i) Lx - ANx *-0,vdim6i(x,A) E (domL\KerL) na 0) x]0,1[; (ii) Nx ~ImL vdim6ix E KerLn a 0 ; (iii) Do(QNkerL,0 n KerL) *-0, vdi: Q :Z ~ Z 1amQtphepchi~u lienwc saGchoKerQ=ImLvaQNKerLla thuh(fpcuaQNtrenKerLn O. thl Lx =Nx, coitnh:1tmQtnghi~mtrongdomLnO, trongdo. L: domLCx ~ Z la anh X;;ttuye'ntinh Fredholmvdi chI sf)zero, 0 c x la mQtt?Pconrod,bi ch?n. B (}d~2. 3 : E>?tF =L - N, vdiN la L - compacttren0 vad?tG : 0 ~ Y la L - compacttren0, vdiY la mQtkhonggianvectdconcuaZ saGcho Z =ImL E9Y (t6ngdC;lisf»),gia sa nhungdi~uki~nsailla xayfa, vdi GKerLlathuh(fpcuaGtrenKerL no, (i) Lx - (1- A)Gx- ANx*-0vdim6i(x,A)E(domLnao)X ]0,1[ ; (ii)Gx*-0vdimQix E KerL nao; (iii)Do(GkerL,on KerL)*-O. thlphu'dngtrlnh Lx =Nx coitnha'tmQtnghi~m.TrongdomLno 55 B6 d~2.4: Gia saH ECL(0) vaF =L - N vdiN: 0 ~ z laL - Compactsao cho: (i) : AFx+(1- A)Hx"*0,vdi (x, A) E (domLnaO) x] 0,1[ ; (ii):DdH, 0) "*0thlphuongtrlnh Lx =Nx co it nha'tmQtmQtnghi~mtrongdomLnO. B6 d~2.5 : B~tF=L - N vdiN: 0 -+Z laL - compactva 1: (domLnao) xI ~ Z, cod"mg: F(x,A)=Lx +G(x,A),vdi G : 0 x I ~ Z laL - compactvar(.,1)=F, giasanhii'ngdi~uki~n saula xayra (i) 0 ~r(domL.naO) x [O,I[); (ii) DL (1 (.,0),0) "*0, thlphuongtrlnh Lx =Nx, co it nha'tmQtnghi~mtrongdomLn0 ChUngminhBd d€ 2. 5. Ne'ux E domLnaO,saochoLx =Nx,thlphepchungminhdahoan thanh, n€u kh6ng,thldungdi~uki~n(i) , . 0 ~r(domLnaO)xI), va dungHnhcha'tba'tbi€n quad6ngluau, cungvdidi~uki~n(ii) chungtaco DL(F, 0) =Ddl(., 1),0) =Da'R., 0),0)"* 0,r6idungHnhcha't17.2,thl phuongtrlnhLx =Nxcoit nha'tmQtnghi~mtrongdomLno v~yb6d~ 2.5 da:du<Jchungminh. 56 Chang minhB6ai 2.4. V di m6i x E domL nOva IvE I, Ne'uH =L - K, vdi K: 0 -»Z la L - compacthl IvFx+(1 - Iv)Hx =Lx - IvNx- (1 - Iv)Kx =Lx +G (x, Iv),d day G : 0 x I ~ Z laL - compact,L +G (.,0)=H vaL +G (., 1)=F, r5iap d1,lngB6 d~2.5tacod~uphai chungminh. ChangminhB6ai 2.3. E>~tH =L - G, khi Z = 1m L EBY la t6ng d(;lisf), thl no cling la t6ng tapa, bdi VI Y la khang gian hii'llh(;lnchi~u. GiasaQ:Z ~ Z la phepchie'nlien tt,lC,saocho ImQ =Y va KerQ = ImL. Thl QG =G va VI v~yHx = 0 ne'uva chi ne'u QHx =0, (I - Q) Hx =0, Nghzala di~unaytu'dngdu'dngvdi Gx =0, Lx =0 Gx =0, X EKerL, Tu (ii), suyra H E CL (0), va dungtinhcha't1.13,taco IDL(H,O)\=\Do(GKerL,OnKerL)\. VI v~ydi~ukit%n(i) va (iii) trongb6d~ du'Qchungminh. Chang minhB6ai 2.2. Chungta,la'ytrongb6d~2.3 Y =ImQvaG=QN,r6rangG la - Compacttren0, tu di~ukit%n (ii) suy fa QNx *0 , vdi mQi x E KefLnaO, luc naydi~ukit%n(ii), (iii) trong b6 d~2.2 clingchinhla di~ukit%n(ii), (iii) trongb6d~2.3 . . Ne'u(2.23)Lx - (1- Iv)QNx - IvNx =0, nhanQ va1- Qv~oca2ve'cua(2.33),chungtaco QNx=0, Lx- IvNx=0, 57 phuongtrlnhdftu,clIngvdidi~uki~n(ii) suyradingx E (domL\ KerL) nan, VI v~yphuongtrlnhhai, mall thu~nvoi di~uki~n(i). Do do Lx - (1- A)QNx- ANx"*0, r6iapdlJngb6d~2.3. B6d~duQchungminh. Bay gio,ta chungminhdinhly 2. 3,chungta se apdl;lngB6 d~ 2.2, vdiphuongtrlnhthugQnFx =Lx - Nx=0 trongdomL c X, domL= {xEX: x thuQcC1},Lx =x', Nx =f(.,x).Bdike'tquacuachuang1,L la anhX';lFredholmvdichis6zero, KerL =ImP,ImL=KerQ, vdi P :X ~ X, Q: X ~ X. IftnluQtduQcdinhnghlabdi (Px)(t)=x(0), (Qx(t)=J x(s)ds,tEl, va I N : X ~ X la L - lientlJCdftydu,dinhnghlar bdi2034r ={x E C (lr, Rll) : x(t) E G, tEl}. Tli tinhcha't2. 7, Suyra di~uki~n(i) cuab6d~ 20 2 duQcthoaman,bay gio d6ngnha't,mQtcacht1;1'nhienKerL va ImQ vdi Rll, chungta, tha'y.dinganh X';lf dirihnghlatrong(2031).Th1,1'ccha'tla s1;1'thuh~pcuaQN trenkerL vading arn KerL d6ngnha'tvdi aGoLa'yx la anhX';lhangbangvdi u E a G, ta'tca t E Ir trongdi~uki~n(iii) cuadinh nghla203chungtaco, Vdi m6itEl, (V~(u),f(t,U))"*0 r6i la'ytichphantrentoanbQI, (V~(U),f(U))"*Ovdi m6i u E aG va VI v~ydi~uki~n(ii) cua b6 d~2. 2 thoa,di~uki~n(iii) cuab6d~2.2 la mQttrongnhii'ng iadinhly 2.3. V~Y dinhly duQcchungminh. H~Qua2.4: Gia sarangt6nt~imQt';lprod,16i,bi ch~nG c Rll,vdi 0 E G, va vdim6iu E a G, mQtphaptuye'nn(u)t~iu cua80, saGcho (n(u),f(t,u))"*0 58 Vai m6itEl vane'u(2.32)xayfa,thlbaitmln(2.1)coit nha'tmQt nghi~mx saochoxCI)c G. ChUngminh Theotinhcha't2.5,G la 1t~pbienng~tcho(2.1),f6i apdl;lngdinh19 2. 3.H~quaduQcchungminh. H~Qua 2.5 : Gia sacomQthamcohuangng~tW cho(2.1),saocho I w(v)I ~CO n€u I v I ~ 00, thlne'u Do(w',B(p))*O bai loan(2.1)coit nha'tmQtnghi~m Chungminh VI W la hamcohuangng~tcho(2.1)nell,vai m6itEl t6nt~ip >0, saocho (2.35)(w'(u),f(t,u))<0, vaimQiu ERn, I U I ~p, Theo tinh cha't(2.6), t6n t~imQtt~pbien ng~tB(p) cho (2.1). Tli (2.35),taco (-w'(u),f(u))>0 Vai mQiu EJ B (p) Theotinhcha't1.16,chungtaco: Do(i, B(p))=Do(-w', B(p)), ma Do(w',B(p))*O Ap dl;lngtinhcha't1. 