bài toán hai điểm biên kỳ dị 8

luận văn trình bày về sự tồn tại nghiệm của một số phương trình vi phân cấp hai thỏa mãn điều kiện biên trên bằng phương phát điểm bất động của ánh xạ compact và lý thuyết phổ của toán tử đối xứng, liên tục hoàn toàn trên không gian Hilbert

! ::::::: 1::': 1.:::.::::::: i:.:::::: i':::.:.: : : r':r:.":" :"ij:::.I, i:," 1,.:'j:::.":'i:j:j::,::'i:::::: :j::jj:, : 'ijL~_llffl ~ w.!$@§Qf 81

CHu'dNG 3 vAl DIEU KI:tN CHO sTj TON Ti}1 NGHI:tM

CUA BAI TOAN HAl DIEM BIEN KY D!

Trang chltdng nay ta xet t6n t~i nghi~m cua phudng trlnh co d~ng :

1 -(py')'= q(t)f(t,y(t),py') tE (0,1)

pet) Thoa man mQt s6 dieu ki~n bien khac nhau khi cac ham pet) , q(t) , f(t,y(t),py')

lrang chudng nay ta gi~?sifchung l¥ :

1) f : [0, l]x RxR ) R

2) q(t) E C(O,1) ; q(t) > 0 tren (0,1).

!

I.XET sTJTON TAl NGHIEM CUA PHUONGTRllifu

1

- (py')' = q(t)f(t, yet), py')

pet)

{

- vy(O)+ ~ limt >O+ p(t)y'(t) = 0 ; v,o,~ ~ 0

t E (0,1)

(1.1 )

Dinh Ly 3.1

Gla sl't' :O' ,

Trang do

+ Ih(t,u,v)l::;;K{lula+ IvlP+1} vdi K, a, ~ la cac h~ng sf{ 1> a, ~ ~ 0 + Ig(t,u,v)l::;;A(t,u)lvI2 +B(t,u); A(t,u) ,B(t,u) bi ch~n tren mQt t~p bi ch~n ;

2

+ ug(t,u,v) ~ clul + diuvi; c, d E R ; d>O ;

Gia sii' them:

7 R lien tl)c

thl khi a'y phltdng trinh 1.1se co nghi<%mn6u : C + - dN(I~ < J.l ;

va J.l=Al 1a gia tri rieng d~u lien cua loan tU' Ly(t)

Chung minh dinh ly:

Xet phltdng trinh 1.2

1

- (py')' = Aq(t)f(t, yet), py') ; pet)

{

- vy(O) + B hmt -+O + p(t)y'(t) = 0

A, t E (0,1)

(1.2)

]

(I

/

I

I

2

SHYfa :

1

0

~ - _[y(l)]2 - -[y(0)]2 + jp(t)q(t) I y(t)h(t, yet), py') I

I

- A jp(t)q(t)y(t)g(t,y(t),py')dt0

<=> i!Y'W+~[y(l)r+ ~[y(O)]2::; [P(t)q(t)ly(t)h(t,y(t),PY')\dt

1

0

Ta co :

+ fp(t)q(t) I yet) II py'lP dt+ fp(t)q(t) I yet) I dt}

~ f p(t)q(t)[Iyea)I+f Iy'(s)Ids] dt

~ f p(t)q(t)[Iyea)I+ II y'(t) lie f )2 ]a+ldt

~ 2a+1 [f p(t)q(t) I yea) la+l+ II y'lla+l f p(t)q(t)dt [(f~)2]"+!

~ K3{1 yea) la+1+ II y'lla+1} ;

K3 = 2a+lKI max{1,K2 T}; K1 = f p(t)q(t)dt; K2 = f p(t)q(t)dt [0 -)2 ]a+1

a+!

p(t)q(t) I yet) la+ldt

p(t)q(t) I yet) IIp(t)y'(t) I' dt ~ f I pet) 121y'(t) 1'1pet) 12 get) I yet) ~t

a

a

~ II y'(t) II' HI p(t)q(t) IIp(t)~q(t) 12-' [I yea) + II y'(t) IIY-'d

a

~ KJ IIy'(t) f' + I yea) IIIy'(t) II']

-Af p(t)q(t)y(t)g(t,y(t),p(t)y'(t))dt ~-cA f

I

- dAf p(t)q(t) I y(t)p(t)y' (t) kit

0 I

0

f

l

1

- dAf p(t)q(t) Iy(t)p(t)y' (t) Idt ~ -dNof Jp(t)q(t) Iyet) l.Jp(t) 1y' (t) Idt

0

~ -dNo ell y'(t) II(II y'(t) W +~[y(1W + v [y(OW)2

~ -dNo c (IIy'(t) W +-[y(l)r +-[y(O)r).

