Luận văn Phương pháp bậc tôpô cho bài toán biên

8Chlidng 2 " " Cae ket qua ehinh 2.1 Truong hQpthu~nnhftt 2.1.1 B~litoan bien thli nh§.t Trong trl1CJnghc;5PA =B =0, (1.1)(1.2) tr0thanh x"(t)= f(t, x(t),F(x)(t),x'(t),H(x')(t)), a(x) =0,x'(I) =O. Dieuki~n(AI) cua f dl1c;5Cs11d\mg: (AI) T6n tl;1icaes6L1 ~0~L2saocho f(t,x,u,L1'W) ~ 0 ~ f(t,x,u,L2'W) vdih§,uh~tt E J vavdiillQi(x,u,w) E [L1,L2;F, H]IRo Xet bai toan x"(t)= J\f*(t,x(t),F(x)(t),x'(t),H(x')(t),J\),J\ E [0,1], a(x) =0,x'(I) =0, f* : IR6 ---tIR, f*(t,x,u,v,w,J\) = J\f(t,x,ii,v,w) +(1- J\)(v- L2), conX,ii, V,w xac dinh nhl1sau (2.1) (2.2) (2.3) (2.4) (2.5) 9A { xIx!:::; L x = Lsign(x) Ixl>L,L =rnax{-L1, L2}, - { u lul :::;p(F, [L1,L2]x) u = p(F, [L1,L2]X)sign(u) lul > p(F, [L1,L2]x), I' L1 V <L1 V = < V L1 :::;V :::; L2 L2 V > L2," = { w Iwl:::;p(H,(L1,L2)x) W = p(H,(L1,L2)x)sign(w) Iwl >p(H,(L1,L2)x). TaCOdanhgianghi~rnellabaitoan(2.3)(2.4)trongb6de2.1. B6 d~2.1. Cia sitx la m{}tnghi~mcua(2.3)(2.4),A E (0,1), con ham] thoadi€u ki~n(A1). Th€ thi Ilxll :::;L, L1 :::;x'(t) :::;L2, Vt E J. (2.6) Chung minh. a)Cia SU:rnax{x'(t),t E J} =x'(to)>£2.V6is6duongx'(to)- £2 ta Urndu£2+~khin ;;?no.R6rangtoi- 1, bdin@uto= 1thl x'(to)=x'(1)=0>£2+ ~ la kh6ngthichh<;:Jp. Do x' lien tl,lC,v6i E = ~ >0,cornQt/5'>0saGchokhi It- tal</5' thl !x'(to)- x'(t)1 x'(to)- E = x'(to)- ~ khi t E [to,to+ /5],/5=~/5'. Bdi x'(to)>L2 +~ lien x'(t) >L2,t E [to,to+ /5],va ta co L2 <x'(t) :::;x'(to),VtE [to,to+/5]. Tfch phantren [to,to+ /5]phuongtrlnh (2.3): l to+8 x"(s)ds = to l to+8 A ]*(s,x(s),F(x)(s),x'(s),H(x')(s),A)ds to l to+8 = A (A](S,X,u,v,w) +(1- A)(v- L2))ds to l to+8 - (A2](s, X,u,v,w)+A(1- A)(v - L2))ds to 10 trongdox =x(s),u =Fx(s),v =x'(s),w =Hx'(s). VI s E [to,to+8]nenv =x'(s) >L2,vadodo 1 to+5 "\(1 - "\)(v - L2)ds >O. to v = x'(s) > L2 dan d@nv = L2.Theocachd~t(x,it,w) E - - [L1,L2;F,H]IR'Th@thi f(s,x,it,v,w) =f(s,x,it,L2'W)~0,va 1 to+5 ,,\2f(s,x,it,v,w)ds ~O. to 1 to+5 guyra ,,\ ("\f(s,x,it,v,w) +"\(1- "\)(v - L2))ds > 0,hayla to 1 to+5 1 to+5 x"(s)ds> 0, trai v6i x"(s)ds=x'(to+8)- x'(to)( O. ~ ~ V~yta phai co di~unguQcl:;ti:max{x'(t),t E J} ( L2 +~. b) Gia s11min{x'(t), t E J} = x'(to) < L1 cling dua d@nmQt mati thuan.TavantimduQcn1E N d~x'(to)<L1- ~khin ~nl. Dotfnh lient\lCcuax',comQt8>0saGchoL1>x'(t)~x'(to),Vt E [to,to+8]. Tfch phan tren [to,to+ 8] phuong trlnh (2.3): 1 to+5 x"(s)ds = to 1 to+5 x"(s)ds - to 1 to+5 ,,\f*(s,x(s), Fx(s), x'(s),H x'(s), "\)ds to 1 to+5 (,,\2f(s,x,it,v,w) + "\(1- "\)(v - L2))ds. to Ta s E [to,to+8]suyrav =x'(s)<L1.Khidov- L2 = (v- L1) + to+5 . (L1-L2) <0,va { "\(1-"\)(v-L2)ds <O.VI v <L1nenv = L1. lto Theocachd~t,(x,it,w) E [L1,L2;F, H]IR,vata co f(s, x,it,v,w) = f(s, x,it,L1,w) ( O.Di~unaydand@n 1 to+5 1 to+5 1 to+5 x"(s)ds =,,\ ,,\f(s,x,it,v,w)ds+(l-"\) (v-L2) <0 ~ ~ ~ 1 to+5 mati thuan v6i x"(s)ds =x'(to+8)- x'(to)~O. to 11 c) Nhu'vi;iy,vdi x la nghi~mcua (2.3)(2.4),t E J tuy y van du ldn ta du'QcL1 - ~ ~ x'(t) ~ L2+ ~.Cho n -+ +00 thl co L1 ~x'(t)~ L2,VtE J. d) Vi a(x) =0liencoCE J saorhox(c)= o. +Ni\ut >c,ta c6x(t) =x(t) - x(c)=l x'(s)ds. Vdi LI ,;; x'(s) ,;; L2 tIll l' LIds';;l' x'(s)ds,;;1'L2dS. Suyral' LIds';;x(t),;;l' L2dshayJa (t - c)L1~x(t)~ (t - C)L2' VIO <t-c <1,L1~0~L2lien(t-C)L1?:L1,(t-C)L2~L2.Do doL1~x(t) ~ L2,var6i Ix(t)1~max{-L1, L2}=L, tacla Ilxll~L. + N~ut <c,ta cox(t) =-(x(c)- x(t))=-1' x'(s)r1s. Viii L] « x'(s) « L, th}1'L]dS« l' x'(s)ds« 1'L'dS,vi<1t eachkhiic:-1c L,ds ~x(t) ~ -1' LIds haylil -(c - t)L2~x(t) ~ (c- t)(-Ld. Ma 0 < c - t < 1,- L2 ~ 0 ~ - L1 lien (c- t)(-L1) ~ -L1, (c- t)(-L2) ?: -L2. guyra -L2 ~ x(t) ~ -L1, vatv do Ix(t)1~ max{-L1, L2}= L, tacla ta co IIxll~L. 0 Tiep theota sechangminhSljt6nt~inghi~mcua(2.1)(2.2).Danh gia (2.6)cho nghi~mcua (2.3)(2.4)du'Qcdung,va ta s11d\mgb6 d~ 1.1d~co slj t6n t~inghi~mcua (2.3)(2.4)khi A = 1.Saudo ta ki~m tra (2.3)(2.4)vdi A =1chfnhla(2.1)(2.2). D~nh1:5'2.1. Gia 811f thoadi€u ki~n(A1). The thi (2.1)(2.2)co nghi~mthoa(2.6). Chung minh. L§.yY = C1(J), Z = L1(J) la cackhonggianBanach vdichu~nthongthu'dng. ft.H1.;;;:TVN~!\~NI THoU.~!~~.t OOlOt)~ J 12 D~tdomD= {x E AC1(J)/a(x) = 0,x'(I) = O},cachamD,N xacdinhnhu belldu6i D : domD ---+ Z X f + X" , N : Y x [0,1] ---+ Z (x,A) f + j*(.,x(.),Fx(.),x'(.),Hx'(.),A), t~pS1nhu sau, n E N: S1={xE Y/llxll <max{L2,-Ld +~, L1 - ~< x'(t) < L2 + ~,t E J}. R6 rangD tuy@ntfnh trenY va N lient\lCtrenY x [0,1]. a)Ta kh~ngdinh D la mQtsonganhtit domDvaoD(domD). D~co di~unayta chI ra D la donanh. + Hamx = 0lamQtphiintu-cuadomDthoaDx = 0nen0 E kefD. + Gia su-x E kefD. Ta coDx= 0 hayla x" = O.Th@thl x"(t) = 0 v6iIDQit E J. guy ra x' la mQthamh~ngtren J, max E domDnen x'(I) = 0,va nhu vi;iyx'(t) = 0 v6i mQit E J: x la hamh~ngtren J. V6ix E domDta coa(x) = O.Theo chli y 1.1 ta tim dU<;1cmQt c E J saochox(c) =O.Dovi;iy,x =0trenJ. Nhu vi;iykefD = {O}:D donanh. b) Ta ki~mtra D-1N : Y x [0,1]-+ Y la anhX<,l,compact:Gia su-A la IDQtt~pconbi ch~ncuaY x [0,1],ta changminhD-1N(A) la mQt t~pcompactb~ngcachsu-d\lngdinh ly Ascoli-Arzela. + D-1N(A) bi ch~nd~u: L§,y(x,A) E A b§,tky. DM y = D-1N(x, A), thl N(x, A) = Dy: N(x, A)(t) = y"(t). Ta c6,v6is E J, 11 y"(r)d-r=y'(l) - y'(s) =0- y'(s) ~ -y'(s), d dayta su-d\lngy'(I) = 0,do y E domD. ClingVIy E domDnena(y)=0vacomQtcE J saochoy(c)=O. Vai t E J ta dU<)cl' y'(s)ds ~ y(t) ~ y(c)=y(t). 