X: IRn xI ~I Rnla ham lien tl;lctren mi~n D.
M= Sup{15+ fte(s)dsl tEl},
Jto (4.4.6)
fX 1
G(x) = J, -ds (c:>O,x>O).
E g(s)
(4.4.7)
Chung minh djnh If.
Tli (4.4.1) va (4.4.2), ta Suy fa:
II yet) - x(t) II::;II y(to) - x(to) II+ (II X(y(s),s) - X(x(s),s) IIds + (II R(y(s),s) IIds
::; IIy(to) - x(to) II+ (Kg(1I yes) - xes) lI)ds
+( e(s)ds (4.4.8)
f)~t u(t) =lIy(t) - x(t)11~ 0, VtEI, va 8=lIy(to)- x(to)l!. Tli (4.4.8), ta duQC:
u(t) ::;M + (Kg(U(S»dS. (4.4.9)
Ap dvng b6 d~ Bihafi, ta duQc: u(t) ::;G-1[G(M) + K(t-to)],
hay lIy(t) - x(t)1I::;G-1[G(M) + K(t-to)].(D)
(4.4.10)
4.4.2. H~ qua 4.1.
Ne'u R(y,t) =0, MEl, thi ta co:
Ily(t) - x(t) II ::;G-1[G( 15)+ K(t-to)].
4.4.3. H~ qua 4.2.
(4.4.11)
Ne'u g(u) =u thi ta co:
Ily(t) - x(t) II ::;15exp[K(t - to)] + (exp[ K(t - s)]e( s)ds. (4.4.12)
MlJ rQllg va Ullgd(l1lgBiJ dl Grollwall-Bellman Hoang Thanh Long
§4.5. SV PHT}THUQC CUA NGHItM THEO
THAM SO
Ta da nghien cUu slf lien tl;lc cua nghi~m theo di~u ki~n delu va theo v~ phai. Bay gio ta nghien CUuslf lien tl;lccua nghi~m theo tham s6.
X6t phudng trlnh vi phan:
x'(t) =X(x(t),t,~), (4.5.1)
x: I ~ IRll; X: IRllxlxlR~ IRll,la ham lien tl;lctheo cac bi~n va theotham s6 Jl, va thoa man di~u ki~n Lipschitz theo bi~n x, nghla la 3L > 0 : tham s6 Jl, va thoa man di~u ki~n Lipschitz theo bi~n x, nghla la 3L > 0 :
IIX(x,t,Jl) - X(y,t,Jl)II::; Lllx - yll, V(x,t), (y,t)ED,VJlEIR. (4.5.2) Dinh Iy.
Ne'u phu{fflg trinh (4.5.1) co ham X(x,t,Jl) thoa man ddu ki~n (4.5.2) thi nghi~m xl/t) =rp(t,p) cila no lien t1;lCrhea tham sf;'J-l..
Chung minh dinh Iy. Ta celn chung torAng:
VE> 0, 38(E,~o) > 0: I~- ~ol< 8 => IIcp(t,~)- cp(t,~o)1I< E.
Th~t v~y, tu (4.5.1), ta co:
cp(t,~o) = cp(to'~o) + rl X(cp(s'~o),s,Jlo)ds.
Jlo (4.5.3)
cp(t,~) = cp(to'~) + rl X(cp(s, Jl),s, Jl)ds.Jlo (4.5.4)
Til (4.5.3) va (4.5.4), ta thu duQc:
II <p(t,f.l) - <p(t,f.lo) II < II <p(to'f.l) - <p(to'f.lo) II
Mi'irfjng va ll'ng d~tngBli di Gronwall-Bellman Hoang Thanh Long
§4.2. Stj LIEN T{)C CUA NGHItM THEO
;:: " ;:: '- "'?
DIEU KIENDAU VA THEOVE PHAI.
Tinh lien t\lC cua mQt ham s6 la ra'"tquail trQng vi dt!a vao tinh lien t\lC ta co th€ xa'"pXl gia tri cua ham s6 ling vdi st! thay d6i nho ban dgu.
Tinh lien wc cua nghi<%mcua mQtphudng trlnh vi phan cling khong phiii
la ngo(;li 1<%.
Xet hai bai loan Cauchy sau:
X'= X(x,t); x(to) = Xo. (4.2.1)
(4.2.2) y'= Y(y,t); y(to) = Yo,