+K6{lly'(t)WH +ly(O)11H]+ly(O)I"+1 +lly'II"+! +l};

~ K6{ IIy'(t) 111H+I yea) 11H]+Iyea) 1"+1+ IIy'II"+1+l} (1)

Ap dvng tinh cha't sail : Ne'u L ~ 0 , 0 ~ k < 2 ,0 <h ~1 my ythl t6n t<;liffiQt h~ng

sao cho :

h

Lxk ~ - X 2 + N

2

do do (1)=> ~II y' (1) 112+ v [y(O)]2 ~ K7 ==H~ng sO'

V~y t6n t<;lih~ng s6Mo sao cho : lIy'(t)11~ Mo , Iy(a)!~ Mo

'Vx > a (**)

~

suy ra 'v't E [0,1] , ly(t)1< M (h~ng so') .(3)

Ta co:

I (p(t)y'(t)),1 ~ p(t)q(t)f(t,y(t),py')

=? f (p( t )y' (t ))'dt ~ KII II y' (t) 112SUPrO,I]p 2(t)q( t) + K12f pet )q(t)dt = KJ3

1

yay lyll=max{sup[O,I] I y(t) I, SUp(0,1)Ip(t)y'(t)I}~M.

Ap d\1ngdint ly 2.1 trong chu'dng I ta co phu'dng trlnh 1.1 co nghi~m

')

yet) E C[O,l] n C~(O,l) , p(t)y'(t) E C[O,l]

II.XET sTj TON T~I NGHIEM CUA PHUONGJ,'RINH~

1

pet)

