Mở rộng và ứng dụng bổ đề Gronwall -BellMan 7

Luận văn thạc sĩ toán học -ngành Toán Giải Tích -Chuyên đề :Mở rộng và ứng dụng bổ đề Gronwall -BellMan

MiJr{}ngva ung d1!ngBli d€ Gronwall-Bellman Hoang Thanh Long

CHUaNG 2

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 Wi neu md rQng mQt 86d(;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

tp = SUp{tEQ Iuq(to) + qJtortb(s)exp[ -q r~a(r )drJds > OJ.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

Mil ri)ng va ring d(tng Bd d~ Gronwall-Bellman Hoang Thanh Long

tf:lC,khong am tren £2 Ne'u

u (t)~a(t)+2b(t) Jtocp(s)u(s)ds, l7tEQ, (2.4 )

thi

1

lu(t)1 ~(a(t)+b(t) rto[cp2(s)+a(s)]exp[ fb(r)drJdsj2, s [7tE£2(2.5)

Chung minh dfnh If 2.1 Di;it

vet) = 2 rt <p(s)u(s)ds, 'v'tEQ

Lffy d~o ham hai vii cua (2.6), ap dl;lng bfft d~ng thlic Cauchy va kiithqp voi (2.4), ta duQc:

v'et) ~ <p\t) + aCt)+ b(t)v(t). (2.7)Suy fa

vet) ~ fto [cp2(S)+ a(s)]exp[ fb(f)dfJiS].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,

MlI ri)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)

-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)

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,

thi

I

u(t) S;fer) + [M<f + q f0qJ(s)ds/q VlEQ, (2.14 )

Chung minh dinh Iy 2.3 E>~t

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:

Llli)n van th{lc sf loan h{Jc Mil nganh : 1.01.01

Mil ri}ng va 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[ qra(r )dr ]ds}q (2.23)

VtE[to,tp), t =Sup{tEQlexp[q fJ Jtorta(s)ds]/q{-q Jtort b(s)ds]}q >c}.

Chung minh dinh If 2.4 Ta chung minh r5 rang nhusau:

D~t vet) la vfi phai cua (2.20) Ta co:

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.

MlJrl)ng va ung dljng Bd dff Gronwall-Bellman Hoang Thanh Long

'r7tE[ta,tp), t p =Sup{tEQI(p-l) J~rta(s)KI-q(s)ds<l},

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 Iffy tkh phan hai vfi tll to dfin t, ta dU<;lc:

vI-get) Socl-q + (1- q) rt[b(s) + a(s)vp-q(s)]ds

£)~t yet) =vi-get) va e = p- q

1-q'Tll (2.31), ta dU<;lc:

a Nfiu e =1, tuc Ia p =1, tll (2.32), ta co:

Ap d1;1ngdinh Iy 1.5, ta du<;lc:

MlIri)ng va llng d~tngBli di Gronwall-Bellman Hoang Thanh Long

yet) ~ cl-q exp[(I- q) Jtort a(s)ds]

+(1- q) (b(s)exp[(I- q)fa(r)dr]ds (2.34)net) ~ exp[ (a(S)dS]

hay net) ~ [ZI-q (t) + (1- p) rt a(s)ds]I-PJto (2.37)

c N€u 8 > 1, tuc la p > 1, ap dl;lng d~nh ly 2.10 vao (2.32), ta duQc:

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,

trang do 0 < a la mQt hang so'va 0 :;;p ::;z!: 1 thi

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 + qfto [b( s) + Jtorl' K( s, T)dT + Jto Jtor r'h( s, T,cr)dTdcr]ds > OJ.

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,

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 Jtort (b(s) + r~K(S,T)dT+ rs r'h(S,T,cr)dTdcr]ds> OJJto Jto Jto

Chung minb djnb ly 2.7 Xem [2].(0)

+ + Ja (Ja ( Ja- Kn(t,tl' ,tn)uP(tn)dtn) )dt]], 'r/tEJ,(2.43)

Lllqn van th{lc sF loan h(JC Mil nganh : 1.01.01

MiJ rl)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 WYi i =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:

wJi mQi ham lien tl;lcw( t) trang J.

