Phương pháp bậc Tôpô cho bài toán biên4_2_2

Luận văn thạc sĩ toán học:Phương pháp bậc Tôpô cho bài toán biên

Dieu ki~n (AI) cua f dl1c;5Cs11d\mg:

(AI) T6n tl;1icae s6 L1 ~ 0 ~ L2 sao cho

f(t,x,u,L1'W) ~ 0 ~ f(t,x,u,L2'W)vdi h§,u h~t t E J va vdi illQi(x, u, w) E [L1,L2;F,H]IRo

Xet bai toan

(2.3)

(2.4)

(2.5)

Ta COdanh gia nghi~rn ella bai toan (2.3)(2.4) trong b6 de 2.1.

B6 d~ 2.1 Cia sit x la m{}t nghi~m cua (2.3)(2.4), A E (0,1), con

ham] thoa di€u ki~n (A 1) Th€ thi

Ilxll :::;L, L1 :::;x'(t) :::;L2, Vt E J. (2.6)

a) Cia SU:rnax{x'(t), t E J} = x'(to) > £2 V6i s6 duong x'(to) - £2 ta

Urn du<;:Jcno E N saG cho x'(to) > £2 + ~ khi n ;;?no R6 rang to i- 1,

bdi n@uto = 1 thl x'(to) = x'(1) = 0 > £2 + ~ la kh6ng thich h<;:Jp

Do x' lien tl,lC, v6i E= ~ > 0, co rnQt/5'> 0 saG cho khi It- tal < /5'

thl !x'(to) - x'(t)1 < E. Tv do, x'(t) > x'(to) - E = x'(to) - ~ khi

V~y ta phai co di~u nguQc l:;ti:max{x'(t), t E J} ( L2 +~.

b) Gia s11 min{x'(t), t E J} = x'(to) < L1 cling dua d@n mQt matithuan Ta van tim duQc n1 E N d~ x'(to) < L1- ~ khi n ~ nl Do tfnh lien t\lCcua x', co mQt 8 > 0 saGcho L1 > x'(t) ~ x'(to), Vt E [to, to+8].

Tfch phan tren [to, to + 8] phuong trlnh (2.3):

Ta s E [to,to + 8] suy ra v = x' (s) < L1 Khi do v- L2 = (v- L1) +

lto

Theo cach d~t, (x, it, w) E [L1,L2;F, H]IR,va ta co f(s, x, it, v, w) =

f(s, x, it, L1, w) ( O Di~u nay dan d@n

c) Nhu'vi;iy, vdi x la nghi~m cua (2.3)(2.4), t E J tuy y va n du ldn

ta du'Qc L1 - ~ ~ x'(t) ~ L2 + ~ Cho n -+ +00 thl co L1 ~ x'(t) ~ L2, VtE J.

d) Vi a(x) = 0 lien co CE J sao rho x(c) = o.

+ Ni\u t > c, ta c6 x(t) = x(t) - x(c) = l x'(s)ds.

Vdi LI ,;; x'(s) ,;; L2 tIll l' LIds';;l' x'(s)ds,;;1'L2dS.

Suy ra l' LIds';; x(t) ,;; l' L2ds hay Ja

vdi chu~n thong thu'dng ft.H1.;;;:TVN~!\~NI

THoU. ~!~~. t

OOlOt)~ J

D~t domD = {x E AC1(J)/a(x) = 0, x'(I) = O}, cac ham D, N

xac dinh nhu bell du6i

R6 rang D tuy@n tfnh tren Y va N lien t\lC tren Y x [0,1].

a) Ta kh~ng dinh D la mQt song anh tit domD vao D(domD). D~ co

di~u nay ta chI ra D la don anh.

+ Ham x = 0 la mQt phiin tu-cua domD thoa Dx = 0 nen 0 E kef D + Gia su-x E kef D Ta co Dx= 0 hay la x" = O Th@thl x"(t) = 0 v6i IDQit E J guy ra x' la mQt ham h~ng tren J, ma x E domD nen

x'(I) = 0, va nhu vi;iyx'(t) = 0 v6i mQi t E J: x la ham h~ng tren J.