{

~~~ p(t)y' (t) = ~~ p(t)y' (t)

t E (0,1)

(2.1)

Dinh Iy 3.2

Giii sU' If(t,y,py')1 ~ ~l(t) + ~2(t)ly(t)lr + ~3(t)lp(t)y'(t)IP

Chung minh dinh Iy

Nh~c l~i: L2pq[O,1]la khong gian cac ham thuQc (C[O,l],R) sao cho :

I

Vy(t) E L:q <=>f p(t)q(t)0 I yet) 12dt < Ct;)

Xet phu'dng trinh 2.2 :

I,:::,:,:::::;:.: :.::::::::::::::::::::::.:::'::::'::::::,:,::",::",::,,:,::,,:::,::'::"::::':'::':.::'::.::.:::.:.:::::::::.: :: ::,- :::::I'wEI::III'.@I~Q$X:: :I:::::::::::::::::' -

-1

- (py')' = Aq(t)f(t, yet),py')

pet)

{

y(O) = y(l)

~~p.pet)y' (t) = ~~IP-p(t)y' (t)

A , t E (0,1)

(2.2)

Ap d\lng dinh 19 2.2 trong chu'dng I ta co nghi~m t6ng quat cua phu'dng trlnh 2.2

y (t) = A 3 1 (t) + B I.Y2(t) + A J 0 y 1 (s)y 2 (t) w~S ~ 1 (t)y ~ (s) q (s)f (s, Y(s), PY') ds

Trong do Yl(t),Y2(t) la cac nghi~m dQc l?p tuye'n tinh cua phltdng trlnh (l/p)(P) Yl,Y2EC[O,l] , PYl' , PY2' E AC[O,l] ,w(s) la ham Wronskian cua tt(;li s Ta co P~

=h~ng s6

A.(II + 12)

[y 2(0) - Y ~(1)]10 - [y I (1) - Y 1(0)]11

" Y2(1)-Y2(0)

)d '

1 YI(s)li~ p(t)Y2'(t)-Y2(s)li~ P(t)YI'(t)

[y 2(0) - Y 2(1)]10 - [y 1(1) - Y I(O)]I}

1

~ I[Y2(0)-Y2(1)]Io -[YI(1)-YI(O)]II I( II}I+II21)

~ K4( 1131+1121)

iI, I ~ 2SUPUEI0.IJYI(t)Y2(s)lfYI(S)Y2(1)-Y2(S)Y}(1)q(s)p(s)1f(s,y(s),Py')lds

10 = lim p(t)YI'(t)-lim1-+0' 1-.1- P(t)YI'(t) ;

I

~ K 3fp(s)q(s) I res, yes), Py') Ids;

0

1131 ~ IY2(1)-Y2(0)I[suPsEI01]IYI(s)lIli~p(t)Y2'(t)1 +

I

+ sup SE[O.,IJIY2 (s) II ~~~ p(t)y 1'(t) 1][q(s)p(s) Ires,yes), Py') I ds

1

~ K J p(s)q(s) Ires,yes), py') Ids;

0

,.::: : I m, :.:,: '::::f t::::::::!",:: ,:: :,:::!ill ':ill.,;:.!;! :::: :.:':' : :;]@mi.: ~'q£ g - itfI ~}

J

V?y IA" I s K4(KJ + Ks)fp(s)p(s) If (s,y(s),py')/ds

0

I

(1

IB" / S IYl(1)-yJO) I/AJ.1+1121

y~(1) - Y2(O)

I

0

; A = K4 (K J + K 5);

I

0 1

I

tu d6: ly(t)1 S (K1A+K2B+KJ)fp(s)q(s)lf(s,y(s),py')lds

0

- A] f p(s)q(s) Ires, yes), py') Ids1

0

S

fp(s)q(s)l~i(s)lds S (fp(s)q(s)ds)2 (fp(s)q(s)l~j(s)12 ds)Z;

Ap dl;1ngbit d~ng thlic:

Suy fa :

Ta co do yea) =y(1) Den t6n tC;titoE(O,l) saG cho y' (to) = y(l )-y(O) = O

VtE(O,l): Ip(t)y'(t) I ~ Ip(to)y'(to)l+fl(p(t)y'(t)),ldt

0 1

0

~ C9 + C10 Ipy'l~lX +Cs I py'l~

~ Ipy'lo ~ C9 +CIO Ipy'I~[3 +cslpy'l~

Ap dl,1ngba"td~ng thlic (*) ta CO:chQn N1, N2 saG cho :

ClOlpy'loaP < 8 Ipy'lo+ N1, Clllpy'll < 8 Ipy'lo + N2 va 0 < 8 <112;

~ 3Mo> 0 : I yll = max{ Iylo, Ipy'lo} ~Mo

Ap dl,1ngdinh ly 2.3 trong chu'dng I ta kC'tlu~n phu'dng trinh 2.1 co nghi~m YEC[O,l] nC1(0,1) ; PYEAC[O,l]

III XET s!1 TON TAl NGHIE M CUA PHUONG TRINH:

1

pet) (py')'= q(t)f(t,y(t),py')

.

{

lim p(t)y' (t) = b

i~~ p(t)y' (t) = a1 >0+

t E (0,1)

(3 1)

Dinh ly 3.3

Giil sa : If(t,y,py')1 ~ ~l(t) + ~2(t)ly(tW+ ~3(t)lp(t)y'(t)IP

Ne'u ~i(t) E L2pq[0,1] va 1> r , p ;:::0 thi phu'dngtrinh 3.1 se co nghi~m

J]

Chung minh dinh Iy :

Nh~c l<:ti:L2pq[0,1] Hi kh6ng gian cac ham thuQc (C[O,1],R) sao cho :

1

'1y(t) E L2pq <=> Jp(t)q(t) Iyet) 12 dt < 00

0

Xet phu'dng trlnh 3.2:

1

pet)

{

~~~p(t)y' (t) = b

1 ->0+

Ap dl,lng dinh 19 2.4 trong chu'dng I ta co nghi~m t6ng quat cua phu'dng trlnh 3.2

(3.2)

y (t) = A 1_)' 1 (t) + BAy 2(t) + A P'l (5)Y 2(t) ~ ~] (1)y 2(5) q (5)f (5, Y (5), py ') ds 0 W 5