13 guyfa y(t) ~ -1' 11y"(r)d'rds=-1' 11N(x,>.)(r)dnls, Vdir E [s,l] ta clingdi;itx = x(r),u = Fx(r), v = x'(r),w = Hx'(r), vaco N(x, "\)(r) = f*(r, x(r), Fx(r), x'(r), Hx'(r)) = f*(r, x, u,v,w), N(x,"\)(r)="\f(r,X,u,iJ,w)+(1- ,,\)(v - £2)' 811dl,lngdi;ictint thli ba cua fEe ar(J x }R4),ta tim dl1<;JcmQt hamrjJE £1(J) 8aacha If(r, x, u,iJ,w)1:::;cp(r),Vr E J. Ta co IN(x, "\)(r) I :::;"\If(r, x, u, iJ, w) I + (1- ,,\)Iv - £21 IN(x, "\)(r)\ :::;"\rjJ(r)+(1- "\)(Ivl+£2)' Vi (x,"\) E A nenx E Y, va co h~ng86M' > 0 8aacha Ix'(t)1:::; M',t E J. Da do IN(x, "\)(r)1 :::;"\rjJ(r)+(1- "\)(M' +£2)'Ma ta co y(t)= -il N(.T,>.)(r)drdsnenly(t)1( ilIN(X, >.)(r)ldrds, va nhti vf!,y thlly(t)l ~II (A4>(r) + (1- A)(M' + L,))drds. Nhting rjJE £1(J), danhgianay8etrd thanh ly(t)1«l' [(AII4>II£'(1)+(1- A)(M'+L2))drds. D@nday ta khing dint ly(t)1bi chi;in:ly(t)1:::;M, M > 0 1amQt h~ng86.Di~unayco nghia1aID-1 N(x, "\)(t)1= \y(t)\ :::;M vdi mQi t E J. Ma (x,"\) E A bilt ky lienD-1N A bi chi;ind~u. + D-1N A donglien tl,lc: Lily (x,"\) E A, va tl,t2 E J bilt ky,coth§ gilt811t1~ t2. DM D-1N(x,"\) = y, ta danhgiaD-1N(x, "\)(tl)- D-1N(x, "\)(t2)' itlVi D-1N(x,"\)(td - D-1N(x,"\)(t2) =y(tl)- y(t2)= y'(s)ds,t2 may'(s)= -(y'(l) - y'(s))= - l' y"(r)drnen itl 1 1 y(t1) - y(t2) =- y"(r)drds. t2 s itl 1 1 itl 1 1 8uy ra ly(td-y(t2)1 :::; ly"(r)ldrds = IN(x,"\)(r)ldrds. t2 s t2 S Lilc nay ta 811dl,lng IN(x, "\)(r)1:::;"\rjJ(r)+(1- "\)(M'+£2),clingvdi 14 danhgiaAII</YII£l(J)+(1- A)(M'+L2))~ 11</YII£l(J)+ M' +L2).DM C =1IO.Ta co i tl 1 1 ~ (A</y(r)+(1- A)(M' +L2))drds i t2tl 1 8 1 ~ (AII</YII£l(J)+ (1- A)(M' +L2)))drds i t2tl 8 i tl ~ C(l - s)ds~ Cds= C(t1- t2)' t2 t2 ly(t1)- y(t2)\ guyra D-1N(x, A) lient~c.B(ji (x,A) E A tuy ynen D-1 N A dong lient~c. + K@thQpD-1NAb! ch[[tnd@uvaD-1N A donglient~c,theod!nh ly Ascoli-Arzela,D-1N la toantv compactrenY x [0,1]. c) Phuongtrinh x"(t) = Af*(t,x(t),Fx(t),x'(t), Hx'(t), A) duQcvi@t thanhDx = AN(x, A). Ta vi@tlC;1it~p0, v6in E N tuy y: 0 = {xE Y/llxll < max{L2,-Ld +~, L1 - ~< x'(t) < L2+~,t E J}. TheobE>d@2.1,v6i A E (0,1), n@ux la nghi~mcuaphuongtrinh x"(t)= Af*(t,x(t),Fx(t),x'(t),Hx'(t),A) thi x ph.:Uthoa L1 ~x'(t)~L2,t E J va Ilxll~ max{-L1, L2}.Noi khacdi, x kh6ngth@n~mtrenBO. Nhu v~y,Dx - AN(x, A) -I 0 v6i ffiQi(x,A) E (domDn BO)x (0,1). Cacdi@uki~ntrongbE>d@1.1duQcthoaman.Dodo,phuongtrinh Dx =N(x, 1)conghi~mtrongdomDn0, haybaitoansailconghi~m trong0: x"(t)= 1.f*(t,x(t),F(x)(t),x'(t),H(x')(t),1), a(x) =0,x'(l) =O. Ta chiconki@mtrar~ngdaychinhla (2.1)(2.2). x langhi~mcuabaitoantrendaythiL1- ~ ~x'(t)~L2+~va Ilxll~ max{-L1, L2}+ ~.Cho n ~ +00 ta co L1 ~ x'(t) ~ L2 va Ilxll~ max{-L1, L2}. 15 V6iy =x(t),u=Fx(t),v=x'(t),w=Hx'(t)taco f*(t,x(t),F(x)(t),x'(t),H(x')(t),1)= = l.f(t,y,u,v,w) + (1-1)(v - £2)= =f(t,y,u,v,w). R6rangv =v =x'(t),Y=y = x(t),u= u = Fx(t), conw = w = Hx'(t). V~yv6i A = 1ta tro l~iphltc5ngtrinh bandati: X"(t) = f(t, x(t),F(x)(t),x'(t),H(x')(t)). Dinh1y2.1duQcchangminh. 0 2.1.2 Bai toan bien thil hai Khi A =B = 0ta co(2.7)(2.8): X"(t)= f(t, x(t),F(x)(t),x'(t),H(x')(t)), a(x) =0,x'(O)=o. (2.7) (2.8) Di~uki~n(A2) cua1: T6n t~i£1 :::;;0 :::;;£2saGcho f(t,x,u,£2,W):::;;0:::;;f(t,x,u,L1'W) v6ihallh@tt E J va v6imQi(x,u,w) E [£1,£2;F, H]IR. Tru6ch@tta xU'1ydi~uki~nbiena(x) =0,x'(O)=O. a)Thayt = 1- s,dM x(t)=u(s).Ta cou(s)= x(1- s). Tti do u'(s) = -x'(1 - s),u'(I) = -x'(1 - 1) = -x'(O) = 0 va U"(s)=x"(1- s) =x"(t). V6ix E X ta d~tx* 1ahamx*(t) =x(l-t), t E J. R6 rangx* E X. Boiu*(t)=u(l-t) =x(t)nenx =u*;vaboi-(u')*(t) = -u'(I-t) = (u(1- t))'(t)=x'(t) ta vi@tx' = -(u')*. D~ta* :X ---+JR.,F*,H* : X ---+X 1acachamdinhboi a*(x) = a(x*), F*x(t) = Fx*(1- i),t E J H*x(t) = Hx*(1 - i), t E 1. 