V6i x E domD ta co a(x) = O Theo chli y 1.1 ta tim dU<;1cmQt

c E J sao cho x(c) = O Do vi;iy,x = 0 tren J.

Nhu vi;iykef D = {O}: D don anh.

b) Ta ki~m tra D-1 N : Y x [0,1] -+ Y la anh X<,l, compact: Gia su- A

la IDQtt~p con bi ch~n cua Y x [0,1], ta chang minh D-1N(A) la mQt

t~p compact b~ng cach su- d\lng dinh ly Ascoli-Arzela.

+ D-1 N(A) bi ch~n d~u:

L§,y (x, A) E A b§,t ky.

DM y = D-1 N(x, A), thl N(x, A) = Dy: N(x, A)(t) = y"(t).

Ta c6, v6i s E J, 11 y"(r)d-r = y'(l) - y'(s) = 0- y'(s) ~ -y'(s),

d day ta su- d\lng y'(I) = 0, do y E domD.

Cling VIy E domD nen a(y) = 0 va co mQtc E J sao cho y(c) = O

Vai t E J ta dU<)c l' y'(s)ds ~ y(t) y(c) = y(t).

guy fa y(t) ~ -1' 11 y"(r)d'rds= -1' 11 N(x, >.)(r)dnls,

Vdi r E [s,l] ta cling di;it x = x(r), u = Fx(r), v = x'(r), w =

Hx'(r), va co

N(x, "\)(r) = f*(r, x(r), Fx(r), x'(r), Hx'(r)) = f*(r, x, u, v, w),

811dl,lng di;ic tint thli ba cua fEe ar( J x }R4),ta tim dl1<;JcmQt

ham rjJE £1(J) 8aa cha 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 nen x E Y, va co h~ng 86 M' > 0 8aa cha Ix'(t)1 :::;

M', t E J Da do IN(x, "\)(r)1 :::;"\rjJ(r)+ (1 - "\)(M' + £2)' Ma ta co

va nhti vf!,y thlly(t)l ~ II (A4>(r) + (1 - A)(M' + L,))drds. Nhting

ly(t)1 « l' [(AII4>II£'(1) + (1 - A)(M' + L2))drds.

D@nday ta khing dint ly(t)1 bi chi;in: ly(t)1 :::;M, M > 0 1a mQth~ng 86 Di~u nay co nghia 1a ID-1 N(x, "\)(t)1 = \y(t)\ :::;M vdi mQi

t E J Ma (x,"\) E A bilt ky lien D-1 N A bi chi;in d~u.

danh gia AII</YII£l(J)+ (1- A)(M' + L2)) ~ 11</YII£l(J) + (M' + L2) DM

C = 1I<pII£l(J)+ (M' + L2) thi C > O.Ta co

+ K@thQp D-1 NAb! ch[[tnd@uva D-1 N A dong lien t~c, theo d!nh

ly Ascoli-Arzela, D-1 N la toan tv compact tren Y x [0,1].

c) Phuong trinh x"(t) = Af*(t,x(t),Fx(t),x'(t), Hx'(t), A) duQc vi@t

thanh Dx = AN(x, A) Ta vi@tlC;1i t~p 0, v6i n E N tuy y:

0 = {x E Y/llxll < max{L2, -Ld +~,

L1 - ~ < x'(t) < L2 +~, t E J}.

Theo bE>d@2.1, v6i A E (0,1), n@ux la nghi~m cua phuong trinh

x"(t) = Af*(t, x(t), Fx(t), x'(t), H x'(t), A)

thi x ph.:Uthoa L1 ~ x'(t) ~ L2,t E J va Ilxll ~ max{ -L1, L2} Noi

khac di, x kh6ng th@ n~m tren BO Nhu v~y, Dx - AN(x, A) -I 0 v6i

ffiQi(x, A) E (domD n BO) x (0,1).

Cac di@uki~n trong bE>d@1.1 duQc thoa man Do do, phuong trinh

Dx = N(x, 1) co nghi~m trong domDn 0, hay bai toan sail co nghi~m

trong 0:

x"(t) = 1.f*(t, x(t), F(x)(t), x'(t), H(x')(t), 1),

a(x) = 0, x'(l) = O.

Ta chi con ki@mtra r~ng day chinh la (2.1)(2.2).

