Mở rộng và ứng dụng bổ đề gronwall bellman

D~I HQC Quac GIA TP. HO CHi MINH
TRUONG DAI HOC KHOA HOC TU NHIEN

HoANG THANH LONG
MORaNGvA UNGDUNG
. .
~ ~
BODE GRONWALL-BELLMAN
Chuyen nganh : Toan Ghii Tich
Ma s6 : 1.01.01
LUA.N VAN THAC 51 ToAN HOC
. . .
NGUOI HUaNG DAN KHOA HOC:
PGS. TS. NGDYEN DINH PHU
_ .
I;Ht.~H.TlI' NHIEN
~.::(."; lTHtJ \t1~N
001103
I.

TP. Hfi CHi MINH - 2005
1
MlJri)ng va ung d~ng Bd d~ Gronwall-Bellman
Hoang Thanh Long
MUCLUC
. .
M1).C 11).c.
Loi Carn do.
Danh rn1).ccac ky hi~u.
Chu'dng 0 -T6ng quail.
Chu'dng 1 - B6 dS Gronwall- Bellman va mQt sf{

m(j rQng d(;lngtuy€n tinh.
Chu'dng 2 - MQt sf{m(j rQng d(;lngphi tuy€n.
Chu'dng 3 - MQt sf{m(j rQng d(;lng ham exponent.
Chu'dng 4 -MQt sf{ling dl;!ng
§4.1. Sl;!'duy nha't nghi~m cua phuong trlnh vi
phan va tich phan.
§4.2. Sl;!'lien tl;!Ccua nghi~m theo diSu ki~n
d~u va theo v€ phai.
§4. 3. £hinh gia tinh b~ch~n cua nghi~m.
Trang
1
3
4
5
7
13
34
38
38
43
45
§4.4. Sai l~ch nghi~m hai phuong trlnh vi phan. 48
§4.5. Sl;!'phl;! thuQc cua nghi~m theo thalli sf{.
A'
§4.6. On d~nh mil trong kh6ng gian Banach.
50
52
Lufjn wIn th[Jc Sf loan h{JC
Mil nganh : 1.01.01
2

Mlf ri)ng va ung d¥ng Btf di Gronwall-Bellman
Hoang Thanh Long
§4.7. On dinh cae h~ tlfa di~u khi€n.
~
§4.8. On dinh h~ kich dQngthu'ongxuyen.
Ke'tlu~n.
Tftili~u tham khao.
56
59
61
62
3
Mi'Jri)ng va ling d~tngBd di Gronwall-Bellman Hoang Thanh Long
, ?
L(J/ CAM (j N
Ddu lien toi xin chan thanh cam an sau sdc den Thiiy
PGS.TS. Nguy~n Dinh Phu:dii t(in tlnh huang ddn toi tit d~
cuang den luc hoan thanh lu(in van.
Toi xin chan thanh cam an haT Thiiy phiin bifn
PGS.TS. Dinh Ngl)c Thanh va Thiiy TS. Nguy~n Thanh
Long dii dQc lu(in van va dang gap nhi~u ykien quy bau.
Toi xin chan thanh cam an cac Thdy Co TruiJngDc;zi
HQc Khoa HQc T1;CNhien, TruiJng Dc;zi HQc Su Phc;zm,
TruiJng Dc;zi HQc Bach Khoa dii t(in tlnh giang dC;Zyva
truy~n dc;ztnhi~u kien thac mal, b6 ich giup tai lam quen
ddn Val vi~c nghien cau khoa hQc.
Tai ciing xin chan thanh cam an gia dlnh, cam an cac
bc;zndii luon luon df)ng vien, giup diJ va tc;zodi~u ki~n v~
mQi m(it di tai hoan thanh to/tvi~c hQc.
Luljn van tlUJCsl loan h{JC

Mil nganh 1.01.01
4
MlJri)ng va u'ng d~tngBlf di Gronwall-Bellman
Hoang Thanh Long
DANH MUC KY HIEU
. .
Trang lu~n van nay co sa dl,mgcae ky hi~u va quy tide cgn thi€t.
1. IRll:Khong gian thl;icn chi~u.
2. IR+= [0,00).
3. Q = [to,tr] c IR.
5. 1.1: Gia tfi tuy~t d6i .
6. 11.11: Chu§'n. Tren IRllta lfiy chu§'n euclide.
7. exp(u) = ell.
8. Sup : C~n tren.
9. inf : C~n dtidi.
10. max : Gia tri IOnnhfit.
11. L : T6ng.
12. A =(aik), i,k = I, ,n, la ma tr~n vuong cfip n.
n
13. IIAII = Sup I laikI, la chu§'n cua ma tr~n A.
k i=l
I
14. IIullL2= (1:,lu(xW dX)2 < 00.
15. Dam : Mi~n xac dinh.
16. Re : Phgn thl;ic.
17. (D) : K€t thlic chung minh.
Lllqn van thl!c si loan h(JC Mil nganh : 1.01.01
5
, MlJri)ngvallngd~tngBd dilGronwall-Bellman
Hoang Thanh Long