16 Vi a*(x)=a(x*)tacoa*(u)=a(u*),mau*=x liena(u*)=a(x). N~ua(x)=0thl phaicoa(u*)=0,hayla a*(u)= O. b)Taki@mchanga* E A, F*,H* E V. +a*tuy~ntfnh,bi ch~nvatang. Thi;itv~y,l~yx,y E X b~tky vac E IRtuyy ta coa*(x+cy)= a(x+cy)*.Ma (x+cy)*(t)= (x+cy)(l - t) = x(l - t) +cy(l - t) nen(x+cy)*(t)= x*(t)+cy*(t).Do a tuy~ntfnhliena(x +cy)*= a(x*)+ca(y*).Suyraa*(x+cy)=a*(x)+ca*(y):a* tuy~ntfnh. a*bi ch~n:Ila*(x)11=Ila(x*)11~M.llx*11=M.llxll,VxE X. Cias11cox,y E X saGchox(t)<y(t)vdit E J. Th~thl x(l - t) <y(l - t), vax*(t) < y*(t).VI a tangliena(x*) < a(y*),hayla a*(x)< a*(y). Do do a* la hamtang. Vi;iya* E A. + Baygidta ki@mtra F* E V. H* E V la hoantoantuongt\!. TrUdCtienta changminhF* lient\lCtrenX. L~y{xn}eX, Xn ---+X E X khi n ---++00.Ta c§,nF*xn ---+F*x khin -t +00. Theocachd~t,Xn---+X E X thl x~---+x* trongX. Taco IIF*xn - F*xll = supIF*xn(t) - F*x(t)l, tEJ IF*xn(t)- F*x(t)1 = IFx~(1- t) - Fx(l - t)l, IFx~(1- t) - Fx*(l - t)1 ~ IIFx~- Fx*ll. Vi F lient\lCtrenX lienkhix~---+x* thl IIFx~- Fx*11---+O. Day ladi~uta dangc§,n:F*Xn---+F*x trongX hayF* lient\lCtrenX. D@co F* bi ch~nta S11d\lngtfnh bi ch~ncua F. Vdi x E ~ b~t ky,~ bi ch~ntrongX, ta co x* E ~ va IIF*xll = IIFx*ll. Ma IIFx*11 bi ch~nlien IIF*xll clingbi ch~n. Ta daki@mtra xongF* E V. H* E V tuongt\!. c) Vdi t = 1- s thl Fx(t) = (Fu*)(t) = (Fu*)(l - s) = F*u(s) va Hx'(t) = H( -(u')*)(t) = H( -(u')*)(l - s) = H*(-(u'))(s). 17 Phuongtrlnh x"(t) = f(t, x(t),Fx(t),x'(t),Hx'(t)) trc;,thanh u"(s)=f(l - s,u(s),F*u(s),-u'(s), H*(-(u'))(s). (2.9) V6iy E X, t E J, d~tFgy(t)= F*y(t) = Fy*(l - i), Hgy(t)= H*(-y)(t) = H(-y)*(l-t) thl Fg,Hgxacdinhva130cacphantv cua t~pV. Ti~pt\lCd~tg(t,x,u,v,w) =f(l - t,x,u,-v, w) v6it E J va x,u,v,wEIR, khidophuongtrlnh(2.9)dU(5cvi~t130 u"(s)=g(s,u(s),Fgu(s),u'(s),Hgu')(s)). Nhuv~y,(2.7)(2.8)dU(5cduav~(2.10)(2.11): u"(s) =g(s,u(s),Fgu(s),u'(s),Hgu')(s)), (2.10) a*(u)= 0,u'(l) =o. (2.11) Tit a),b),c) ta k~tlu~nn~ubaitoan(2.10)(2.11)conghi~mu thl baitoanbandati(2.7)(2.8)seconghi~mx =u*,x(t) =u(l-t), t E J. Svt6nt~inghi~mcua(2.7)(2.8)dU(5cth@hi~ntrongdinhly 2.2. D!nh 15'2.2. Cia 871f thoadi€u ki~n(A2). The thi (2.7)(2.8)co nghi~mthoa(2.6). Chung minh. Ta apd\lngdinhly 2.1chobai toan(2.10)(2.11)b~ng cachki@mchllngdi~ubell du6i g(t,x,u,-L2,w) ~0~g(t,x,u,-L1,w) v6ihallh~t E J vav6imQi(x,u,w)E [-L2,-L1; Fg,Hg]]R. Tru6ctienta chllngminhp(Fg,[-L2, -L1]x) ~p(F,[L1,L2]x). L~yy E [-L2, -L1]x b~tky. Ta co Ilyll ~ max{-L1, -( -L2)} = max{-L1, L2}, ma lIy*11= Ilyll nen Ily*11~ max{-L1, L2}, tllC 130 y* E [L1,L2]x. Honmia,v6i t E J, Fgy(t)= F*y(t) = Fy*(l - i), IFgy(t)1= IFy*(l - t)j ~ IIFy*ll. VI y* E [L1,L2]x nen IIFy*11~ p(F, [L1,L2]x). Tit do, IFgy(t)1~p(F,[L1,L2]x),r6i suy ra IIFgyll ~p(F,[L1,L2]x). 17 Phuongtrlnhx"(t)= f(t,x(t), Fx(t),x'(t),Hx'(t)) trCJ.th~1llh u"(s)= f(l - s,u(s),F*u(s),-u'(s), H*(-(u'))(s). (2.9) VdiY E X, t E J, di;itFgy(t)= F*y(t) = Fy*(l - i), Hgy(t)= H*(-y)(t) =H( -y)*(l - t) thl Fg,Hgxacdinhvala cacph§,ntv cua t~pD. Ti~pt\lCdi;itg(t,x,u,v,w) = f(l- t,x,u, -v,w) vdit E J va x,u,v,w E ~,khidophuongtrlnh(2.9)duc;Jcvi~tla u"(s)=g(s,u(s),Fgu(s),u'(s),Hgu')(s)). Nhuv~y,(2.7)(2.8)duc;Jcduave(2.10)(2.11): u"(s)=g(s,u(s),Fgu(s),u'(s),Hgu')(s)), (2.10) a*(u)=0,u'(l) =o. (2.11) Tl1a),b),c) ta k~tlu~nn~ubaitoan(2.10)(2.11)conghi~mu thl baitoanband§,u(2.7)(2.8)seconghi~mx =u*,x(t) =u(l- t), t E J. Sv ton ti;1inghi~mcua (2.7)(2.8)duc;Jcth~hi~ntrongdinh15'2.2. D~nhly 2.2. Gia sV:f th6adi€u ki~n(A2). Th~thi (2.7)(2.8)co nghi~mth6a(2.6). Chung minh. Ta apd\lngdinh15'2.1chobai toan(2.10)(2.11)b~ng cachki~mchangdieubell dudi g(t,x,u, -L2,w):::;;0:::;;g(t,x,u, -L1,w) vdih§,uh~t E J vavdimQi(x,u,w)E [-L2,-L1; Fg,Hg]rn;. Trudctien ta changminhp(Fg,[-L2, -L1]x):::;;p(F, [L1,L2]x). Lfty y E [-L2, -L1]x bftt ky. Ta co Ilyll :::;;max{-L1, -( -L2)} = max{-L1, L2},ma Ily*11= Ilyll lien Ily*11:::;;max{-L1, L2}, tac la y*E [L1,L2]x. Hon nfi'a,vdi t E J, Fgy(t) = F*y(t) = Fy*(l - i), IFgy(t)1 = IFy*(l - t)1 :::;;IIFy*ll. VI y* E [L1,L2]x lien IIFy*11:::;;p(F, [L1,L2]x). Tl1d6, IFgy(t)l:::;;p(F, [L1,L2]x),roi guy ra IIFgy11 :::;;p(F, [L1,L2]x). 18 IIFgyl1~ p(F, [L1,L2]x) v6i y E [-L2, -L1]x btit ky. Nhu th@thl p(Fg,[-L2, -L1]x) ~ p(F, [L1,L2]x). Vi~ckH~mtra p(Hg,(-L2, -L1)x) ~ p(H,(L1'L2)X) la tuongtV'. Ltiyz E (-L2, -L1)x tuyy,tachangminhduQcIIFgZl1~ p(H, (L1,L2)x). Baygid,trd11;1ibaitoandangxet,ltiy(x,u,w) E [-L2, -L1; Fg,Hg]JR btitky thl secolxl ~ max{-L1,-(-L2)} = max{-L1,L2)}, lul ~ p(Fg,[-L2, -L1]x) va Iwl ~ p(Hg,(-L2, -Ldx). Cling v6i hai k@t quatrenday ta khiingdinh ding n@u(x,u,w) E [-L2, -L1; Fg,Hg]JR thl (x,u,w) E [L1'L2;F, H]JR. Llicnaytheodi~uki~n(A2)cuahamf ta co f(1 - t,x,u,L2,w) ~0~ f(1 - t,x,u,L1,w). Ma g(t,x,u,-L2,w) = f(l- t,x,u,L2,w), g(t,x,U,-L1'W) - f(1-t,x,u,L1,w) nen g(t,x,u, -L2,w) ~ 0 ~ g(t,x,u, -L1,w). Cacdi~uki~ncuadinhly 2.1duQcthoaman.Bai toan(2.10)(2.11) conghi~mu thoa Ilull ~ max{-L1, L2}, -L2 ~ u'(s) ~ -L1, S E J. Nghi~mcuabaitoan(2.7)(2.8)la x =u*,x(t)= u(1- t),t E J. Do lIu*11=Ilullvax'(t)= -u'(1 - t) nennghi~mx naythoa(2.6): Ilxll ~max{-L1, L2},L1~x'(t)~L2,t E J. Ta changminhxongdinh ly 2.2. 