Ilxll ~ max{ -L1, L2} + ~.Cho n ~ +00 ta co L1 ~ x'(t) ~ L2 vaIlxll ~ max{ -L1, L2}.

2.1.2 Bai toan bien thil hai

Khi A = B = 0 ta 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~u ki~n (A2) cua 1: T6n t~i £1 :::;;0 :::;;£2 saG cho

f(t,x,u,£2,W):::;; 0:::;;f(t,x,u,L1'W) v6i hall h@tt E J va v6i mQi (x, u, w) E [£1, £2; F, H]IR.

Tru6c h@tta xU'1y di~u ki~n bien a(x) = 0, x'(O) = O.

a) Thay t = 1- s, dM x(t) = u(s) Ta co u(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).

V6i x E X ta d~t x* 1a ham x*(t) = x(l-t), t E J R6 rang x* E X.Boi u*(t) = u(l-t) = x(t) nen x = u*; va boi -(u')*(t) = -u'(I-t) =

(u(1 - t))'(t) = x'(t) ta vi@tx' = -(u')*.

D~t a* : X -+JR.,F*, H* : X -+X 1a cac ham dinh boi

a*(x) = a(x*),

H*x(t) = Hx*(1 - i), t E 1.

Vi a*(x) = a(x*) ta co a*(u) = a(u*), ma u* = x lien a(u*) = a(x).

N~ua(x) = 0 thl phai co a(u*) = 0, hay la a*(u) = O.

b) Ta ki@mchang a* E A, F*, H* E V.

+ a* tuy~n tfnh, bi ch~n va tang.

Thi;itv~y, l~y x,y E X b~t ky va c E IRtuy y ta co a*(x + cy) =

nen (x + cy)*(t) = x*(t) + cy*(t) Do a tuy~n tfnh lien a(x + cy)* =

a(x*) + ca(y*) Suy ra a*(x + cy) = a*(x) + ca*(y): a* tuy~n tfnh.

a* bi ch~n: Ila*(x)11 = Ila(x*)11~ M.llx*11= M.llxll,Vx E X.

Cia s11co x, y E X saGcho x(t) < y(t) vdi t E J Th~ thl x(l

-t) < y(l - t), va x*(t) < y*(t) VI a tang lien a(x*) < a(y*), hay la

a*(x) < a*(y) Do do a* la ham tang

Vi;iya* E A.

+ Bay gid ta ki@mtra F* E V H* E V la hoan toan tuong t\!.

TrUdCtien ta chang minh F* lien t\lC tren X.

L~y {xn} eX, Xn -+X E X khi n -++00 Ta c§,n F*xn -+F*x

IFx~(1- t) - Fx*(l - t)1 ~ IIFx~- Fx*ll.

Vi F lien t\lC tren X lien khi x~ -+x* thl IIFx~- Fx*11 -+O Day

la di~u ta dang c§,n:F*Xn -+F*x trong X hay F* lien t\lC tren X.

D@co F* bi ch~n ta S11d\lng tfnh bi ch~n cua F Vdi x E ~ b~t

ky, ~ bi ch~n trong X, ta co x* E ~ va IIF*xll = IIFx*ll Ma IIFx*11

bi ch~n lien IIF*xll cling bi ch~n

Ta da ki@mtra xong F* E V H* E V tuong t\!.

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).

x, u, v, wEIR, khi do phuong trlnh (2.9) dU(5cvi~t 130

u"(s) = g(s, u(s), Fgu(s), u'(s), Hgu')(s)).

Nhu v~y, (2.7)(2.8) dU(5cdua v~ (2.10)(2.11):

u"(s) = g(s, u(s), Fgu(s), u'(s), Hgu')(s)), (2.10)

Tit a), b), c) ta k~t lu~n n~u bai toan (2.10)(2.11) co nghi~m u thl bai toan ban dati (2.7)(2.8) se co nghi~m x = u*, x(t) = u(l-t), t E J.

Sv t6n t~i nghi~m cua (2.7)(2.8) dU(5cth@hi~n trong dinh ly 2.2.

D!nh 15' 2.2 Cia 871f thoa di€u ki~n (A 2) The thi (2.7)(2.8) co nghi~m thoa (2.6).