CHUaNG 0
~
TONG QUAN
Trang loan hQc t6n t~i mOt sf{phuong trlnh va ba-t phuong trlnh ra-t
quail trQng. Chung mang nhi~u
ynghla th1fcti~n cho nhi~u ling dvng khac
nhau. Ba-td~ng thuc Gronwall hay B6 d~ Gronwall-Bellman la mOttrong
sf{do. B6 d~ nay tuy co d~ng h€t suc ra-tdon gian, nhung duQcling dvng
ra-thi~u qua dS chung minh s1fduy nha-tnghi~m cua phuong trlnh vi phan,
dung danh gia s1fsai l~ch nghi~m, dung dS tlm di~u ki~n du cho mOt sf{
bai loan 6n dinh nghi~m, VI v~y doi h6i phai hoan thi~n ba-t d~ng thuc
nay nhu mOt di~u ta-tnhien cua quy lu~t phat triSn. Va co nhi~u lac gia ma
rOng theo
ytuang va mvc dich khac nhau (xem [1, 2, 3, 4, 5, 6]). Tuy theo
mvc dich giai quy€t bai loan ma cac lac gia ma rOng khac nhau. Theo
chung Wi v~n can ra-tnhi~u d~ng, nhu d~ng lilYthua, d~ng ham exponent,
cgn duQc ma rOng.
Mf:lCdich cua lu4n van la t6ng k€t cac d~ng cua B6 d~ Gronwall-
Bellman, d6ng thai ti€p tvc ma rOng va trlnh bay mOt sf{ling dvng cua
chung.
Lu~n van duQc chia lam nam chuang.
Chuang 0 - T6ng quail.
Chuang 1 - B6 d~ Gronwall-Bellman va mOt sf{ma rOngd~ng tuy€n
tinh. Chuang nay g6m B6 d~ Gronwall-Bellman va 9 dinh ly 1.1-1.9.
Chuang 2 - MOt sf{ma rOng d~ng phi tuy€n. Chuang nay g6m mOt b6
d~ b6 trQ va 23 dinh ly 2.1-2.23.
Luljn van lh{lc si loan h(JC
Mil nganh : 1.01.01
6
. Mli rf)ng va ling dl!-ngBd di Gronwall-Bellman

Hoang Thanh Long
Chuang 3 - MQt 86 md rQng d~lllg ham exponent. Chuang nay g6m
illQtb6 d€ va 04 dinh 19 3.1-3.4.
Chuang 4 - MQt 86 ling dvng. Chuang nay g6m 8 lInh v1,1'cling dvng
khac nhau. Nhling ling dvng nay 1a: Chang minh sf! duy nh{{t nghi~m cua
phuong trinh vi phan va rich phan; Sf! lien tl:lCcua nghi~m rhea di~u ki~n
ddu va rhea vi phdi; Danh gia tfnh hj chc;incua nghi~m; Sai l~ch nghi~m
><,7
cua hai phuang trinh vi phan; On djnh mil trang khong gian Banach; Sf!
d d
phl:l thuf)c cua nghi~m rhea tham s6; On djnh cac h~ tf!a ddu khiin; On
djnh cac h~ kich df)ng thuiJng xuyen.
Cac md rQng cua B6 d€ Gronwall-Bellman con co thS tie'p tvc phat
triSn nhi€u bon nlia. Nhi€u ling dvng cua chung co thS ap dvng cho vi~c
giai quye't cac bai loan 6n dinh cac h~ tu~n hoan, Nhi€u vfin d€ lien
quail de'n B6 d€ nay khi md rQng81,1'phv thuQc cua bfit d&ngthlic vao hai
hoi:icnhi€u bie'n 86, hoi:ic 81,1'phi tuye'n da dc;mgcua cac m6i quail h~ cac
ham trong bfit d&ng thlic. MQt d~ng khac cua B6 d€ cling rfit duQc quail
tam 1acac d~ng roi r~c (xem [2, 3, 6, 8, 9]).
Lufjn van th{lc sl loan h(JC
Mil nganh : 1.01.01
7
MlJ rfJng va lIng d(tng Bd di Gronwall-Bellman
Hoang Thanh Long
CHUaNG 1
N ~ ,
BO DE GRONWALL-BELLMAN VA
" K ? " K "
MOT SO MO RONG DANG TUYEN TINH
.

Trang B6 d~ Gronwall-Bellman co nhi~u d?i luQng tham gia. Ne'u
chung ta 19nluQt thay d6i cac d?i luQng nay chung ta se co nhi~u ma
fQng. Cac ma rQng nay co ra't nhi~u ling d\lng va duQc phat biSu dudi
d?ng cac dinh ly. Cac ma rQng chli ye'u la cac d?ng ba't d~ng thlic tkh
phan.
Trudc lien chung ta xem l?i B6 d~ Gronwall-Bellman dudi d?ng ba't
phudng trlnh vi phan va ba't d~ng thlic tkh phan.
I. B6 d~ Gronwall-Bellman.
1.1.1 B6 d~ 1.
Gid sa u(t) la ham sf;'khd vi tren n. Ne'u t6n tc;zicac hang so'k, c 7: 0
saD cho:
u'(t) S curt) + k, b1En,
(1.1)
thi ta co:
k
u(t) S u(to)exp[c(t - to)] +-(exp[c(t - to)] - l}, 't;1'tEn.
c
( 1.2)
1.1.2. B6 d~ 2.
Gid sa u(t), art) la cac ham so'lien tl;lc,khong am tren n. Ne'u t6n tc;zi
hang so'k ;::0 saD cho:
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MlJrf)ng va lIng d~mgBtl d€ Gronwall-Bellman
Hoang Thanh Long
u(t)~a(t)+k f' u(s)ds, VtEQ,
Jto
(1.3)
the ta co:

u(t)~a(t)+k
rta(s)exp[k(t-s)Jds, VtEo.
Jto
(1.4)
1.1.3. H~ qua.
N€u aCt) =a =constant,\ftEQ, thl ta co:
u(t) ~ aexp[k(t - to)]'\ftEQ.
(1.5)
II. MQt sf) md rQng d~ng tuye-n Hnh.
1.2.1. Djnh Iy 1.1 (Xem[3 D.
Gid sa u(t) la ham so'lien tl;lc,khong am tren 0. N e'u tbn tC:licac hang
so' a ~O, k ~O, c > 0 saD cho:
u(t) ~ a + rt[cu(s) + kids, VtEQ,
Jto
( 1.6)
the ta co:
k
u(t) ~ aexp[ crt - to)J+ -{exp[ crt - to)J -I}, VtEo.
c
(1.7)
Chung minh djnh Iy 1.1.
Ta co th6 chung minh bang cach ap dl;lngb6 d~ 1 nhusau.
Di:it
v(t)=a+ rt[cu(s)+k]ds,\ftEQ.
Jto
(1.8)
TO'(1.6) va (1.8), ta suy fa u(t) ~ vet), veto) =a.
K€t hQp Iffyd~o ham hai v€ (1.8), ta duQc:
v'et) ~ av(t) + k, \ftEQ. (1.9)
Lllqn van th(lc si loan h(Jc

Mil nganh : 1.01.01
9
MlJri)ng va ung dlJng Bd di Gronwall-Bellman
Hoang Thanh Long
Ap dlJng b6 a~ 1, ta au'qc (1.7).(0)
1.2.2. Djnh Iy 1.2 (Xem [2]).
Gid sit u(t), k(t) la cac ham so'lien tl;lc,khong am tren Q. Niu tbn tqzi
hang so'a ;?0 saD cha
u(t)~a+ rt k(s)u(s)ds, b1EQ,
Jto
(1.10)
thE
u(t)~aexp[ rtk(s)ds], \1tEQ.
Jto
(1.11)
1.2.3. Djnh Iy 1.3 (Xem [6]).
Gid sit u(t), a(t), k(t) la cac ham so'lien tl;lc,khong am tren Q. Niu
u(t)~a(t)+ 1:k(s)u(s)ds, b1EQ,
(1.12)
thE
u(t)~a(t)+ 1:a(s)k(s)exp[ fk(r)dr]ds, b1EQ.
Chung minh djnh Iy 1.3. Xem [6].(0)
(1.13)
1.2.4. Djnh Iy 1.4 (Beesack, Xem [3]).
Gid sit u(t), a(t), b(t), k(t) la cac ham lien tl;lc,khong am tren Q.
a). Niu
u(t) ~ a(t) + b(t) 1:k(s)u(s)ds, b1EQ,
(1.14)
thE
u(t) ~a(t)+b(t) 1:a(s)k(s)exp[ fb(r)k(r)dr]ds, \1tEQ. (1.15)

b). Kef qua a) vdn dung ntu thay dau H~' bJi dau H;?"trang (1.14)
va (1.15).
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Md rqng va ung dljng Bd di Gronwall-Bellman
Hoang Thanh Long
c). Ke'tqua a) va b) win dung neu thay r biJi r' va
S
t biJi r'.
Jto Jt s Jt
Chung minh dinh Iy 1.4. Xem [3].(0)
1.2.5. Dinh Iy 1.5 (Xem [4D.
V6i cac gid thie/; nhu djnh ly 1.3. Va gid sit a( t) la ham khd vi tren Q.
Neu
u(t)~a(t)+ rt k(s)u(s)ds, btEQ,
Jto
(1.16)
thE
u(t)~a(to)exp[ (k(s)ds] + (a/(s)exp[ fk(r)dr]ds, tltEQ.(1.17)
Chung minh dinh Iy 1.5. f)~t vet)la vii phai cua (1.16), ta co:
v'et) = alet)+ k(t)u(t).
~ a'(t) + k(t)v(t).
(1.18)
Suy ra
vet)
~ a(to)exp[ ( k(s)ds] + (a'(s)exp[ fk(r)dr ]ds
(1.19)
Tli b~t d~ng thlic nay, ta du<;5c(1.17).(0)
1.2.6.Dinh Iy 1.6 (Xem [4D.

Gid sit u(t), kef) la cac ham lien tf:lC,kh6ng am tren Q; art), bet) la cac
ham duCfng,khd vi tren Q. Neu
u(t)~a(t)+b(t) (k(s)u(s)ds, btEQ,
(1.20)
thE
u(t)
~ betHc(to)exp[ (b(r )k(r )dr]
+rc'(s)exp[ fb(r)k(r)dr]ds), tltEQ.
~ s
(1.21)
Lll{jn van thlJc Sf loan hQc
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Mil ri)ng va ung d1!ngBd di Gronwall-Bellman
Hoang Thanh Long
trang do crt) =art)
b(t) .
Chung minh djnh Iy 1.6.
Chia hai v6 cua (1.20) cho bet), ap dlJngdinh ly 1.5.(0)
1.2.7.Djnh Iy 1.7 (Xem[10], tr.191-192).
Cho u(t), art) la cac ham lien tl:lc,khong am tren n. Gid sa K(t,s) 2!O,
gi6i nQi v6i to::{s::{t ::{t]va K(t,s)
= 0 v6i to::{t < s ::{t]. Ne'u
u(t)~a(t)+ rl K(t,s)u(s)ds, \?tEn,
J/o
( 1.22)
thl
u(t) ~ rp(t), \?tEn,
trong do rp(t)la nghi~m cua phuong trlnh
rp(t)

