X: IRn xI ~I Rnla ham lien tl;lctren mi~n D.
Chung minh dinh Iy 1.1.
Gia sa x(t), yet) la hai nghi<%mcua phuong trlnh (4.1.1) thoa di~u
ki<%ndftu x(to) =y(to) =Xo.Ta co
x(t) =Xo+ fl X(x(s),s)ds, VtEI.
Jlo (4.1.3)
yet) =Xo+ fl X(y(s),s)ds, VtEI.
Jlo (4.1.4)
Tli (4.1.3) va (4.1.4), ta duQC:
IIyet) - x(t) II ~ II( X(y(s),s) - X(x(s),s)ds II.
~ fl K IIyes) - xes) IIds.
Jlo (4.1.5)
E>~tu(t) =Ily(t) - x(t) II2 0, ap dlJng dinh ly 1.1 chuang 1, ta duQCu(t)
=0, vdi t thoa man It - tal < rM-1. Vi d\l. Xet bai loan sail:
y"(x) + ay'(x) + ~y(x) = f(x), (4.1.6)
MiJ rfjng va ling dlJng Bfl dff Gronwall-Bellman Hoang Thanh Long
y(xo) =a; y' (xo) =b, (4.1.7)
Ta chung minh phuong trlnh (4.1.6) voi di~u ki~n d~u (4.1.7) conghi~m duy nh~t tren [xo, Xo+ T], voi T > 0 n~lOdo. nghi~m duy nh~t tren [xo, Xo+ T], voi T > 0 n~lOdo.
Th~t v~y, gicl sli'phuong trlnh tren co hai nghi~m Yl, Y2khcl vi lien Wc Wi b~c hai tren [xo, Xo+ T].
D:)t w=Yl - Y2. Khi do, w thoa man:
w"(x) + aw'(x) + pw(x) = O.
w(xo) = w' (xo) = O.
(4.1.8)(4.1.9) (4.1.9)
Ta chung minh: w(x) =O,'v'XE[Xo, Xo+T].
Nhan hai vii cua (4.1.8) voi w' va rut gQn, ta duQc: 1 d
--{w'\x) + pw\x)} + aw'\x) =O.
2 dx (4.1.10)
L~y tich phan hai vii (4.1.10) tuxodiin x, ta duQc: W'2(X)= _pW2(X)- 2a Ix w'2(s)ds.
Xo (4.1.11)
M:)t khac, ta l?i co:
W2(X) ~ T Ix w'\s)ds.
Xo (4.1.12)
Tuphuong trlnh (4.1.11), ta suy fa:
I w'\x) I ~ IPII w\x) I+21 a IfoIW'2(S) Ids. (4.1.13)
Thay (4.1.12) vao (4.1.13), ta duQc:
Iw'\x) I ~ {IpiT + 21 a I}foIw'\s) Ids. (4.1.14)
Ap d\lng b6 d~ 1 chuang 1 (c=21al + IpIT, k =0, u(xo) =0 ), ta thu
duQc w'\x) ~ O. Suy ra
Mi'Jri)ng va ling d(tng B6 dl Gronwall-Bellman Hoang Thanh Long
w(x) = c, 'v'XE[Xo,xo+T], C la m9t h~ng sf{nao do. Do (4.1.10) nen ta duQc:
w(x)=0, 'v'XE[Xo,xo+T].
4.1.2. Dinh Iy 1.2.
Ne'u ~~ tbn t[Ji, lien tl:lctren D thz hai roan Cauchy cho phuang trenh
(4.1.1) tren It - tol < rM-1co duy nhat nghi~m.
Ti€p theo la m9t sf{k€t qua v~ st! duy nha't nghi~m cua phuong trlnh
tkh phan.
£)~tLl= {(t,S)E QxQlto~s~t~td.
4.1.3. Dinh Iy 1.3.
Cho K: L1-f IR la mQt ham lien tl:lc.fJi,it F ={f If Q -f IR lien tl:lc}.
Gid sit art) EF va J-lE IR la mQt hdng so: Khi do phuang trenh rich phan Volterra tuye'n tinh tren F