0 2.1.3 Bai toan bien thil ba Khi A = B = 0ta co(2.12)(2.13) x"(t)= f(t,x(t),F(x)(t),x'(t),H(x')(t)), (2.12) x(O)=0,x(l) =O. (2.13) 19 a) Cia S11L1 :( 0 :( L2,L3 :( 0:( L4,L3 L2. Khi do co noE]\I saGcho L2 + 2 L3. Ta xac dinh hamhnvai~ ~ . n) nonhu sail hn(t,x,u,v,w) = j(t, X,ii, L4,w) j(t,x,ii,v,w) j(t, X,ii, L2+ ~,w)+ +g(L2,~,v) j(t, X,ii, L2,w) j(t,x,ii,v,w) j(t, X,ii, L1,w) j(t,x,ii,L1 - ~,w)- -g(L1' -~, v) j(t,x,ii,v,w) j(t, X,ii, L3,w) L4 <V L2 +1 < v :( L4n L2 +1. <v :( L2 +1n n L2 <V :( L2 +1.n L1 :( V :( L2 L1 - ~ :( v <L1 L1- 1 :::::.v < L 1 - 1.n :::. n L3 :( V < L1 - 1n v < L3 (2.14) ddayg(Li,k,v) = (j(t, x,ii,Li,w)- j(t, x,ii,Li+k,W))(Li+k- v)n, caegiatrt cuai la 1va 2, L4 x> L4 x = <.X L3 :( X :( L4 . \.. L3 X <L3, it = { :(F, (L3,L4)x)sign(u) W={ ;(H, (L",L4)x)sign(w) Ch6.y 2.1. hn E Car(J x }R4),n du ldn. lul :( p(F,(L3,L4)x) lul >p(F,(L3,L4)x), Iwl :( p(H,(L3,L4)x) [wi> p(H, (L3,L4)x). Chung minh. Ta ki~mtra l§,nlu<;:JtcaetinhchatcuahamCaratheodory. Truaehetta khiingdtnh,khi n du lOn,hn(t,x,u,v,w) = j(t, X,ii, V,w) vai mQit E J, (x,u,v,w) E }R4.ThM v~y,l&yt E J, (x,u,v,w) E }R4 tuyy,n ) no, xet eae trudng h<;:Jpsail. Nell v) L4 thl hn(t,x,u,v,w) =j(t,x,ii,L4,w)=j(t,x,ii,v,w). 20 N~uv:( L3 thl hn(t,x,u,v,w) =j(t,x,u,L3,w)=j(t,x,u,v,w). N~uL2 <V < L4thlluonco j(t,x,u,v,w) = j(t,x,u,v,w).Vi 2 / Hrn- =0 nenv6isodl1dngv - L2 ta tlm dl1<;1cnl ;::nosaochon-++oon 1 < v - L2 khi n ;:: nl, hay la L2 + 1 < v khi n ;:: nl. Va nhl1n n v~y,n~ulayn ;::nl thl ta cohn(t,x,u,v,w) = j(t, x,U,v,w), tac la hn(t,x,u,v,w)= j(t,x,u,v,w), n;:: nl. N~uL1 :( V :( L2 thl co ngayhn(t,x, u,v,w) = j(t, X,U,v,w)v6i n ~novanhl1v~yhn(t,x,u,v,w)= j(t,x,u,v,w) khin;::no. N~uL3 < V < L1 thl j(t,x,u,v,w) = j(t,x,u,v,w), ta co mOt 86n2;::nosaocho1 v khin n n ~n2.Layn ;::n2thl tacohn(t,x,u,v,w)= j(t,x,u,v,w), tacla hn(t,x,u,v,w)= j(t,x,u,v,w),n;::n2. Nhl1v~yta dachangminh,khi n dli 16n,v6imQit E J, (x,u,v,w) E JR4thl hn(t,x,u,v,w) =j(t,x,u,v,w). Vadodo,VI j E Car(J x }R4)nenv6i moin dli Wnthl: + hn(.,x, u,v,w) do dl1<;1ctren J; + N~uK C }R4compactthl sup hn(t,x, u,v,w) E L1(J). (x,u,v,W)EK TachIconki~mtrav6imoin dli Wn,hn(t,.,.,.,.) lient1.lCtren}R4 voihftuh§tt E J. Lay (x,u,v,w) E }R4batky.Giltsit {xd, {ud,{vd,{Wk}la cac dayIftn111<;JthOit1.lv~x, u,v,w, ta changminhhn(t,Xk,Uk,Vk,Wk)~ hn(t,x,U,v,W) khi k ~ +00. D~Y 13.dang xet n kha Wn, ta co th~vi§t hn(t,Xk,Uk,Vk,Wk) = j(t, Xk,Uk,Vk,Wk)vahn(t,x,u,v,w) = f(t, x,u,v,w). Do do,d~sitd1.lngdl1<;1ctinh lient1.lCclia j(t,., .,.,.) ta phlticoXk~ X,Uk~ U,Vk~ V,Wk~ w. ChangminhXk ~ X,Uk~ U,Wk-+w hoantoantl1dngtv changminhVk~ V. Ta vi§t lq.iky hi~uf): A y= L4 Y L3 y> L4 L3 :( Y :( L4 Y < L3' 21 + N@uv >L4 thl V = L4.Do Vk-7 v lientont9>is5k1saDcho Vk>L4,k ~ k1.Nhuv~yVk=L4=V,k ~ k1. + N@uv = L4 thl v = v = L4- L§,yE >0 r§,tnho. Do Vk -7 V =L4 lien ton t9>is5 k1saDcho IVk- L41 <E,k ~k1- Ta co IVk- vi < E,k ~ k1. Th~tv~y,gQiC1= {k~ k1: L4 <Vk <L4+E}vaC2= {k~ k1: L4 - E < Vk ~ L4}. V6i k ~ k1 tuy y, n@uk E C1 ta co IVk- vi = IL4- L41= 0<E,conn@uk E C2 thl Ivk- vi = Ivk- vi <E. +N@u L3 <V <L4 thl v = v. ClingdoVk-7 v lien ta Urn du<;1c mOtk1saDchoL3 < Vk< L4,k ~ k1.Ta co Vk= Vk,k ~ k1.R6 rang A A Vk =Vk -7 V =V. + Trudngh<;1pv =L3 gi5ng trudng h<;1pv =L4, conv < L3 gi5ng V> L4- Ta khiingdtnhdu<;1Cn@uVk-7 v thl Vk-7 v. Ta dachangrninhxonghnE Car(J x JR4)v6i rnoin dli 16n. D Chli Y 2.2. Neu{xn},{un},{vn},{wn}la caedayhamIan lurt h(Ji t'l),vt x,u,V,w trongX th'ita cov(jihauhett E J ~ - hn(t,xn(t),un(t),vn(t),wn(t))-7 f(t, x(t),u(t),v(t),w(t)) khin -7 +00. Chung minh. Theochliy2.1,ta tlrndU<;1Cn1d~khi n ~n1 thlseco hn(t,X,U,V,W) = f(t,X,U, 11,W) v6irnQit E J, (X,U,V,W) E JR4tuy y. Do do, n@un ~ n1 thl hn(t,xn(t),un(t),vn(t),wn(t)) =- - f(t, xn(t),un(t),vn(t),wn(t)). Vi Xn -7 x trong X lien xn(s) -7 x(s) v6i rnQis E J. Trong chli y 2.1ta dachIra n@udays5{an}hOit\l v@a thl anhOit\l v@a. Sitd\lng- ---- di@unayv6ian=xn(s),a=x(s)tacoxn(s)-7 x(s), s E J b§,tky.----- TucJngtv, un(s) -7 u(s), vn(s)-7 v(s), wn(s) -7 w(s) v6is E J b§,t ky. 22 Li;1ico j(t,.,.,.,.) lien t\IC tren JR4v6i hiiu h@tt E J. Th@thl- ~ - j(t,xn(t),un(t),vn(t),wn(t))-+ j(t,x(t),u(t),v(t),w(t)): ~ - hn(t,xn(t),un(t),vn(t),wn(t))-+j(t, x(t),u(t),v(t),w(t)). 0 b) Di~uki~n(A3) cua j duqcsU'd\Ing (A3)T6nti;1iL1 ~ 0 ~ L2,L3 ~ 0 ~ L4 saocho j(t,x,u,L1,w) ~ 0 ~ j(t,x,u,L2,w), j(t,x,U,L41W) ~ 0 ~ j(t,x,u,L3,w) v6ihiiu h@tt E J va v6i mQi (x,u,w) .E (L, M; F, H)ITJ!.,trong do L = min{Ll,L3}conM = max{L2,L4}. c)Xetbaitmin(2.15)v6idi~uki~nbien(2.13),A E [0,1],n ) no: x"(t) =Aj~(t,x(t),Fx(t),x'(t),Hx'(t),A), (2.15) hamj~ bi@udiennhu sauv6i t E J, (x,u,v,w) E JR4,A E [0,1], j~(t,x,U,V,W,A)=Ahn(t,x,u,v,w)+(1- A)p(V), (2.16) hamhnxacdinh d a), vap : JR-+ JR lien t\IC thoa p(v) ) 1,v E [L3 - l.., L3] U [L2'L2 + l..],no no p(v)~ -l,v E [L1- ;0,L1]U [L4,L4+;0]' Bd d~saucho danh gia v~nghi~mcua bai toan (2.15)(2.13)khi L3 < L1,L2 < L4. Nhiic li;1i,Slj t6n ti;1icua no EN, la s6 saocho L2+;0L3,daduqckh~ngdinhtru6cdo. B5 d~2.2. Cia SV:j thoadi€u ki~n(A3), L3 <L1,L2<L4,vabai loan(2.15)(2.13)conghi~mx vrJiA E (0,1)van) no.Tht thi ta co dankgiasau,t E J, n ) no, 1 1 1, 1 L3 - - ~x(t)~L4+-, L3 - - ~x (t) ~L4+ -.n n n n (2.17) Chungminh. Theodi~uki~nbien(2.13),x(O)= x(l) = 0, ta tlm duqca E (0,1) saochox'(a) =O. a)GiasU'max{x'(t)jtE [0,an =x'(to)>L2+* sedaTId@nmQtdi~u matithuan. 