Chung minh. Ta ap d\lng dinh ly 2.1 cho bai toan (2.10)(2.11) b~ngcach ki@mchllng di~u bell du6i

v6i hall h~t t E J va v6i mQi(x, u, w) E [-L2, -L1; Fg,Hg]]R.

Tru6c tien ta chllng minh p(Fg, [-L2, -L1]x) ~ p(F, [L1,L2]x).L~y y E [-L2, -L1]x b~t ky 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.

Hon mia, 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).

Phuong trlnh x"(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)

Vdi Y E X, t E J, di;it Fgy(t) = F*y(t) = Fy*(l - i), Hgy(t) = H*(-y)(t) = H( -y)*(l - t) thl Fg,Hg xac dinh va la cac ph§,n tv cua

t~p D Ti~p t\lC di;itg(t,x,u,v,w) = f(l- t,x,u, -v,w) vdi t E J va

x, u, v, w E ~, khi do phuong trlnh (2.9) duc;Jcvi~t la

u"(s) = g(s, u(s), Fgu(s), u'(s), Hgu')(s)).

Nhu v~y, (2.7)(2.8) duc;Jcdua ve (2.10)(2.11):

Tl1a), b), c) ta k~t lu~n n~u bai toan (2.10)(2.11) co nghi~m u thlbai toan ban d§,u (2.7) (2.8) se co nghi~m x = u*, x( t) = u(l- t), t E J.

Sv ton ti;1inghi~m cua (2.7)(2.8) duc;Jcth~ hi~n trong dinh 15'2.2

D~nh ly 2.2 Gia sV: f th6a di€u ki~n (A2). Th~ thi (2.7)(2.8) co nghi~m th6a (2.6).

Chung minh. Ta ap d\lng dinh 15'2.1 cho bai toan (2.10)(2.11) b~ngcach ki~m chang dieu bell d udi

g(t,x,u, -L2,w):::;; 0:::;;g(t,x,u, -L1,w)

vdih§,uh~t t E J va vdi mQi(x, u, w) E [-L2, -L1; Fg,Hg]rn;.

Trudc tien ta chang minh p(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) Tl1 d6, IFgy(t)l:::;;p(F, [L1,L2]x), roi guy ra IIFgy11 :::;;p(F, [L1,L2]x).

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~c kH~mtra p(Hg, (-L2, -L1)x) ~ p(H, (L1' L2)X) la tuong tV'.Ltiyz E (-L2, -L1)x tuy y, ta chang minh duQc IIFgZl1~ p(H, (L1, L2)x).

Bay gid, trd 11;1ibai toan dang xet, ltiy (x, u, w) E [-L2, -L1; Fg, Hg]JR

btit ky thl se colxl ~ max{-L1,-(-L2)} = max{-L1,L2)}, lul ~

p(Fg, [-L2, -L1]x) va Iwl ~ p(Hg, (-L2, -Ldx). Cling v6i hai k@t

qua tren day ta khiing dinh ding n@u (x, u, w) E [-L2, -L1; Fg, Hg]JR thl (x, u, w) E [L1' L2; F, H]JR.

Llic nay theo di~u ki~n (A2) cua ham f 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)

g(t,x,u, -L2,w) ~ 0 ~ g(t,x,u, -L1,w).

Cac di~u ki~n cua dinh ly 2.1 duQc thoa man Bai toan (2.10)(2.11)

co nghi~m u thoa Ilull ~ max{ -L1, L2}, -L2 ~ u'(s) ~ -L1, S E J.

Nghi~mcua bai toan (2.7)(2.8)la x = u*,x(t) = u(1 - t), t E J Do

lIu*11= Ilullva x'(t) = -u'(1 - t) nen nghi~m x nay thoa (2.6):

Ilxll~ max{-L1, L2}, L1 ~ x'(t) ~ L2,t E J.

2.1.3 Bai toan bien thil ba

Khi A = B = 0 ta co (2.12)(2.13)

x"(t) = f(t, x(t), F(x)(t), x'(t), H(x')(t)), (2.12)

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 + 1n < v :( L4

L2 + 1.n < v :( L2 + 1n

L2 < V :( L2 + 1 n L1 :( V :( L2

Chung minh. Ta ki~m tra l§,nlu<;:Jtcae tinh chat cua ham Caratheodory

Truae het ta khiing dtnh, khi n du lOn, hn(t, x, u, v, w) = j(t, X, ii, V, w) vai mQi t E J, (x, u, v, w) E }R4.ThM v~y, l&y t E J, (x, u, v, w) E }R4tuy y, 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).

