tài liệu tham khảo ứng dụng phương pháp bậc tôpô trong nghiệm tuần hoàn của phương trình vi phân, kết luận
CHUaNG NGHlltM TUAN HoAN CUA PHUONG TRINH VI PHAN TmJONG 2.1 Dfnh ly chinh : La"y 1=[0, 1], va f: I x RD~ RD, thoa nhii'ng di~u ki~n caratheodory, va ky hi~u x la chu§:nEuclide cua I I x E RD,va (x, y) la tich va htidng cua x va y Trong chtiang nay, chung ta se chung minh st! t6n tC;iinghi~m cho bai tmin (2.1) x1(t)= f(t,x(t)) , tEl { x(o) = x (1) nhii'ng nghi~m se dtiqc gQila tu~n hoan Chung ta ky hi~u X la khang gian cua C (1, RD)ma nhung ph~n tll'cua no thoa di~u ki~n thu hai (2.1) vdi chu§:nd~u thtiong dung la Ixl = maxlx(t)\ tEl ' Z =L (I, RD),vdi chu§:nthtiong dung la IIxiiI = fixet)ldt I va domL la khang gian cua X, ma nhung ph~n tii' cua no la lien t\lC tuy~t dot Anh x~ L va N l~n ltiqt dtiQCxac dinh tren domL va X, bdi (L.x)(t) = x'(t), (Nx) (t) =f(t, x(t)) Vdi tEl, thl L va N la'ynhung gia tri Z, va bai loan (2.1) ttiang duang vdi vi~c giai phuong trlnh thu gQn Lx = Nx 25 ngoai fa, tll slj ma ta chuang 1, = {x E dam KerL IrnL d = {z EZ day Px L : xCi) = x(o), Vt E I} : fZ(I)dl = x(o), Qx = ImP =o} = KerQ = fz(t)dt I VI the'L la anh x'] Fredholm vdi chi sO'zero va tinh chat 1.5, ta co N la L - hoan loan lien tl;lCtrong X Ta c~n m(>tb6 d€ d€ chung minh slj t6n t'].inghi~m B6 d~ 2.1 : Cho r > va V E Cl(Rn, R), thoa man V' (x) * 0, vdi x I I =r ? d day V' la gradient cua V, va Iffy G :X (Gx) (t) va H =L - Z, duQcxac dinh bdi ~ =- V' (x(t)), tEl G, vdi X, Z va L duQCxac dinh nhutren Thl H E CL(B(r)) va !DL(H,B(r))! =\Do(V',B(r))\ Chung minh Ta xet anh x'] H :XxI~Z (x, A) H Lx - AGX- (1 - A) QGx Vdi Q duQc xac dtnh d ireD Thl H la L - hoan toaD lien tl;lcireD X x I Ne'u (x, A) E X x I, saG cho H (x, A) =0, thl x la lien wc tuy~t d6i - tu~n hoan va 26 (2.2)x'(t)=- AV'(X(t))-(l-A)fV'(X(S))ds, I '\itEI Do x' lien wc tren I va liy tich vo huang hai v~ (2.2) vai x'(t), tich phan tren I va dung Hnhchit - tu~n ho~mcua x, ta duQc fix' (012dt = I Vi v~y x(t) x'(o) =x(o), '\it E I, bdi vi = -AV'(x(o))- (1- A)fV'(x(s))ds I Bdi (2.2), x(o) thoa phudng trlnh V'(x(o)) =0 f)i~u guy ding \xlo = Ix(o)\"* r Theo tinh chit bit bi~n d6ng luau cua ly thuy~t b~c, ta co : DL(H,B(r)) = DL(H(.,l),B(r)) = DL(H(.,O), B(r)) = DL(L - QG,B(r)) Nhung QG : X ~ ImQ, vai z= ImL~ImQ Vi v~y, bdi tinh chit (1.13), ta co \DL(L - QG,B(r))1 = \Do(-QGKerL,B(r)nKerL)1 =IDo(V',B(r))! d day B(r) duQc ky hi~u qua c~u tam 0, ban kinh r B6 d~ duQc chung minh Bay giC1ta chung minh dinh ly chinh cua chudng Binh Iy 2.2 Gia sa ding nhung di~u ki~n sail day xay 27 (i) co mQt V E C1 (Rn, R+), vdi : Vex) ~ + 00, ne'u I x I ~ 00 va a ELI (I, R+), thoa man s aCt) (2.3) (V' (x),f(t,x») Vdi mQi x ERn, mQi tEl (ii) T6n t~i r > va W E C1 (Rn \ B(r), R), cho (V' (x), w' (x» > VdimQix, Ixl ~r,va (2.4) f(W'( x(t)),f( t,x(t)))dt I s0 Vdi mQi anh x~ lien tl,lctuy~t d6i - tugn hoan x : I ~ Rll, vdi minlx(t)1~ r tel Thl bai toan (2.1) co it nha't IDQtnghi~ID ChUng mink Ta mu6n ap dl,lngdinh 19 (1.17) vdi F =L - N va H = L - G nhtt b6 d~ (2.1) va tinh cha't cQng tinh, cling vdi V' (x) x E Rn vdi I x ~r I "* 0, cho IDQi H eCL(B(p») vdi mQip ~ r £)gu