=art) + rl K(t,s)rp(s)ds.
J/o
(1.23)
Dinh ly 1.8, 1.9 sau day duqc ap dlJngra'thi~u qua trong vi~c khao
sat cac bai loan 6n dinh. No la h~ qua cua dinh ly 1.5.
1.2.8.Djnh Iy 1.8 (Xem [3]).
Cho u(t) la ham lien tl:lc,khong am tren n va thoa man bat dcing thac
u(t) ~ exp[ -art - ta)]u(ta)
+ r (au(s) + b)exp[-a(t-s)]ds, \?tEn,
J/o
(1.24 )
trong do a, 0 < a, 0 < b la cac hling so: Khi do, ta co:
u(t) ~ exp[ -( a - a)(t - ta)]u(ta)
+b(a - at! [1- exp[ -(a - a)(t - to)]}' \?tEn. (1.25)
Chung minh djnh Iy 1.8. Tlnh loan tnjc ti6p tu dinh ly 1.5 ho~c chung ta
Luqn van th{lc sf loan h(JC
t)H.~H.TtfNH'EN
THtr\lIEN
.
Mil nganh : 1.01.01
12
MlJrQngva u'ngdlJng Bii d€ Gronwall-Bellman
Hoang Thanh Long
c6 thS chung minh nhl1 sail:
B~t
x(t)
=u(t)exp( at).
Khi d6, tu (1.24), ta dl1Qc:
(1.26)
x(t) S Keto)+ i~ [axes) + bexp(as)]ds,

Ap dl;lngdinh 1:91.5, ta dl1Qc:
(1.27)
x(t) S x(to)exp[a(t - to)] + bexp(at) rt exp[(a - a)s]ds.
Jto
S x(to)exp[a(t - to)]
+ b(a - arl exp(at){ exp[(a - a)t] - exp[(a - a)to]}' (1.28)
V~y u(t) S u(to)exp[ -(a - a)(t - to)]
+ b(a - arl{l- exp[-(a - a)(t - to)]} .(D)
1.2.9. Djnh Iy 1.9.
Cha u(t), art), b(t) fa cac ham lien tl;lc, khong am tren Q. Ne'u
u(t) S exp[ -art - to)]u(to)
+ 1:[a( s)u( s) + b(s)] exp[ -art - s)]ds ,\ftED., (1.29)
trang do a fa hang so: thEta co:
u(t) S u(to)exp[ -art - to)+ rta(s)ds]
Jto
+ rb(s)exp[-a(t-s) + fa(r)dr]ds,\ftED (1.30)
to s
Chung minh djnh Iy 1.9. Tl1ong tl! chung minh dinh 1:91.8.(D)
Binh 1:91.9 t6ng qu:H h6a dinh 1:91.8. Binh 1:91.8 dl1Qcsuy ra tu dinh
1:91.9 trong trl1dng hQp a, b 1a cac ham h~ng,
Lllqn van thfJc sl loan h{Jc
Mii nganh : 1.01.01
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MiJr{}ngva ung d1!ngBli d€ Gronwall-Bellman
Hoang Thanh Long
CHUaNG 2
" ,," ? " ""
MOT SO MO RONGDANG PHI TUYEN
.
Trang chuang 1 chung Wi da trlnh bay mQt 86 k€t qua md rQng d(;mg

tuy€n tinh d6i voi ham u(t). Trang chuang nay chung Wineu md rQngmQt86
d(;lngphi tuy€n d6i voi ham u(t).
2.1. B6 d~ b6 trQ (Xem [5]).
Cho u(t) la ham duong, khd vi tren Q, a(t), b(t) la cac ham lien tl:lc
tren Q va p 20 la mQt hdng so: Gid si/:co bat dcing thuc
u'(t)sa(t)u(t)+b(t)uP(t), 'r7tEQ. (2.1)
Khi do tuy theo p, ta co cac ket qua sau:
a. Ne'up
=1 thi
u(t) S u(t )exp[
([a( s) + b( s)Jds, 'r7tEQ.
(2.2)
b. Ne'u p :;z!:1 thi
I
u(t)sexp[ rta(s)ds]{uq(to)+qrtb(s)exp[-q r~a(r)drJds/J, (2.3)
J~ J~ J~
'r7tE[to ,tp), q =1- p va
tp =SUp{tEQ Iuq(to) + q rtb(s)exp[ -q r~a(r )drJds > OJ.
Jto Jto
Chung minh b6 d~ b6 trQ. Xem [5].(0)
2.2. Dinh Iy 2.1.
Cho u(t) la ham lien tl:lCtren Q. Gid si/:art), bet), cp(t)la cac ham lien
Luijn van thlJc si loan h{Jc
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14
Mil ri)ngva ring d(tng Bd d~ Gronwall-Bellman Hoang Thanh Long
tf:lC,khong am tren £2 Ne'u
2 rt
u (t)~a(t)+2b(t) Jtocp(s)u(s)ds, l7tEQ,
(2.4 )