10,suyfa Do(-w', B(p))*0, nghlala Do (i, B(p))*0 f6iapdl;lngdinh192.3. H~quaduQcchungminh. Cu6i clingchungta duafa IDQtgia thie'tlien quaild~danhgia t~p nghi~m,matinhcha'tcuaham86r~tde nh~nbie't,clingnhttsl;l'rangbuQc cuat~pnghi~m. 59 Dinh ly 2.4 : Chof: [0,1]x Rn~ Rn, thoaman. (i) E>i~uki~ncaratheodory (ii)T6nt~iR >0saochovdimQix ERn, IIxii ~R , ~0 (3i)If(t,x)1~aCt)+b(t)lIxllvdi a,b~L1[0,1] Va 1b(t)dt=k <1 Khi d6bailoan { x1(t)=f(t,x(t)) t EI =[0,1] (I) x(o)=x(l) , C6 nghi~m ChUngminh Bu'8c1: Xetbailoan { Xi(t)=(1- A)a.x(t)+M(t,x(t)) . (It.) . , AE[O,I] x(o)=x(l) . trongd6:0<a <1- k . Tli (2i)suyra(I0 khongth€ c6nghi~mxthoamanIx(o)1>R. Th~tv~y Giasaxlanghi~mcua(I;.)thoamanIx(o)l>R. Nhanvohuanghaiv€ phu'dngtrlnhdftucuabailoan(I;.)vdix(t), tac6: (x'(t),x(t))=(l-A)alx(t)12+A(X,f(t,x)) , do x lient\lctuy~td5ivaIx(0)1>R , d~t J= {tE [0, 1], Ix(s)I~R, "Is E [O,t]}, thi J la t~pd6ngtrong[0,1]. (2) 60 Tachangminh J =[0,1] GhisaJ ~[0,1] d~t1=maxJ,thlt1>0, richphanhaivecua(2)tren[0,tIJ Taco ~[IX(t1)12-\x(o)12J=(l-A)a. £1Ix(t)12dt+A£lO Surra Ix(t})1>Ix(o)1 Do x lien1l;1cnent6nt~it2>t1sacchoIx(t)!>R, t e[t1,t2]. Mau thuftnvdi t1=maxJ V~yJ =[0,1] Surra 1 [ 2 . 2 J2 Ix(I)1-lx(o)1 =(1- A)a.!IX(1)12dt+i !f(t,x)dt>0 mallthuftnvdi x(l) =x(o) Bude2 : Giii sii't~pnghi~meua(1:0khongbi eh~n,t6nt~iday(An)ntrong[0, 1]saoehoXnIanghi~meua(lA.n)vaIlxnll>n Do buGe1nenxn(l)=xn(o)va I Xo(o)I ~R, vdi mQin. GQitoe [0,1]saoeho\xn(tn)I=llxnll>n Taco: Xn(t) = (1 - A) a.£n Xn(t)dt+A £n (t,xn(t))dt+Xn(0) . 61 Chiahaiv€ cho Ilxnll,taco : 1= Ixn(tn)1~(1- A)al tn Ixn(1)1dt+ _II A II fn[a(t)+b(t)lxn(t)l]dt Ilxnll 0 X t) Xn 0 Ixn(o)1+- llxnll Cho n ~ 00,taco : 1 ~(1- A)a +A lb(t)dt < (1 - A) (1 - k) +A < 1 mall thu~n. V~yt~pnghi~m(II..)bi ch~nvoi A E [0,1]. E>~tRl >0 saocho Ilx)..!!<Rl, voi mQiA E [0,1] Do tinhbit bi€n quad6ngluau,taco: DL (aI, B (0,R1))=DL (F, B(o,RI)) tfongdo F7L-N ling voi bai tmln { XI(t)=f(t,x(t)), t E[O,I] x(o)=x(1) Va Falingvdibaitmln,Fa=L - No,(Nox)(t)=a x(t) { X'(t)=a x(t) .. , t E[0,1] x(o)=x(l) DoKefFa={O},nenconghit$mduynhitlax(t)=0, \it E [0,1] Suyfa IDL(Fo,B(O,Rd)1=1 IDL(F,B(O,Rl))\ =1 Dodo V~y bai tmln(I) conghit$m 62

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