tien chung ta chI ding nhii'ng nghi~m - tugn hoan cua hQ nhii'ng phu'dng trlnh x' (t) =- (1 - A) V' (x(t» + Af (t, x(t», tEl, AE10,1[ la mQt ti~n bi ch~n Ne'u di~u kh6ng Kay fa, se co mQt day (An)nEN*vdi An E ]O,l[ 28 va mQt day (xn)n EN*, vdi IXnlo2 n va xn la mQtnghi~m - tu~n hoan cua (2.5): x~(t)=-(l-An)V'(Xn(t))+Anf(t,xn(t)), tEl, n EN* Ngoai fa, vdi m6i tEl, n E N*, dung di~u ki~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 fQng Xnva a fa R bdi - tu~n llOan, ta du'Qc vdi m6i T E R va m6i t E [T, T + 1] v( xn (t)):S;V(xn (T)) + Jf r a(s)ds Do tinh - tu~n hoan cua Xn,di~u suy fa ding (2.6) max V( Xn(t)) :s;mill V( xn (t)) + Ilal!l tEl tEl Bay giO, n€u tn E I cho \Xn (tn)1 = maxlxn (t)1= \Xnl n tEl Ta co max V(Xn(t)) V(xn (tn)) tEl va bdi di~u ki~n (i) V(Xn(tn))~oo n€u n ~oo Ngoai fa bdi (2.6) mill V(xn(t))~ tEl 00 n€u n ~ 00 Hoan loan tu'dng tt,I'suy fa ding minlxn(t)1 ~ 00ne"u n ~ 00 tEl 29 La'y nl E N* cho, voi m6i n ~ nl minlxn (t)j ~ r tEl Bi~u sail duQc suy tti' (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, bdi tinh - tuftn hoan cua Xn,va voi n ~ nl, ta suy tti' (ii) = -(1- An) f(V'( xn (t)),W'(xn (t)))dt + An J(f(t,xn(t)), W'(xn (t)))dt < I I di~u mall thu~n VI v~y nhG'ng nghi~m cua hQ nhG'ngphuong trlnh 1a ti~n bi ch~n, vOi ti~n bi ch~n duQc ky hi~u bdi p Chung ta co th€ chQn p chop ~ r Bay gio dung b6 d~ (2.1), gia dinh (i) va tinh cha't (1.15), chung ta co IDL(H,B(p))! = IDo(V', V(p)~ = Va bdi dinh 1y (1.17), dinh 1y duQc chung minh Binb Iy 2.2' : Gia sa nhG'ng di~u ki~n san day Kay (i') T6n t~i V cl (Rn, Rr), voi Vex) ~ + 00 ne'u Va a I x I ~ 00 ELl (I, R+) cho (2.3') (V' (x), f(t, x)) ~ -aCt) voi mQi x ERn, mQi tEl (ii') T6n t~i r > va W E Cl (Rn\B(r), R), saG cho (V'(x), Wi(x)) > 0-, voi mQi x, Ix! ~rva 30 (2.4') f(W'(x(t)),f(t,x(t)))dt?:: I Vai mQi X : ~ Rn lien t~c tuy~t d6i - tuftn hO~lllvai minlx(t)l?::r tEl ChUng minh D6i bi~n, bftng cach d~t t = - T Vai mQi tEl co TEL Luc niiy(2.3') (2.3 ') vii (2.4') trd thiinh (- V'(x(1- T)),f(l- T,x(l- T)))?::-a(l- T) ~ (2.4 ) (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 Bftng cach d6i bi~n mQt Iftnnua - T = 't Ta co (2.3') vii (2.4') trd thiinh (V'(x('t) ),f( 't,x( 't))) ~ a('t) f(W'(X('t)),f('t,X('t))}t't ~ I Ap d~ng dinh 19 (2.2), dinh 19 duQc chung minh 2.2 Ung d\lng cua dinh Iy 2.2 Trong phftn niiy, chung ta se cho mQt viii ung d~ng thu vi cua djnh 19 2.2, vai sl;l'll;l'achQn d~c bi~t Ung d~ng dftu lien Iii cho phuong trinh vo huang (n = I) H~ qua 2.1 : Gia sa n = vii a.e tEl, f(t,.) Iii khong tang Thl biii loan (2.1) co 00 mQt nghi~m n~u vii chi n~u t5n t~i y E L (I, R) cho 31 (2.7) J f( t,y(t) )dt = I Chung minh * Di~u kit%nc~n : Ta Iffy Y Hi mQt nghit%mcua (2.1), suy fa Jf( t,yet))dt = Jy' (t)dt = y(l) - yea) = I I * Di~u kit%ndu : VI f(t,.) la khong tang nen ta co, voi mQi x E R va a.e tEl, xf(t,x) ~ x f(t,O) ~ \f(t,O)I.(lxl + 1) Do di~u kit%n(i) cua dinh ly (2.2) thoa man voi V(x) aCt)= f(t, 0) I =G) ( ( ul 1-1 +1) du I Bdi VI Vex) ~ + 00 ne'u x I I ~ 00 Va a ELI (I, R+), saG cho (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 Iffy x E domL la ph~n tii'tily Y saG cho : minlx(t)1~ Ilyll = f tel