thi
1
lu(t)1 ~(a(t)+b(t) r[cp2(s)+a(s)]exp[ fb(r)drJdsj2, [7tE£2(2.5)
to s
Chung minh dfnh If 2.1. Di;it
vet) = 2 rt <p(s)u(s)ds, 'v'tEQ.
Jto
(2.6)
Lffy d~o ham hai vii cua (2.6), ap dl;lngbfft d~ng thlic Cauchy va kiit
hqp voi (2.4), ta duQc:
v'et) ~ <p\t) + aCt)+ b(t)v(t).
(2.7)
Suy fa
vet) ~ f[cp2(S) + a(s)]exp[ fb(f)dfJiS].
to s
Thay vao (2.4) va Iffy din, ta duqc (2.5).(0)
(2.8)
2.3. Dfnh If 2.2.
Cha u(t), b(t), <P(t) la cac ham lien tf:lC,khong am tren £2 0 ::;p :;z!:1, a
la cac hang so: Gid sit b(t) la melt ham khong gidm va khd vi tren £2 Ne'u
u(t)~a+b(t) rt<p(s)uP(s)ds, l7tEQ,
Jto
(2.9)
thi
I
u(t) ~ b(t){[ ~Jq + q r<p(s)bP (s)dsj,q l7tE[to,tp),
b(to) to
(2.10)
trang do tp =SUp{tEQ I[~Jq + q r cp(s)bl'( s)ds > OJva q = 1- p.
b(to) Jto

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15
MlIri)ng va ung d~tngBif d€ Gronwall-Bellman Hoang Thanh Long
Chung minh dinh Iy 2.2. E>~tvet) la v€ phili cua (2.9). Khi do, ta co:
b'(t)
viet) S -[ vet) - a] + b(t)qJ(t)vP(t).
bet)
S b'(t) vet) + b(t)qJ(t)vP(t).
bet)
(2.11)
Ap dlJng b6 d~ b6 trq, ta duqc:
b
'
(
)
b
'
( )
I
t S t s r -
vet) s;exp[ r -ds]{aq + q rqJ(s)b(s)exp[-q r -dr]ds}q (2.12)
Jto b(s) Jto Jto her) .
I
V~y vet) S;betH [~]q + q rt lp(s)bP(s)ds}4(D)
beta)
Jto .
2.4. Dinh Iy 2.3.
Cho u(t),fer), qi.t)la cac ham lien t¥c, khong am tren Q. 0 ~p <1 la
m(jt hling so: q =1 -p. Ne'u

u(t) S;f (t) +
rtlp(s)uP(s )ds, VlEQ,
Jto
(2.13)
thi
I
u(t) S;fer) + [M<f + qfqJ(s)ds/q VlEQ,
0
(2.14 )
trang do M =Sup{f(t) E IRI tED}.
Chung minh dinh Iy 2.3. E>~t
vet) = rt lp(s)uP(s)ds.
Jto
(2.15)
La'y dC;lOham hai v€ va chu yrAng u(t) S;f(t) + v(t), ta thu duqc:
v'et) s;qJ(t)[f(t) + v(t)]P.
v'(t) s; lp(t)[M + v(t)]P.
(2.16)
Lufjn van th{lC si loan h(Jc Mil nganh : 1.01.01
16
MiJri)ng va ung d(tng Bd di Gronwall-Bellman Hoang Thanh Long
Chia hai v€ cua (2.16) cho [M +v(t)]P, ta dtiQc:
viet)
[M + v(t)]P ~ lp(t).
(2.17)
Lgy tich phan hai v€ tu tod€n thai v€ bgt d~ng thuc, ta dtiQc:
[M + v(t)]q ~ Mq + q
rtlp(s)ds.
Jto
(2.18)

Suy ra
vq(t) ~ Mq + q rtlp(s)ds.
Jto
(2.19)
Lgy din hai v€, ta dtiQc:
I
vet) ~ {Mq + q ( lp(s)ds}q (D)
Dinh ly tren khong c~n tinh don di~u, kha vi cua ham f. Tuy nhien ta
c~n phai tinh Sup{f(t)EIR ItEn}.
2.5. Dfnh ly 2.4 (Xem [3]).
Cha u(t), a(t), b(t) la cac ham lien tl;lc,khong am trenD, p 20, c la
cac hang so'saa cha
u(t) ~c+ rl [a(s)u(s) + b(s)uP(s)]ds, 'rItEil.
Jlo
(2.20)
Khi do tuy thep p, ta co cac kit qua sau:
a. Niu 0 sp <1 thi
I
u(t) ~ exp[ (a(s)ds]{ cq + q (b(s)exp[ -q 1:a(r)dr]dsii (2.21)
lit ED, trang do q = 1 -p.
b. Niu p =1 thi
u(t)~cexp[ ([a(s)+b(s)]ds], 'rItEQ
(2.22)
Llli)n van th{lc sf loan h{Jc
Mil nganh : 1.01.01
17
Milri}ngva ung d1!ngBfl di Gronwall-Bellman Hoang Thanh Long
c. Ntu p >1 thz
1
u(t)::; c{exp[ q 1: a(s)ds] + c-'1q1: b(s)exp[ q ra(r )dr ]ds}q (2.23)