23 +Trudctienta chungmint comQtkhaang[" v] C (to,a) saGcha x'(v)=£2,x'(,) =£2+~ vavdit E b, v]thl £2~x'(t)~ £2+ ~. Phuongtrlnhx'(t)- £2 - ~ =0(theabi§nt) conghi~mtl E (to,a) bdix'(to)>£2+ ~ vax'(a)=0<£2+~. L§.y, = max{t2E (to,a) : X'(t2) = £2 + ~}.Khi do x'(,) = £2+ ~,x'(,) > £2. K§t hQpvdi x'(a) = 0 < £2e), phuongtrlnh x'(t)=£2co nghi~mt2E (" a). L§.yv = min{t2E (" a) : X'(t2)= £2}.R6 rangb, v] c (to,a), x'(v)=£2,x'(,) = £2+~;honmia,vdit E b, v]thl £2~x'(t)~ £2+~. Ta ki@mchungkhing dint nay. Giasut6nt:;tit E b, v] :x'(t)> £2+~.Duongnhient i- "t i- v. Bdix'(a) = 0 < £2+~ nen ta Urn duQcmQtt3 E (t,a) saGeha X'(t3)= £2+~. Da, = max{t2E (to,a) : X'(t2)=£2 +~}nent3~,. DayIa di~uvo If VI t3E (t,a) va t E b, v] thl phai co , < t3.Vf!}.yta eox'(t)~£2+ ~,t E b, v]. x'(t)~£2,t E b, v]duQcchungmint tuongtv. + Khi cokhaangb, v] c (to,a) saGchax'(v) = £2,x'(,) = £2+ ~, £2~x'(t)~£2+ ~,t E b, v],thl ta duQe j vx"(s)ds =x'(v)- x'(,) =-~ <O.~ n Trangkhi do, thea(2.15),(2.16) [X"(S)dS = A1"f~(s,x(s),Fx(s), x'(s),Hx'(s),A)ds - A1" (Ahn(s,x(s),Fx(s),x'(s),Hx'(s))+ +(1- A)p(x'(s)))ds lTa dangxet£2 > O.N~u£2 =0 thl ta van tlm dll<;iCmQtkhoangb,v] E (to,a) saDcho x'(~)= £2+ ~,x'(v)=£2, d6ngthai £2 :::;x'(t) :::;£2 + ~,t E b,vl. Th~t v~y,lily v = min{tE (to,aI,x'(t)= O},viq = max{tE (to,v),x'(t)=£2+~= ~},thl b, v]litkhoangd.ntill. Chid.n ki~mchung£2:::;x'(t) :::;£2+~,t E [" vI.N~ucot E [" v]saGchox'(t) > £2+~thl do x'(v) =0 nencotl E (t,v): x'(td = £2+~, mallthuanvoi~= max{tE (to,v),x'(t)= £2+~= ~}.N~u cot E b,v] saGchox'(t) < £2thldox'(T)=£2+ ~nencomQtt2E (T,t): x'(t2)= £2=0,mall thuanvoiv =min{tE (to,a],x'(t)=O}. 24 - >,'l" hn(s,:£(8),Fx(s),X'(S),Hx'(s))ds+ +-X(1- -X){p(x'(s))ds. DM X = X(S),U = Fx(s),v = X'(S),W= Hx'(s), ta CO(X,it,W)E (L3,L4;F, H)]R.VI s E [r, v] nen£2 (; x'(s) (; L2 + ~.Thea (2.14), hn(s,x,u,v,w) = f(s, x, it,L2,w). Dung di@uki~n(A3) cua f, v6i s E [r,v], taduQchn(s,x,u,v,w)) O.M~tkhac,L2 (; x'(s) (; L2+~, vatfnhch§,tcuahamlient1,1Cp sedaTIdenp(x'(s)) ) 1.guy ra .{ x"(s)ds;, A(l - A).{ p(x'(s))ds;, A(l - A)(v- ,,) >o. + Dendayxu§,thi~nmOtmatithuan,gia Sltban d§,usai,va phai co max{x'(t)jtE [0,an (; L2+ ~,hayla x'(t) (; L2 +~,Vt E [0,a). b)min{x'(t)jtE [0,an ) L1 - ~ duQcchU'ngminh tuong t\!o c) Nhu v~yv6i t E [0,a), L1 - ~ (; x'(t) (; L2+ ~. d) Baygidv6it E (a,1]ta chU'ngminhL3 - ~ (; x'(t) (; L4+~. Cia Sltmax{x'(t)jt E (a,I]) = X'(t4)> L4 + ~.Gi6ngnhUtrong ph§,na) trenday,ta tIm duQcmOtkhaang[r,v] C (a,t4) saGcha L4(; x'(t)(; L4+~,VtE [r,v]vax'(,) =L4,x'(v)=L4+~. 1v x" (s)ds = x' (v) - x'(,) = ~> O.I n Trangkhi do, thea(2.15),(2.16), 1"x"(s)ds = >'.ff~(s,x(s),Fx(s),x'(s),Hx'(s),>')ds - >.'1" h,,(s,x(s), Fx(s), "'(8),H x'(s))ds+ +.\(1-.\) 1" p(x'(s))ds. NeudMx =x(s),u=Fx(s),v=x'(s),w=Hx'(s)thl (x,it,w)E (L,M; F, H)]R.V6i s E [r,v] thl L4 (; x'(s) (; L4+ ~.Thea (2.14), hn(s,x,u,v,w)=f(s,x,it,L4,w). Da di@uki~n(A3) cua f, hn(s,x,u,v,w) (; O.Tv L4 (; x'(s) (; 25 L4+ ~ tacop(x'(s)),;; -1. Suy fa i" x"(s)ds < 0, mauthuh viii 1" x"(s)ds> 0 trllOCd6.Vi;iyphaic6max{:c'(t)/lE (a,II} ,;;L4+ ~. Tl1ongtv, min{x'(t)jtE (a,I]}~L3- ~. e)Toml~i,n~ux la nghi~mdm (2.15)(2.13),n E N,n ~no,thl: L1 - ~~x'(t)~L2+~,t E [0,a), L3 - ~ ~x'(t)~L4+ ~,t E (a,1]. Tv day,VI L3 <L1,L2 <L4vax'(a)=0,v6is E J ta co L3 - ~ ~x'(s)~L4+~. f) Tfch phanL3 - ~ ~x'(s)~L4 +~ tv 0d~nt,t E J: 1 it 1(L3- -)t ~ x'(s)ds~ (L4+ -)tnon hayla (L3- ~)t~ x(t) - x(O)~ (L4+ ~)t.Da x(O)= 0 nensuyra (L3- ~)t~x(t)~ (L4+~)t,haylaL3- ~ ~x(t)~L4+~. B6de2.2chvngminhxang. D H~ qua 2.1. Gid su:co di€u kitjn (A3)) L3,L1 ~ 0 ~ L2,L4; L3 -I L1,L2 -I L4; va3noEN: I L3 - Lll >2, I L4 - L2 1 > 2. N€u bailoanno no (2.15)(2.13)conghitjmx v{Ji,\E (0,1)van ~no)thzta co: 1 1 1, 1 L - - ~x(t)~M +-, L - - ~x (t)~M +-, t E J,n n n n (2.18) trongdoL =min{L1,L3},M =max{L2,L4}. Chung minh. Ta chiara nhieutrl1dngh<,5pd@xet. a) L3 < L1: L =L3. N~uL2 <L4,M = L4,ta congay(2.18)theab6 de2.2.N~uL2 > L4,M = L2.Trangdinhnghiahn (2.14)ta chIc§,n thayd6ivi tri cuaL2 va L4. Tl1ongtv trangchvngminhb6 de2.2, L3 - ~ ~u'(t) ~ L2 +~,'lit E J. b) L1 < L3: L = L1. N~uL2 < L4,M = L4. Trang dinh nghiahn (2.14)ta thayd6i vi tri cuaL1 va L3' Theab6 de2.2,(2.18)thoa: L1 - ~~u'(t)~L2+~,'litE