Trong do Yl(t),Y2(t) la cac nghi~m dQc l~p tuy€n Hnh cua phu'dng trlnh

A2A3 - A4A]

B.=b-A).A3

] y](s)limp(t)Y2'(t)-Y2(s)limp(t)YI'(t)

A3=limp(t)YI'(t)1 .0+ ; A4=1i1~p(t)YI'(t)1 ->1 ;

Ta co \it E [0,1] , A E(O,l) :

IA I I aAo -bA1

I I

1

II A

" ::;; AA AA2341 + AA234 -A A 51::;;

IB).I ::;; I-I+I-IIAAI ::;; Ks+K9IA,,1

K4+K51A51

1

1

0

Vdi Ks, K6 la cae h~ng s6 V?y vdi K7=Ks.~ thl:

I 1 + 2 sup [0,1]jYI(t)Y2(t)I!Zp(s)q(s)!f(s,y(s),py')ldS

1 1

~ KJIA"I+K2IB," I+K3J~p(s)q(s)lf(s,y(s),Py')lds ;

0 c

I

0

1

0 I

0

I

0

BI +B2 f p(s)q(s) If(s, y(s),py') I ds ~BI + B2 (f p(s)q(s) I~I (s) Ids +

+ f p(s)q(s) I ~2 (s) IIyes) la ds + f p(s)q(s) I ~3 (s) IIpy'l~ ds)

£)~ t Iylo= sup[o,l]ly(t)1 ; Ipy' 10= sup(o,l)lp(t)y' (t)1 ta co:

~ B i , i = 3,,5 ( B i = H~ngs6 );

== C1+C2IYlo"+C3Ipy'loP

Ta co:

p( t)y' (t) =A).p( t)y I' (t) + B).p( t)y 2' (t) + Af0 y 1 (S)p( t)y 2' (t) ~ ~(t)y W S I' (t)y 2 (S) q (S)f(s, y(S), py ')

Ip(t)y'(t)1 :::; IA) ISUPSE(O,I) Ip(t)Y2'(t)I+IB) ISUPseIO,I] Ip(t)Y2'(t)1 +

+ ~SUPs,tE(O,I)Ip(s)y2'(s)llp(t)YI'(t)lsuPs,tE[O,J) IYI(t)Y2(s)lfq(s)p(s)lf(s,y,pY')ld~

}

I

0

I

o

}

Ip(t)y'(t)J :::; C4+c5fp(s)q(s)I~/s)lds

0

Ap dvng tinh chfft (*) trong phgn II ta co:

1

2

=> I y 10 ,:::; B9 + BlOI py'loP (2)

(l)=> Ipy'lo:::; C6+C7(B9+BlOlpyloP)"+Cslpy'loP

,:::; C6+2"C7B9"+2"B10" IpYloP" +Cslpy'loP

,:::; C9+ClOlpYloP"+Cslpy'loP,

=> Ipy'lo':::; CII+!lpy'lo

2

<=> I py'lo':::; 2CII ==H~ng 56.(3)

i :;!:.'!:!:::! :!:!:!:!:.:! : ;;;:!::::.! :!:::!.!:!::!::::.::!.;:: ';::::'::.: :.:.:!:;.1:::!:: 1:1';:,"':'.:::.::::: :1:;;;:.:::1::::11"1-.1':!11':g;ff:£$@§Q1:.:;.:! :::::::::::::::::::':illi:,.:::::::::.:: 21

Tli (2) va (3) suy ra t6n t(;lih~ng s6 Mo > 0 saG cho :

IV.XET STjTON T~I NGHI]tM CUA PHu'(1NGTRINH:

1

-(py')' + A,r(t)y(t) - Ily(t) = f(t,y(t),py') ; A,,1l E R, t E (0,1)

p(t)q(t)

{

Giii thi€t them:

1) r(t) > 0 va bi ch~n tren (0,1).

2) p(t)~q(t) bi ch~n tren[O,l].