1 I
VtE[to,tp), t =Sup{tEQlexp[q rta(s)ds]/q{-q rtb(s)ds]}q >c}.
fJ Jto Jto
Chung minh dinh If 2.4. Ta chung minh r5 rang nhusau:
D~t vet)la vfiphai cua (2.20). Ta co:
v'(t)::; a(t)v(t) + b(t)vP(t).
(2.24 )
a. Nfiu 0 ::;p <1, thl ap dl;mgb6 d~ b6 trQ, ta duQc (2.21).
b. Nfiu p
=1, tu (2.20), ta co:
u(t)::; c + ([a(s) + b(s)]u(s)ds.
Ap dlJng dinh 1y 1.2, ta duQc (2.21).
c. Nfiu p > 1, ap dlJng b6 d~ b6 trQ cho (2.24), ta duQc (2.23).(0)
2.6. Dinh If 2.5.
Cho u(t), a(t), b(t) la cac ham lien tl;lc,khong am tren Q, c ;:::0, p ;:::0,
0 ::;q ::;1 la cac hang so: p ;:::q. Ntu
u( t) ::;c + rt a( s)uP (s)ds + rt b( s)14'1(s )ds, tit ED.
Jto Jto
(2.25)
Khi do tuy theo p, ta co cac kef qua sau:
a. Ntu p
=1 thz
u(t)::;exp[ rta(s)ds]
Jto
1
(Cl-'1 +(1-q) rt b(s)exp[(q-1) r~a(r)dr]dsp-'1 (2.26)
Jto Jto
b. Ntu p < 1 thz
Lu{jn van th{Jc si loan h(Jc
Mii nganh : 1.01.01

18
MlJrl)ng va ung dljng Bd dff Gronwall-Bellman
Hoang Thanh Long
I-p I
u(t) So[ZI-q(t)+(1-p) rta(s)dst-p,
Jto
(2.27)
trong do Z(t)
=Sup{K( s) I s E[ta,t]}.
c.Ne'u p > 1 thi
I p-l I
u(t) SoK1-q(t)[1+(1-p) rta(s)K~(s)dsJ'-p,
Jto
(2.28)
p-l
'r7tE[ta,tp), t =Sup{tEQI(p-l) rta(s)KI-q(s)ds<l},
p J~
trong cae bat dcing thac tren K(t) =c1-q + (1- q) rt b(s)ds.
Jto
(2.29)
Chung minh djnh Iy 2.5. £)~t vet) Ia vfi phiii cua (2.25). Khi do, ta co:
v'et) = a(t)uP(t) + b(t)uq(t),
Soa(t)vP(t) + b(t)vq(t),
So[a(t)vp-q(t) + b(t)]vq(t).
(2.30)
Chuy~n vg(t) sang tnii va Iffytkh phan hai vfi tll todfin t, ta dU<;lc:
vI-get) Socl-q + (1- q) rt[b(s) + a(s)vp-q(s)]ds.
Jto
(2.31)
£)~t yet) =vi-get)va e= p - q

1-q'
Tll (2.31), ta dU<;lc:
yet) SoK(t) + (1- q) (a(s)y8(s)ds.
(2.32)
a. Nfiue =1, tuc Ia p =1, tll (2.32), ta co:
yet) SoK(t) + (1- q) (a(s)y(s)ds.
(2.33)
Ap d1;1ngdinh Iy 1.5, ta du<;lc:
Lluln van th{lc si loan h{JC
Mii nganh : 1.01.01
19
MlIri)ng va llng d~tngBli di Gronwall-Bellman
Hoang Thanh Long
yet) ~ cl-q exp[(I- q) rt a(s)ds]
Jto
+(1- q) (b(s)exp[(I- q) fa(r)dr]ds.
(2.34)
net) ~ exp[ (a(S)dS]
1
{cH + (1- q) rtb(s)exp[(q -1) rsa(r)dr]ds}l-q (2.35)
J~) J~
b. N€u 8
< 1, tuc la p < 1, ap dl;lngd~nh ly 2.7 trong truong hQp d~c
bi~t vao (2.32), ta duQc:
I-p I-q
yet) ~ [Zl-q (t) + (1- p) rt a(s)ds]I-P,
Jto
(2.36)
I-p I
hay net) ~ [ZI-q (t) + (1- p) rt a(s)ds]I-P

Jto
(2.37)
c. N€u 8 > 1, tuc la p > 1, ap dl;lngd~nh ly 2.10 vao (2.32), ta duQc:
p-I I-q
y(t)~K(t)[l+(I-p) rta(s)KI-q(s)ds]I-P,
Jto
(2.38)
I p-I I
hay net) ~ KI-q (t)[1 + (1- p) rt a(s)K 1=4(s)ds]1=P.(0)
Jto
2.7. Dinh Iy 2.6 (Xem [2]).
Cia sit u(t), b(t), K(t,s), h(t,s,o-) la cac ham lien tf:lc,khong am trang
to :;;() :;;s :;;t :;;tJsaG cha
u(t)~a+ rtb(s)uP(s)ds+ rt r~K(s,1:)uP(1:)d1:ds
Jto Jto Jto
+ rt r~ rt h(s,1:,a)uP(a)dad1:ds, VtEQ,
Jto Jto Jto
(2.39)
trang do 0 < a la mQt hang so'va 0 :;;p ::;z!: 1 thi
Luqn van th{lc si loan h(Jc
Mii nganh : 1.01.01
20
M1Jri)ng va ung d(tng Bd di Gronwall-Bellman
Hoang Thanh Long
I
u(t)~{aq+qr[b(s)+ rK(S,T)dT+ rs r'h(S,T,cr)dTdcr]dsjq,(2.40)
Jto Jto Jto Jto
tit E{to ,tp),
tp
=SUp{tEQ Iaq + q f[b( s) + rl' K( s, T)dT + r r'h(s, T,cr)dTdcr]ds > OJ.