N~u v:( L3 thl hn(t,x,u,v,w) = j(t,x,u,L3,w) = j(t,x,u,v,w).

N~u L2 < V < L4 thlluon co j(t,x,u,v,w) = j(t,x,u,v,w). Vi

N~u L1 :( V :( L2 thl co ngay hn(t, x, u, v, w) = j(t, X,U,v, w) v6i

N~u L3 < V < L1 thl j(t,x,u,v,w) = j(t,x,u,v,w), ta co mOt

86 n2 ;:: no sao cho 1n < L1 - v khi n ;:: n2, tac la L1 - 1n > v khi

Lay (x,u,v,w) E }R4bat ky Gilt sit {xd, {ud, {vd, {Wk}la cac

day Iftn 111<;JthOi t1.lv~ x, u, v, w, ta chang minh hn( t, Xk, Uk,Vk, Wk) ~

th~ vi§t hn(t, Xk, Uk, Vk, Wk) = j(t, Xk,Uk,Vk,Wk) va hn(t, x, u, v, w) =

f(t, x, u, v, w) Do do, d~ sit d1.lngdl1<;1c tinh lien t1.lCclia j(t,., ,.,.) ta

phlti co Xk~ X,Uk~ U,Vk~ V,Wk ~ w Chang minh Xk ~ X,Uk ~

U,Wk -+ w hoan toan tl1dng tv chang minh Vk~ V.

Ta vi§t lq.i ky hi~u f):

A

y=

L4 Y L3

y> L4 L3 :( Y :( L4

Y < L3'

+ N@uv > L4 thl V = L4 Do Vk-7 v lien ton t9>is5 k1 saD cho

L4 - E < Vk ~ L4} V6i k ~ k1 tuy y, n@u k E C1 ta co IVk - vi =

IL4- L41 = 0 < E, con n@uk E C2 thl Ivk - vi = Ivk - vi < E.

+ N@u L3 < V < L4 thl v = v Cling do Vk-7 v lien ta Urn du<;1c

mOt k1 saD cho L3 < Vk < L4, k ~ k1 Ta co Vk = Vk, k ~ k1 R6 rang

Vk = Vk -7 V = V.

+ Trudng h<;1pv = L3 gi5ng trudng h<;1pv = L4, con v < L3 gi5ng V> L4-

Ta khiing dtnh du<;1Cn@uVk -7 v thl Vk -7 v.

Ta da chang rninh xong hn E Car(J x JR4)v6i rnoi n dli 16n. D

Chli Y 2.2. Neu {xn}, {un}, {vn}, {wn} la cae day ham Ian lurt h(Ji t'l),vt x, u, V, w trong X th'i ta co v(ji hau het t E J

-hn(t, xn(t), un(t), vn(t), wn(t)) -7 f(t, x(t), u(t), v(t), w(t))

khi n -7 +00.

Chung minh. Theo chli y2.1, ta tlrn dU<;1Cn1 d~ khi n ~ n1 thl se co

JR4tuy y Do do, n@u n- - ~ 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 rnQi s E J Trong chli y

2.1 ta da chI ra n@uday s5 {an} hOi t\l v@a thl an hOi t\l v@a Sit d\lng-

di@unay v6i an = xn(s), a = x(s) ta co xn(s) - -7 x(s), s E J b§,t ky.