J. 26 N§u L2 > L4,M = L2. Trangdinhnghiahn (2.14)ta haand6i L1 v6iL3,L2v6iL4.Theab6de(2.3)thoa: L1 - ~ ( u'(t) ( L2 +~,Vt E J. D B6 d~2.3. Cia s71:j thoadi€u ki~n(A3). Tht thi vdin E N dil ldn, baitoan (2.15)(2.13) khi ,\=1 co nghi~mthoa (2.19), 2 2 2, 2 L - - ( x(t)( M +-, L - - (x (t) ( M +-, t E 1.n n n n (2.19) Chung minh. + Trudngh<;5pL3 i- L1,L2 i- L4 ta dungh~qua2.1.L~yno E N saDcha\L3- L11>;0' IL4- L21 > ;0' Khi n?: novax lamOtnghi~m cua(2.15)(2.13),v6it E J thl L - 1 ~ X(t) ~ M +1 L - 1 ~ x'(t) ~ M +1.n ~ ~ n' n ~ ~ n Khi dotaapd\mgb6de1.1v6iD : domD-+ Z,Dx=x",trang dodomD= {xE AC1(J)jx(0) =x(l) = O},conN : Y x [0,1]-+ Z, N(x,'\) = j~(.,x(.),Fx(.),x'(.),Hx'(.),'\), cungv6i Q ={xE YjL - ~ <x(t)<M +~, L - ~ ( x'(t) ( M +~,t E J}, vak§tlu~n(2.15)(2.13)conghi~mkhi ,\ = 1. +Khi L1=L3=L, tttdieuki~n(A3)cuaj tacoj(t, x,u,L, w) = 0v6it E J, (x,u,w) E (L,M; F, H)'Rf..Trudngh<;5pnayta thi§t l~phn nhusau, hn(t,x,u,v,w) = j(t, x, ii, L4,w) j(t,x,ii,v,w) j(t, X,ii,L2+~,w)+ +g(L2'~'v) j(t, X,ii,L2,w) j(t,x,ii,v,w) j(t,x,ii,L,w) L4 <V L2 + l <v ( L4n L2 +1 <v ( L2 + ln n L2 <V ( L2 +1n L ( v ( L2 V <L. V6in du Wn,phuongtrlnhX"(t)= hn(t,x(t),Fx(t),x'(t),Hx'(t)) comOtnghi~mt:1mthudngx(t) = Lt, nghi~mnayduongnhienthaa 27 (2.19). Tuong t\l khi L2 = L4 = M, thi@tI~phn cho thfchh<;jp,bai tmin(2.15)(2.13)conghi~mtitmthudngx(t) = Mt, vanghi~mnay phli h<;jpvdid{mhgia (2.19).Do do n@utrudngh<;jpL1 = L3 ho1;tc L2 =L4 xiwfa, baitoan(2.15)(2.13),vdiA = 1,vanconghi~mthoa (2.19). 0 S\l t6n tC;1inghi~mcuabai toan (2.12)(2.13)du<;jcchI ra trongd!nh Iy sauday,d!nhIy 2.3. D~nhIy 2.3. Cia sv:coditu ki~n(A3) cuaf. Bili loan (2.12)(2.13) conghi~mx thoa L:::; x(t):::;M,L:::; x'(t):::;M,t E J (2.20) Chung minh. Theob6 d~2.3,bai toan(2.15)(2.13)khi A = 1 co nghi~mXnthoa(2.19),n E N duldn: x~(t) = f~(t,xn(t),Fxn(t),x~(t),Hx~(t),1) - hn(t,xn(t),Fxn(t),x~(t),Hx~(t)). Theo chli y2.1,khin duldn,hn(t,x,u,v,w) = f(t,X,ii,V,w)vdi t E J, (x,u,v,w) E JR4.Vi f E Car(J x JR4) lien co ~E L1(J) saocho If(t,x,ii,v,w)1 :::;~(t)vdihituh@t E J. Do v~y,khi n dli ldn,ta vi@t Ihn(t,x,u,v,w)1:::;~(t),t E J. l t2 Vdi t1,t2E J (giaS11t1 :::;t2)ta co X~(t2)- x~(t1)= x~(s)ds: tl l t2 IX~(t2)- X~(tl) I = I f*(s,xn(s),Fxn(s),x~(s),Hx~(s),1)ds tl l t2 -, hn(s,xn(s), Fxn(s), x~(s),Hx~(s))ds tl l t2 = Ihn(s,xn(s),Fxn(s),x~(s),Hx~(s))lds tl l t2 :::; ~(s)ds. tl Do do, IX~(t2)- x~(tdl :::;It2- tll.II~lIu(J)' Tli day suy ra {x~} 28 d6nglien tl,lc. l t2 l t2 Ta 1l;1ico IXn(t2)- xn(tl)1=I x~(s)ds:::;; Ix~(s)1ds. 1 ~~ ltlt2 2Dung (2.19),x thaybdi xn: Ix~(s)1ds :::;; (M + -) ds. lt2 2 tl tl nVI (M+-)ds:::;; (M+2).lt2-tll lientl n IXn(t2)- xn(tl)1:::;;(M +2).lt2- tll Dieunaydand~n{xn}d6nglientl,lc. {xn(t)}bi chi;indelilahi~nnhienbdi{xn}langhi~mcua(2.15)(2.13) thoa(2.19),trongdocodanhgiaL - ~ :::;;xn(t) :::;;M +~ vdi t E J. Vf).,y,theodinh ly Ascoli-Arzela,A = {xn}la t~pcompactrongX. Day {xn}chaatrongA compactlienseco mQtdaycon {xnk}hQit\) vemQtphl1ntu x trongX. VI {x~}d6nglien tl,lClien {x~J cling d6nglien tl,lc.{x~J thoa mandanhgiaL - ;k :::;;X~k(t) :::;;M +;k vdit E J lien {X~k(t)}labi chi;indeli. Do do theodinh ly Ascoli-Arzelat~pB = {x~J compact trongX vata chQndl1QcmQtdaycon{x~km}cua{X~k}hQit\)vemQt y trongX. Ta changminhx' t6n tl;1ivay = x'. D~dongianvemi;itky hi~uta giaiquy~tviLlidesail: Cho day {xn}trong Cl (J) saochoXn ~ x va x~~ y trong X, thi x' t6n tl;1iva x' =y (tren J). Lily <pE C~ (J) hiLtky.VI x~t6n tl;1i,ta co [ x~(t)rp(t)dt=- [Xn(t)rp'(t)dt. VI Xn~ x, x~~ y trongX =C(J) lienguyra [ y(t)rp(t)dt=- [X(t)rp'(t)dt. TU'do,x' t6n tl;1i,x' = y trongLl(J). Ma y E X = C(J) lien x'(t)=y(t)vdimQit E J, x'E X. viLli dedl1QCgiaiquy~t.Ta changminhxongx' t6n tl;1i, x'(t)=y(t),t E J. 29 TrC!li;tibaitoan.Lticnay,x' t6nti;tivax' EX. Ta cox E c1(J). Ta ki@mchangx thoa(2.12)(2.13),nghlala phajchangminhx" t6nti;ti,x"(t)= f(t, x(t),Fx(t),x'(t),Hx'(t))vax(O)=0,x(l) =O. Tavi~txnkmla nghi~mcua(2.15)(2.13)khi A= 1: x~ (t) =hnk(t,xnk(t),Fxnk (t),x~ (t),Hx~ (t)),km m m m km km xnkm(0) = 0,xnkm(1) = O. 811dl,mgxnkm---+X trong X khi m ---++00, tr116ctien ta co x(O) = x(l) = 0, v~yx thoa di§u ki~nbien (2.13).Va vi xnkmthoa danh gia (2.19)nenkhi chom ---++00 ta co L ~x(t)~M, t E J, L ~x'(t)~M, t E J. Ta chic§,nxet trl1dngh<;jpL = L3,M = L4,bC!icactrl1dngh<;jp khacla tl1ongtv khicothayd6ithichh<;jptrongbi@uthaccuahnva x,ii,w. D@ti~nchovi~ctrlnhbay,ta vi~t,v6in du ldn, x~(t)= hn(t,xn(t),Fxn(t), x~(t),H x~(t)), xn(O)= 0,xn(1)= 0, va dangco Xn lien t1,1Cd~ncftp1,x~E L1(J), Xn ---+x,x~---+x' trong X =C(J). D~tzn(t)= hn(t,xn(t),Fxn(t),x~(i), F[x~(t)).Theochliy2.2taco-~- Zn---+z trong L1(J), z(t) = f(t,x(t),Fx(t),x'(t),Hx'(t)). V6imQi<pE Or:(J) ta co [ x~(t)<p(t)dt=-[ X;,(l)<p'(t)dt. Vetraih(>itl.1yel' z(t)<p(t)dtconYephiiihQitv Ye-11 x'(t)<p'(t)dt, non[z(t)<P(t)dt =-[ x'(t)<p'(t)dt,'if<pE 0;:'(1). Suy fa z =x" trongL1(J): -~- x"(t)=f(t, x(t),Fx(t),x'(t),Hx'(t)). 