to Jto Jto Jto
Chung minh djnh ly 2.6. Xem [2].(0)
2.8.Bjnh ly 2.7 (Xem [2]).
Cho u(t), b(t), K(t,s), h(t,s, a) la cac ham lien tl;lc,khong am trong to::;
, .? ?
a::; s ::;t ::;tj va gza sa
u(t)~a(t)+ rt b(s)uP(s)ds+ rt r~K(S,T)UP(T)dTds
Jto Jto Jro
+ rt r r'h(s,T,a)uP(a)dadTds, 'r/tEQ,
Jto Jto Jto
(2.41)
trang do a(t) 20 la mqt ham so' lien tl;lc,khong gidm tren Qva O::;p;z:Jla
ml)t h!:ingso:Ta co:
]
u(t)~{Aq(t)+q rt[b(s) + rK(S,T)dT+ r r'h(S,T,cr)dTdcr]dsjQ,(2.42)
Jto Jto Jto Jto
tltE[tO,tp),A(t) =Sup{a(s)lsE[to,tJ},
tp= Sup{tEQIAq(t)+q rt (b(s) + r~K(S,T)dT+ rs r'h(S,T,cr)dTdcr]ds> OJ
Jto Jto Jto Jto
Chung minb djnb ly 2.7. Xem [2].(0)
£)~t Ii ={ (tl,h, ,ti)EIRI Ia ~ ti ~ ~ t1 ~ t ~ ~}, i =1, ,n.
2.9.Bjnb ly 2.8 (Xem [2]).
Cho u(t), b(t) la cac ham lien tl;lc,khong am trong J =[ a,p] va
u(t) ~ b(t)[ a +
IK](t,tj)uP(tl)dtl
r rl rtl1 I
+ + Ja(Ja ( Ja- Kn(t,tl' ,tn)uP(tn)dtn) )dt]], 'r/tEJ,(2.43)
Lllqn van th{lc sFloan h(JC
Mil nganh : 1.01.01
21

MiJrl)ng va ling d~tngBd di Gronwall-Bellman
Hoang Thanh Long
trang doa > 0 va 0 S'p :/=1la cac hang so: Ki (t,t], ,U la ham so'lien tl;lc,
khong am trang Ji WYii =1, ,n. Gid sit a:i ton tc;zi,khong am va lien tl;lc
trang Ji wJi i =1, ,n. Khi do, ta co:
I
u(t)~b(t)[alJ + q((R[bPJ(s) + Q[bPJ(s))dsjq, WE[a,fJ]), (2.44)
trang do q =1 - p,
fJ]
= SupftEJlalJ+q(fR[bPJ(s)+Q[bP](s)}ds>O},
R[wJ(t) = KJt,t)w(t) + (K2(t,t,t2)W(t2)dt2
n I 12 Ii I
+~
I
(
I
(
1
- KJt,t,t2, ,t)w(t)dt) )dt2, (2.45)
~ a a alii
1=3
I
laK
Q[w](t) = 1-(t,ti)W(ti)dti
a at
~
I
I
i
ll

i
l;-I aK,
+L ( ( 1-(t,tp ,t)w(t)dt) )dtp (2.46)
, a a a
at
1 1
1=2
wJi mQi ham lien tl;lcw(t) trang J.
Chung minh dfnh ly 2.8. Xem [2].(0)
Dinh 19sau day Iiih~ qua cua dinh 192.8.
2.10. Dfnh ly 2.9 (Xem [2]).
Cha u(t) la ham lien tl;lc,khong am tren J
=[a,fJ]. Gid sit
u(t)~a+ (K(t,s)uP(s)ds+ ({h(t,s,i:)uP(r)di:ds, WEJ,
(2.4 7)
trang do a > 0 va 0 S'p :/=1 la hang so:'K(t,s) va h(t,s,r) la cac ham lien
kh
A A ,.
< < <
<fJ,
aK ,ah ~ .
kh
A A
I
.A
tuc, an
g
am val a - r - s - t - ,. -va - tan tal, an
g
am, len tUG