G9i /"1va ~ll19n 1u<;5tla hai gia tri rieng dgu tien cua hai loan tU'

X6t phudng trinh:

-(py')' + A,r(t)y(t) - Ily(t) =0 ; A,1l E R, t E (0,1)

ay (1) + b lim p(t)y' (t)t-+ I' = 0, a, b ;?:0 , a 2 + b 2 > 0

Dinh If 3.4.

N€u :

2) I~I+I~I < 1

Thi khi a'y m(>ttrong hai di€u Sail se xti'yfa:

a) phltdng trinh 4.1 se co nghi~m duy nhfft

b) phlfdng trinh 4.2 se co nghi~m thlfc.

Chung minh dinh ly

Do lid < Iq lien ta co lId=1= ~ Vi trong do ~ la cac gia tri rieng cua loan tit

Ly(t)=-(l/pq) (py')' Den ap dvng mvc II phffn 2 chudng II ta co

Phudng trinh :

1

- (p(t)y' (t))' + Ar(t)y(t) = 0 .

p(t)q(t)

- ay(O) + p Jim p(t)y' (t) = 0,I~O+

AE R, t E (0,1)

a> O,p 2::0

chi co nghi~m tffm thuong (chu yr~ng rei) bi ch?n lien r(t)EL2pq [0,1] va

L2pq[0,1] C L2pqr[O,l]).V~y nghi~m t6ng quat cua phudng trinh 4.1 co th~ du<;5c

viet dudi dC;lng:

]

yet) = f G(t, s)[f(s, y, py') + /-ly(s)]ds

0

YJ(S)Y2(t) w(s)

w(s)

thoa man lffn lu<;5thai di€u ki~n bien (SL)o D?t :

\11(t) = f G(t, s)f(s, y, py')ds

0

=> 1\I1(t)1 ~ fIG(t,s)f(s,y,py')lds ~ fIG(t,s)~Js)q(s)lds +

+ fI y2 (S)YI (t) II ~l (S) I p(S)q(s)ds

~ - SUp IYI(S)Y2(t)l+ fl~Jt)lp(s)q(s)ds

C "tE[O,I] 0

I

0

0

I

0

* (py')'= pq[-Ar(t)y + ~y]y + pqf(t, y,py')

=>If(py')'ydtj ~fpq2Iyllf(t,y,py')ldt+IAlfpqlr(t)llyI2 dt+I~lfpqlyI2dt

11y'112+-[y(1)]2+-[y(O)]2~ IAlfpqlr(s)llyI2ds+I~lfpqlyI2ds +

1

0

* IIIIf pq Iy I:dt ~ !l:l (IIy'II2 + ~ [y (1)]2 + a [y (0)] 2)

~ AI[y(0)]+A211 J'll

* JI$2(S)llyla+lp(S)q2(S)ds ~ JI$2(s)lIyla+lp(s)q(s)ds

( do q E (0,1) )

~ A3[y(0)]a+1+A4I1y'lla+1

* JI$3(s)lIpy'I~lylp(s)q2(s)ds ~ JI$3(S)llpy'I~lylp(s)q(s)ds

~ AJy(0)]~+I+A61Iy'II~+1

=> (1-8J-~)llyII12 +a.(1_l2J_~)[y(0)]2 ~ A7[y(0)]a+1 +Aslly'II"+1

Ap dvng b(t d£ng thlic (**) ta CO:

1

0

D~t:

1

0

A .. 13pq'[01] ~13 [01]pq' .

1

(1) ~y(t) - J.!JG(t,s)y(s)ds = \!f(t)

nghi~m cua phu'dng trinh 4.3 ma day la phu'dng trlnh Fredholm h(;tch

Hermite Den ta co th6 du'a ra k€t lu~n san:

trlnh 4.1luon co nghi~m duy nh(t

2) Khi ~l= l/cj Phu'dng trlnh yet) = Ay(t) co nghi~m khae kh6ng tue la phlfdng

trlnh 4.2 co nghi~m khae kh6ng Trong tru'ong h<;5pnay mu6n phu'dng trlnh 4.1 ec

nghi~m thl dieD ki~n e§n va du la ham \V(t) tnje giao vdi ta"tea cae vce td rieng

~n (t) eua roan tU'A (ehu y ding cae vce td rieng nay l?p thanh mQt cd sa tnfe ehuffn trong L2pq[O,1])