30 --- --- V6i cacky hi~unhl1bandatithl f(t, x(t),Fx(t), x'(t),Hx'(t)) = f(t, x(t),Fx(t), x'(t),Hx'(t)). Dinh1y2.3chungmintxong. 0 2.2 Tru'ong h<Jpkhong thu~nnhftt Ta vi@t19>ibai toan x"(t)= f(t,x(t),(Fx)(t),x'(t),(Hx')(t)), (2.21) a(x)=A,x'(I) =B, a(x)=A,x'(O)=B, x(O)=A,x(l) =B. (2.22) (2.23) (2.24) 0dayta sechi ra cacdint 1yt6nt9>inghi~mchobai toanbien kh6ngthuannh§,t(2.21)(2.i)v6i i =22,23,24. 1 Ch6 Y 2.3. Ham <p(t)= a(l) (A - Ba(l)) +Bt,t E J, vdi l(t) = t,t E J, th6acaeditukiiJnbien(2.22)va (2.23). a) CachdM ham<pnhl1tren1ach§,pnh~ndl1Qc,bdi a(l) i= O.Di~u naydl1Qckhiingdint ngaysauchliy 1.1. b)Tath§,yc.p'(t)= B v6imQit E J. Tavi@tdl1Qc.p'(O)=c.p'(I)=B. D~ki~mtra ham<pthoaca hai di~uki~nbien (2.22)(2.23),ta chi canki~mchunga(c.p)=A 1axong.Ta co 1 a(c.p)= a(a(l) (A - Ba(I)).l +B.l). VI a :X ---+IRtuy@ntint nen 1 a(c.p)= a(l) (A - Ba(I)).a(l) +B.a(I) hay1aa«p) = (A - Ba(I)) +Ba(I) =A. Ch6 Y 2.4. Vz1(8)= 8,8E J, nen0 ~ 1 ~ 1 va beJia E A taco 0 ~a(I) ~a(l). Buyra 0 ~ :~~~ 1. Dodovait E J tacodun" . , a(l) gw It - a (l) I ~ 1. 31 2.2.1 Bai toan bien thli nh§.t a) DM rp(t)= a~l)(A - Ba(I)) +Bt,t E J, nhu trong ehli y 2.3, thl rp'(t)= B v6i t E J va rpnhu th§ thoa di@uki~nbien (2.22): ex(rp)=A, rp'(1)= B. Til biJu thdc cua 'P ta co 'P(t) = 11)+B (t- :i~D;nhl1 v~y, v6i moi t E J: . IAI a(I) j<p(t)1~a(l) +IBI. t - a(l) ,. Theo eM. y2.4, It- :i~~ ~ 1,vata co Itp(t)1~ ~~I)+ lEI. guy IAI Ilrpll ~ ex(l)+IBI. ra (2.25) b) Ta duabai toanv@d<;1ngthu§,nnh~tv6ibi§n z bfLngbi§n d6i x(t)= z(t)+rp(t). (2.26) Tru6etien,tv (2.26)ta eo a(z) =a(x - rp)=ex(x)- a(rp)= A - A =0, z'(t) =x'(t) - rp'(t) =x'(t) - B, z'(1)=x'(1)- rp'(1)= B - B =O. z'(t)=x'(t)- rp'(t)=x'(t)- B cland§nz"(t)=x"(t). Fx(t) = F(z +rp)(t). Hx'(t) = H(z' +rp')(t). f(t, x(t),Fx(t), x'(t),Hx'(t)) = = f (t,z(t) + rp(t),F (z + rp)(t),z'(t) + B, H (z' + B) (t)). V6i t E J, (x,u,v,w)E }R4,d~t g(t,x,u,v,w) = f(t,x+rp(t),u,v+B,w), va v6i 'YE X dM F* ('Y) = F ('Y+ rp),H* ('Y) = H ('Y+ B). Th§ thl f(t, x(t),Fx(t), x'(t),Hx'(t)) = = 9(t, z (t), F (z + rp)(t) , z' (t), H (z' + B) (t )) =g(t,z(t),F*z(t),z'(t),H*z'(t)). - 32 Phuongtrlnh theox band§,utrd thanh z"(t)=g(t,z(t),F*z(t),z'(t),H*z'(t)), (2.27) a(z) = 0,z'(1)=O. (2.28) R6 rangbaitm1n(2.21)(2.22)conghi~mx khivachikhibaitmin (2.27)(2.28)conghi~mz. Dinh ly 2.4. Cia SV:j thoadi~uki{?nsau (H1) T8n tQ,iA, B, L1,L2 E JR saGchoL1 ~B ~L2 vii j(t,x,u,L1,w)~0~ j(t,x,u,L2,w) vdihauh~tt E J vii m9i (x,u,w) E [A,B, L1,L2,a; F, H]JR. (2.21)(2.22)co nghi{?mx saGcho IIxll~max{B- L" L2- B}+ ~1;)+ IBI, vii L1 ~ x'(t) ~ L2,t E J. (2.29) Chung minh. Ta se ap d1).ngd!nh1y2.1cho bai toan thu§,nnhat (2.27)(2.28).Ki~m chvng F*, H* E D 1ade dang.Chi con di xac nh~nham 9 thoa man di@usail day,t E J va v6i mQi (x,u,w) E [Ll - B, L2- B; F*, H*]JR, g(t,x, u,L1 - B, w) ~ 0 ~ g(t,x, u,L2 - B, w). a) Lay t E J,(x,u,w) E [L1- B,L2 - B;F*,H*]JR bat ky thl (x+ <p(t),u,w) E [A,B, L1,L2,a; F, H]JR,tvc 1a Ix + <p(t)I ~max{L2- B, B - Ld + ~1i)+ IBI, lul ~p(F,[0,max{ILl- BI, IL2- BI}+~111)+IBI]x), va Iwl ~ p(H, (L1,L2)x). Khing d!nhnayduQcchvngminhngaysail day. (x,u,w) E [L1- B,L2 - B;F*,H*]JR chota Ixl ~max{L2- B,B - Ld, lul~p(F*,[Ll - B,L2 - B]x), Iwl ~p(H*,(L1- B, L2 - B)x). 33 Biitdftutli Ixl ~max{L2-B,B-LI}.Ta colx+~(t)1~ Ixl+I~(t)1 ma1~(t)1~ ~~I)+IBInenIx+~(t)1~max{L2-B,B-LI}+ ~~I)+IBI= max{IL1- BI, IL2- BI}+ ~~I)+ IBI. V6i lul ~ p(F*, [L1- B,L2 - B]x) ta phaiki@mtra lul ~p(F,[0,max{IL1- BI, IL2- BI}+ ~111)+ IBI]x). Liiy y E [L1-B, L2-B]x biitky thillyll ~ max{L2-B, B-LI}. Luc do,theo(2:25),taco y+~ E [0,max{IL1-BI, IL2-BI}+ ~~I)+IBI]x. Ta danhgia F*y: IIF*yll = IIF(y+~)II~p(F,[0,max{IL1-BI,IL2-B!}+ ~~I)+IBI]x). Dieunayxayra v6imQiy E [L1- B, L2 - B]x biit ky nenta dl1<;:Jc p(F*, [L1- B,L2 - B]x) ~ p(F,[0,max{IL1- BI, IL2- BI} + ~111)+ IBlJx). Do do n§u co lul ~ p(F*, [L1- B, L2 - B]x) thl lul ~ p(F, [0,max{IL1- BI, IL2- BI} +~~I)+ IBI]x). Bay gidIiiy y E (L1 - B, L2 - B)x biit ky ta coy + B E (L1,L2)x. Theocachd~t,IIH*yll= IIH(y+B)II ~p(H,(L1,L2)x)VIY+B E (L1'L2)x. Nhl1th§ thl p(H*, (L1- B, L2- B)x) ~p(H,(L1,L2)x),va n§ucolwl ~ p(H*, (L1-B,L2-B)x) thl secolwl ~ p(H, (L1'L2)x). b) Liiy t E J, (x,u,v,w) E [L1 - B,L2 - B;F,H]IR biit ky, ta co g(t,x,u,L1- B, w) = f(t, x +~(t),u,L1,w) vag(t,x,u,L2- B,w)= f(t,x + ~(t),u,L2,w);ma (x + ~(t),u,w) E [A,B,L1,L2,a;F,H]IR nentli dieuki~ncuaf ta dl1<;:Jcf(t, x +~(t),u,L1,w) ~ 0 ~ f(t, x + ~(t),u,L2,w).Nhl1vi;1y,v6it E J, (x,u,v,w) E [L1-B, L2-B; F*,H*]IR biitkythl g(t,x,u,L1- B,w) ~0~g(t,x,u,L2 - B,w). c) TheodinhIf 2.1,baitoan(2.27)(2.28)conghi~mz saocho IIzll~max{B- L1,L2 - B}, L1 - B ~z'(t)~L2- B v6it E J. e)x(t)=z(t)+~(t)Ianghi~mcua(2.21)(2.22). Tachangminhx =z+~thoa(2.29). V6i L1 - B ~z(t) ~L2- B vax'(t)=z'(t)+~'(t)=z'(t)+B: 34 L1 :::;x'(t) :::;L2. Theo (2.26), IIxll :::;IIzll+II'PII. Thea (2.25), 11'1'11~~1~)+lEI,vaIlzll~ma.x{E- L" L2+E}thj I!xl! .:; max{B - L" L,+ B}+~~I)+IBI. x 180nghi~mcua (2.21)(2.22)thoa(2.29). Dinh ly 2.4changminhxong. 