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

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