TucJng tv, un(s) -7 u(s), vn(s) -7 v(s), wn(s) -7 w(s) v6i s E J b§,t

ky

Li;1ico j(t,.,.,.,.)- lien t\IC tren JR4 v6i hiiu h@t t 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~u ki~n (A3) cua j d uqc sU'd \Ing

(A3) T6n ti;1iL1 ~ 0 ~ L2, L3 ~ 0 ~ L4 sao cho

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) v6i hiiu h@t t E J va v6i mQi (x, u, w) E (L, M; F, H)ITJ!.,trong do

L = min{Ll, L3} con M = max{L2, L4}.

c) Xet bai tmin (2.15) v6i di~u ki~n bien (2.13), A E [0,1], n ) no:

x"(t) = Aj~(t,x(t),Fx(t),x'(t),Hx'(t),A), (2.15)

ham j~ bi@udien nhu sau v6i t E J, (x,u,v,w) E JR4,A E [0,1],

ham hn xac dinh d a), va p : JR-+ JR lien t \IC tho a

p(v) ) 1, v E [L3 - l , L3] U no [L2' L2 + l no] ,

Bd d~ sau cho danh gia v~ nghi~m cua bai to an (2.15) (2.13) khi

L3 < L1, L2 < L4 Nhiic li;1i,Slj t6n ti;1icua no EN, la s6 sao cho

L2+ ;0 < L4,L1 - ;0 > L3, da duqc kh~ng dinh tru6c do.

B5 d~ 2.2 Cia SV:j thoa di€u ki~n (A 3), L3 < L1,L2 < L4, va bai

loan (2.15)(2.13) co nghi~m x vrJiA E (0,1) va n) no Tht thi ta co dank gia sau, t E J, n ) no,

L3- - ~ x(t) ~ L4 + -, L3 - - ~ x (t) ~ L4 + -.

Chung minh Theo di~u ki~n bien (2.13), x(O) = x(l) = 0, ta tlm

duqc a E (0,1) sao cho x'(a) = O

a) Gia sU'max{x'(t)jt E [0,an = x'(to) > L2+ * se daTI d@nmQt di~umati thuan

+ Trudc tien ta chung mint co mQt khaang [" v] C (to, a) saG cha x'(v) = £2,x'(,) = £2 + ~ va vdi t E b, v] thl £2 ~ x'(t) ~ £2 + ~.

Phuong trlnh x'(t) - £2 - ~ = 0 (thea bi§n t) co nghi~mtl E (to,a)

bdi x'(to) > £2 + ~ va x'(a) = 0 < £2 + ~

L§.y, = max{t2 E (to, a) : X'(t2) = £2 + ~} Khi do x'(,) =

£2 + ~,x'(,) > £2 K§t hQp vdi x'(a) = 0 < £2e), phuong trlnh

x'(t) = £2 co nghi~m t2 E (" a).

L§.y v = min{t2 E (" a) : X'(t2) = £2} R6 rang b, v] c (to,a),

£2 +~ Ta ki@mchung khing dint nay

Gia su t6n t:;tit E b, v] : x'(t) > £2 + ~ Duong nhien t i- "t i- v Bdi x'(a) = 0 < £2 + ~ nen ta Urn duQc mQt t3 E (t, a) saG eha

X'(t3) = £2 +~ Da, = max{t2 E (to, a) : X'(t2) = £2 +~} nen t3 ~ ,.

Day Ia di~u vo If VI t3 E (t, a) va t E b, v] thl phai co , < t3 Vf!}.yta

eo x' (t) ~ £2 + ~, t E b, v].

+ Khi co khaang b, v] c (to, a) saG cha x'(v) = £2, x'(,) = £2 + ~,

lTa dang xet £2 > O N~u £2 = 0 thl ta van tlm dll<;iCmQt khoang b, v] E (to, a) saD cho

x'(~) = £2 + ~,x'(v) = £2, d6ng thai £2 :::;x'(t) :::;£2 + ~,t E b,vl Th~t v~y, lily v = min{t E

(to,aI,x'(t) = O},viq = max{t E (to, v), x'(t) = £2 +~= ~}, thl b, v] lit khoang d.n till Chi d.n ki~mchung £2 :::;x'(t) :::;£2 +~, t E [" vI N~u co t E [" v] saGcho x'(t) > £2 +~thl do x'(v) =0

nen co tl E (t,v): x'(td = £2 +~, mall thuan voi ~ = max{t E (to,v),x'(t) = £2 + ~= ~} N~u

cot E b,v] saGcho x'(t) < £2 thl do x'(T) =£2+ ~nen co mQt t2 E (T,t): x'(t2) = £2 = 0, mall thuan voi v = min{t E (to,a],x'(t) = O}.