2.21. Dinh Iy 2.20 (Xem [3]).
Cho u(t), art), b(t) la cac ham duClng, lien tl:lCtren [c,dJ; k(t,s) la ham khong am, lien tl:lCwJi C ::;s ::;t ::;d; 0 < f( u) la ham lien tl:lC,tang ngc;it; 0
< g(u) (u > 0) la ham lien tl:lC,khong gidm.
Niu A(t) = Supa(s), B(t) = Supb(s), K(t,s) = Supk(cy,s) thEta
,,:5,,:5, ,,:5,,:5, ,,:5a:5'
f(u(t))S;a(t)+b(t) fk(t,s)g(u(s))ds, l7tE[c,dJ, (2.97) ta co:
MiJ rl}ng va ung d(tng Bll di Gronwall-Bellman Hoang Thanh Long
u(t) ::;f-I [G-I (G( A(t)) + B(t) fK(t,s)ds} J, (2.98)
V'tE[c,d') va G(u)= r ~~ ' (E>O, u>O),
E g(f (cr)) (2.99)
d' =max{r E [c,dJ I G[ A(r)J + B(r) fK(r,s)ds::; G[ f(oo)]}.
Chung minh djnh IS'2.20. Ta co th~ chung minh r6 rang nhu sail:
E>~tvet) la vii phai cua (2.97).
Voi m6i TE[C,d], xet c::; t::; T::; d. Ta co:
v(t)::; A(T) + B(T) fK(T,s)g(u(s))ds. (2.100)
Liy d~o ham hai vii, ta duQc: v'(t)::; B(T)K(T, t)g(u(t)),
::;B(T)K(T, t)g[CI (v(t))]. (2.101)
Chuy~n vii, liy tich phan hai vii tu c diin t va d6i biiin, ta duQc:
G(v(t))::; G(v(c)) + B(T) fK(T,s)ds. (2.102)
Suy ra
v(t)::; G-I {G(A(T)) + B(T) fK(T,s)ds}. (2.103) Cho t =T va liy ham nguQc,ta duQc(2.98).(0)
2.22. Djnh IS'2.21 (Xem [3]).
Cha u(t) la ham duong, lien t1;lCtren [c, d]. Gid silvdi u > 0, Y(u) la ham tang ng(it; vdi u > 0, g(u) la ham lien t1;lC,duang va kh6ng gidm. Ne'u
Y(u(t)) ::;f(t) + fcp(s)g(u(s))ds, t7tE[C, dJ, (2.104)
trang do f( t), rp(t) thoa man di~u ki~n trang djnh ly 2.17, thEta co:
MiJ ri}ng va zing d(tng Bli di Gronwall-Bellman Hoang Thanh Long
U(t)~y-l[G-1{G(F(t»+ I[a(s)+f'(s)]ds}], 'rItE[c,d'), (2.105)
r' ds
wJi F(t)=Supf(t), G(u)= J, -I ' (8>0, u>O),
cO;so;t E g(Y (s»
d' =max{rE[c,d] IG[F(r)] + fcp(s)dssG[Y(oo)]}.
(2.106)
Chung minh djnh Iy 2.21. Tlidng tlj chung minh dinh 1:92.17 .(D)
2.23. Djnh Iy 2.22 (Xem [3]).
Cho u(t), fer), F(t,s) thoa man cac diJu ki~n:
a. u( t), f( t), F( t,s) la cac ham dU:rJnglien tl;lc tren IR+ va s ::; t.
h aF(t,s) I, h' khA A
lOA
. a am ong am, zen tuc.
at 0
c. g(u) la ham dU:rJng,lien tl;lc,kh6ng gidm tren (0, ro). d. h(z) > 0, kh6ng gidm va lien tl;lctren (0, ro).
Ntu
u(t) ~ fer) + h(f~F(t,s)g(u( s))ds) , (2.107)
the v6i tEl, ta co:
u(t) ~ f(t) + h(G-1[G( f~F(t,s)g(u(s))ds) + f~rjJ(s)ds]), (2.108) trong do ru dcr G(u) = J, ' (u > 0, E:> 0), E g(h(cr)) ~(t)=F(t,t)+ rtaF(t,s) ds, Jo at (2.109) (2.110)
1= (t E (0,00) IG(f~F(t,s)g(u( s))ds) + I~(s)ds s G(oo)}.
Chung minh djnh Iy 2.22. Xem [3].(D)
MlJ r~ng va ung dlJng Bd d€ Gronwall-Bellman Hoang Thanh Long
Dinh 1y sau 1a h~ qua cua dinh 1y 2.22.
2.24. Djnh ly 2.23. (Xem [3]).
Cha u(t), f(t), F(t) thoa man cac di~u ki?n:
a. u(t), f( t), F(t) la cac ham du(JJ1glien tl;lctren (0, co)va s :::;t. b. g(u) la ham du(JJ1g,lien tl;lc,khong giam tren (0, co).
c. h(z) > 0, khong giam va lien tl;lctren (0, co).
Ne'u
u(t)~f(t)+h[ S;F(s)g(u(s))dsJ, Vt6(0,CO), (2.111)
thi wJi t 61, ta co:
u(t)~f(t)+h(G-1[G( S;F(s)g(u(s))ds) + S;F(s)dsJ), (2.112)
trang do G djnh nghza nhu (2.109) va
1= {tE (0,00)I G(S;F(s)g(u(s))ds) + S;F(s)ds ~ G(00n.