0 2.2.2 Bili toan bien thil hai Ta vansli d\mgbiend6ix =z +<p,<p(t) = at!) (A - BOI(1))+Bt, t E J, d@co(2.30)(2.31): z"(t)=g(t,z(t),F*z(t),z'(t),H*z'(t)), (2.30) a(z) =0,z'(O)=o. (2.31) Caehamg,F*,H* gi5ngnhl1khi dl1abai toanbientha nhatve d~ngthu§,nnhat: +g(t,x,u,v,w) = j(t,x+'P(t),u,v+B,w), t E J, (x,u,v,w) E IR4, + F*y = F(y +cp),Y E X, + H*y = H(y +B), y E X. Xet dieuki~n(H2): T6nt~iA, B, L1,L2E IRsaDchoL1 :::;B :::;L2va j(t,x,u,L2,w):::; 0:::;j(t,x,u,L1,w) vdih§,uh@tt E J, vdi mQi(x,u,w) E [A,B, L1,L2,a; F, H]JR. Djnh ly 2.5. Cia sitj th6adi€u ki~n(H2).Khi do (2.21)(2.23)co nghi~mx thoa(2.32), IIxll~max{B-Ll, L,-B}+ ~~I)+IBI,t E J, vaL, ~x'(t)~L" t E 1. (2.32) 35 Chung minh. D~ap dl).ngdinh ly 2.2,ta chi c§,nki~mtra ham9 thoadi@uki~nsau,khi t E J va(x,u,w) E [L1- B, L2- B; F*, H*]]R: g(t,x,u,L2 - B,w) ~0~g(t,x,u,L1 - B,w). Trangdinh ly 2.4ta da lammQtdi@utuongtv nhu v~yrai. D 2.2.3 Bai tmin bien thli ba D~tcp(t)= A(l - t) + Bt,t E J thl cp(O)= A,cp(l)= B. Dung (2.26),x = z + <p,ta th~yz(O)= x(O)- <p(0) = A - A = 0 va z(l) =x(l) - cp(l)= B - B =O.Vdibi@nd6inay,(2.21)trdthanh z"(t)=f(t, z(t)+cp(t),F(z +cp)(t),z'(t)+B - A,H(z'+B - A)(t)). Vdi y E X d~tF*y = F(y +cp),H*y = H(y +B - A), convdi t E J, (x,u,v,w) E JR4d~tg(t,x,u,v,w) = f(t, x+<p(t),u v+B-A, w) ta cophuongtrlnh z"(t)=g(t,z(t),F*z(t),z'(t),H*(z')(t)). Bai taan (2.21)(2.24)dU<;:icdua v@d~;lllgthu§,nnh~t (2.33)(2.34): z"(t) =g(t,z(t),F*z(t),z'(t),H*z'(t)), (2.33) z(O)=0,z(l) =O. (2.34) x la mQtnghi~mcua(2.21)(2.24)n@uvachin@uz la nghi~mcua (2.33)(2.34). Trudch@t,vdiham/',/,(t)= cp(t)-B+A= 2A-B+t(B-A), t E J, taconh~nxetmin{2A-B,A}~/,(t)~max{2A-B,A},t E J. Th~t v~y,ta xet l§,nlU<;:ittangtrudngh<;:ip: + N@uA = B thl /,(t) = 2A - B = A vamin{2A- B,A} = max{2A- B, A}; + N@uA >B thl VIB - A <0ta co/,(t)= 2A - B +t(B - A) ~ 2A-B =max{2A-B,A}va/,(t)=2A-B+t(B-A) ~2A-B+ (B - A) =A =min{2A- B,A}; 36 +N~uA 0taco,(i) =2A- B +t(B - A) ~ 2A - B =min{2A- B, A}va,(i) =2A- B +t(B - A) ~2A - B + (B - A) =A =max{2A- B, A}. Xet di~uki~n(H3), T6nt1;1iA, B, L1,L2,L3,L4 E JRsaochoL1 ~ B - A ~ L2, L3~B - A ~L4va J(t,x,U,Ll'W) ~0~ J(t,x,U,L2'W), J(t,x,U,L4,W) ~0 ~ J(t,x,U,L3,W) vdih§,uh~tt E J, vdimQi(x,u,w) E (A,B,L,M;F,H)]R, va L = min{L1,L3},M = max{L2,L4}. Ta codinhly 2.6chobaitoaD(2.21)(2.24). D!nh ly 2.6. Cia sitcodituki~n(H3).(2.21)(2.24)conghi~mx thoa L+min{2A-B,A}~x(t)~M+max{2A-B,A},L ~x'(t)~M,VtE J. (2.35) Chung minh. Ta seapd\mgdinhly 2.3cho(2.33)(2.34). a) Ta ki@mtra rAngvdit E J, (x,u,w) E (L - B +A,M - B + A;F*,H*)]Rthl (x+~(t),u,w) E (A,B,L,M;F,H)]R, hay L +min{2A- B,A}~x+~(t)~M +max{2A- B,A}, lul~p(F,(L +min{2A- B,A},M +max{2A- B,A})x), vaIwi~p(H,(L, M)x). (x,u,w) E (L - B +A, M - B +A;F*, H*)]Rnghlala L - B +A ~x ~M - B +A, lul ~p(F,(L - B +A,M - B +A)x), vaIwi~p(H,(L - B +A,M - B +A)x). N~uL-B+A ~x ~M -B+A thlL-B+A+~(t) ~x+~(t)~ M-B+A+~(t) tticlaL+,(t) ~x+~(t) ~ M+,(t). Vdinh~n xetnhlltrendayv~ham,(t) =~(t)- B +A ta sedll<}C L +min{2A- B,A}~x+~(t)~M +max{2A- B,A}. 37 Ltiy YE (L-B+A, M -B+A)x btitky ta coL-B+A ~ y(t) ~ M - B +A vasuyray+'PE (L +min{2A- B,A},M +max{2A- B, A})x. Di@unaydaTIdenIIF*Yll = IIF(Y+'P)II~p(F,(L+min{2A- B, A},M + max{2A- B, A})x). Vf}.yco p(F*, (L - B +A, M - B + A)x) ~p(F,(L +min{2A- B,A},M +max{2A- B,A})x) va nhl1 the,lieUlul ~p(F*,(L - B +A,M ---B +A)x) thi lul ~p(F,(L +min{2A- B, A},M +max{2A- B, A})x). NeuY E (L-B+A,M -B+A)x btitky thi y+B-A E (L,M)x vaIIH*yll= IIH(y+B - A)II ~p(H,(L,M)x) suyrap(H*,(L - B + A, M - B + A)x) ~ p(H, (L, M)x). Vf}.ykhi Iwl ~ p(H*,(L - B + A, M - B + A)x) thi Iwl ~ p(H, (L, M)x). b) Ltiy t E J,(x,u,w) E (L - B + A,M - B + A;F*,H*)]R,theo cachdM thi g(t,x,u,L1- B +A,w) = j(t,x +<p(t),U,Ll'W)va g(t,x,u,L2-B+A,w) = j(t,x+'P(t),u,L2,w),ma(x+'P(t),u,w)E (A,B, L, M; F, H)]Rlientheodi@uki~ncua j ta thu dl1<;c j(t,x+'P(t),u,L1,w) ~ 0 ~ j(t,x+<p(t),u,L2,w). Tv dog(t,x,u,L1- B, w) ~0~g(t,x,u,L2- B, w). Tl1ongtv, g(t,x,U,L3 - B, w) ~0~g(t,x,U,L4- B,w). c) Theodinhly 2.3,bai toan(2.33)(2.34)co nghi~mz, saochov6i t E J, L-B+A ~z(t)~M -B+A vaL-B+A ~z'(t)~M -B+A. R6rangx = z +'Pla nghi~mcho(2.21)(2.24)vax thoa(2.35): +DanhgiaL- B+A ~z(t) ~M - B+A keotheoL+min{2A- B, A} ~x(t) ~ M +max{2A- B, A}. + Vi x'(t) = z'(t) +<p'(t), maL - B +A ~z'(t) ~ M - B +A va 'P'(t)=B - A lien L ~x'(t)~M. Ta chl1ngminhxongdinhly 2.6. 0