. at at' .
Lt«fn 1'0'1 tlt(lC sf todn It(lC
Mti nganlt IJllJJI
22
MlJr(mg va ung dl.lngBd di Gronwall-Bellman Hoang Thanh Long
, . < < < <
j3.
Kh
'
d
' ,
vO'l a - 'f - S - t 1 0, ta co:
I
u(t)::=;;[ail + q L (R( s) + Q(s))dsjq, b1E[a,fJJ),
(2.48)
WJiq
=1 - p va fJJ=Sup{t E J Iail + q L (R(s) + Q(s))ds > O},
R(t)=K(t,t)+ Lh(t,t,1:)d1:,
(2.49)
I
t aK
I
t
I
sah
Q(t)= -(t,1:)w(t)dt+ -(t,s,1:)d1:ds.
a at a a at
(2.50)
2.11. Djnh ly 2.10 (Xem [2]).
Cho u(t), b(t) la cae ham lien tl:le,khong am tren 12 K(t,s), h(t,s, (J)la

cae ham lien tl:le,khong am trong to::; (J::;s ::;t ::;tJ va gid SU:
u(t)::=;;a(t)+ r b(s)uP(s)ds+ rt C'IK(s,1:)uP(1:)d1:ds
Jto Jto Jto
+ rt rs r' h(s,1:,a)u"(a)dad1:ds, L7tEQ,
Jto Jto Jto
(2.51)
trang do a(t) ;:0 la melt ham so'lien tl:le,khong gidm tren Q, 1 < p la hang
so: Ta co:
J
I
t -~
u(t)::=;;a(t)[1-r B(s)ar(s)dsj , b1E[to,fJp),
to
(2.52)
vJi
B(t)=b(t)+ rtK(t,s)ds+ rt r'h(t,s,1:)d1:ds,
Jto Jto Jto
(2.53)
fJp= Sup{tEOll-rrB(s)d'(s)ds>O}var=p-l.
Jto
Chung minh djnh ly 2.10. Xem [2].(0)
2.12. Djnh ly 2.11 (Xem [2]).
Cho u(t), b(t), K(t,s), a(t) la cae ham lien tl:le,khong am trong to::;s::;
Luijn van th{lc si loan h(Jc
Mil nganh : 1.01.01
23
MlJr{}ngva ung d~tngBd d~ Gronwall-Bellman
Hoang Thanh Long
t :::::tj.Ne'u
u(t) S;a(t){ a +

i
t b(s)uP(s)ds +
i
t
i
s K (s, T)UP('r:)dTds), t7t6.0,(2.54)
to to to
trang do a :? 0, P :? 1 fa cae hling so: thi ta co:
I
u(t) s; aa( t)[1- rar
i
t B( s)ar (s)ds ] ;, t7t6[ to,f3r),
to
(2.55)
vcfi
B(t)=b(t)+
fK(t,s)ds,
to
(2.56)
J3r =SUp{tEQ. I rd' rtB(s)ar(s)ds<l}var=p-l.
Jto
Chung minh dinh Iy 2.11. Ta chung minh ro rang nhu sail: D~t
vet) = rt b(s)uP(s)ds + rt rs K(s, T)uP('r:)d'T:ds.
Jto Jto Jto
(2.57)
La'y dqo ham hai vS (2.57), ta du<;jc:
v'(t) = b(t)uP(t) + rt K(t, 'T:)uP('T:)d'T:.
Jto
s;B(t)aP(t)[a+v(t)]p.
S;R(t)a+R(t)v(t).

vdi R(t) = B(t)aP(t)[a+v(t)]P-l.
(2.58)
(2.59)
Tu (2.58), ta suy fa:
vet)+ as; aexp[ rt R(s)ds].
Jto
(2.60)
La'y lUy thua hai vS va nhan r vao hai vS ba't d£ng thuc, ta du<;jc:
fRet) S;rB(t)af (t)af exp[r rtR(s)ds].
Jto
(2.61)
Nhan hai vS (2.61) vdi exp[-r r R(s)ds] va la'ytkh phan hai vS tu to
Jto
Lllqn van th[Jc Sfloan h{Jc
Mii nganh : 1.01.01
24
MlJrQng va ling d1!ngBd d~ Gronwall-Bellman
Hoang Thanh Long
dSn t, ta duQc:
1 - exp[ -r rt R(s)ds]:::;far rt B(s)crP(s)ds.
Jto Jto
(2.62)
Vdi to:::;t:::;PP' tu (2.62), ta suy fa:
1
exp[ rt R(s)ds]:::;[1 - far rt B(s)crP(s)dsf~.
J~ J~
(2.63)
Thay vao (2.60), ta duQc:
I
t

vet) + a:::;a[l- far rB(s)crP(s)ds] r.
Jto
(2.64)
)
t
V~y U(t):::;acr(t)[l- far r B(s)crP(s)ds] r .(0)
Jto
2.13. Dfnh Iy 2.12 (Xem [2]).
Cha u(t) ;;:0, art) ;;:0, b(t) > 0, la cac ham lien tl;lctrang J =[a, 13],
Gid sit aCt) la mot ham tang trang J,
b(t) .
u(t) :::;a(t) + b(t)[ IK/t,tl)UP (tl)dt)
rl r/l rn I
+ + Ja ( Ja ( Ja - Kn(t,tl' ,tn)uP(tn)dt,J )dt1J, MEJ,(2.65)
trang do p > 1 la m(Jt hiing so: Ki (t,tj, ,ti) la ham so'lien tl;lc,khong am
trang Ji wJi i =1, ,n, va a:i tbn t(li, khong am va lien tl;lctrang Ji wJi i =
1, ,n. Khi do, ta co:
1
u(t):::;a(t)[l- r r[a( s)T( R[bP](s) + Q[bP](s))dsJ ;., ME[a ,131),(2.66)
a b(S)
trang do r
=p -1,
Llli)n van th(lc sf loan hf)C
Mil nganh : 1.01.01