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
38 trang |
Chia sẻ: maiphuongtl | Lượt xem: 1810 | Lượt tải: 0
Bạn đang xem trước 20 trang tài liệu Luận văn Ứng dụng phương pháp bậc TôPô trong nghiệm tuần hoàn của phương trình vi phân, để xem tài liệu hoàn chỉnh bạn click vào nút DOWNLOAD ở trên
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