Chung minh dfnh ly 2.8 Xem [2].(0)

Dinh 19sau day Iii h~ 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

MlJr(mg va ung dl.lng Bd 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)

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

+ rt rs r' h(s,1:,a)u"(a)dad1:ds, L7tEQ,

trang do a( t) ;:0 la melt ham so' lien tl:le,khong gidm tren Q, 1 < p la hang so: Ta co:

Cho u(t), b(t), K(t,s), a(t) la cae ham lien tl:le,khong am trong to::;s::;

MlJ r{}ng va ung d~tngBd d~ Gronwall-Bellman Hoang Thanh Long

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)

Nhan hai vS (2.61) vdi exp[-r r R(s)ds] va la'y tkh phan hai vS tu toJto

MlJrQng va ling d1!ngBd d~ Gronwall-Bellman Hoang Thanh Long

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,

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:

M1Jri)ng va ung d(tng B6 di Gronwall-Bellman Hoang Thanh Long

vcii mqi ham lien tZ:tcw( t) trang 1.

Chung minh djnh ly 2.12 Xem [2].(0)

Dinh 1y sau 1a h~ qua cua dinh 1y 2.12

2.14 Djnh ly 2.13 (Xem [2]).

Cha u(t) 2:0, aCt)2:0, bet) > 0, la cac ham lien tZ:tctrang J = [a,/3J.

Gid sit aCt) la mot ham tangtran g 1 Ne'u

Suy fa tll dinh 1y 2.12 khi P =n, K1(t,tl)= K(t), Kj = 0, i ~ 2.(0)

Mi'Jrf)ng va ung dz.mgBd di Gronwall-Bellman Hoang Thanh Long

2.15 Dinh If 2.14.

Cha u(t), art), b(t,s) la cac ham lien tf:tC,khong am trang to:5:s :5:t:5:

tJ Gid sit

u(t)s,c+ rta(s)uP(s)ds+ r r~b(s,'t)u([)d'tds,WED,

trang do c ~ 0, p la cac hling so: q =1 - p Khi do tily rhea p, ta co cac kef

Chung minh dinh If 2.14 Tuong tv chung minh dinh 1y2.4.

Di\ltvet) 1Av€ phai cua (2.71) Khi do, ta co:

v'et) s, a(t)vP(t) + vet) rt bet, 't)d't

Ap dl:mg b6 d~ b6 trQ, ta duQc a, b, c.(D)

Mli ri)ng va ung dljng Btl di Gronwall-Bellman Hoang Thanh Long

2.16 Dfnh Iy 2.15.

Cha u(t), art), b(t,s) la cac ham lien tl;lc,khong am trang tossststj

, ? ?

va gza sa

u(t)sc+ fl a(s)u(s)ds+ r r~b(S,1)U]J('t)d1ds, btEQ,

trang do 0 :::;c, p la cac hang sa, q =1- p Khi do tuy thea p, ta co cac kit

C < (exp[ q flo+h a( s )ds J) -q {-q flo+h flo+hb( s, r )drds] if.

wJi h > 0 new do thi wJi to.st sto + h, ta co:

Cha u( t), qX t) la cac ham lien tl;lc,khong am tren Q va thoa man

u(t) s M + fl cp(s)g(u(s))ds, btEQ,

Mli rf)ng va u'ng d~tngBd de' Gronwall-Bellman Hoang Thanh Long

trang do M la hang so' khong am va g: IR+ -f (0, ro) la ham tang, lien tl;lC Khi do, ta co:

Gid sa u(t) la ham lien tl;lc,khong am tren Q saD cha

Mll ri)ng va ung d(tng Bii di Gronwall-Bellman Hoang Thanh Long

f~(t)

u(t) sf(t)+ K(t,s)g(u(s))ds, btEn,

v6'i cac hamf(t), fjJ(t),K(t,s), g(u) thoa man cac ddu ki~n sau:

a f(t) la ham khd vi, khong am va khong gidm tren 0;

b fjJ(t)la ham khd vi, khong gidm tren ova fjJ(t)::;t, fjJ(ta)=ta;

c 0 < g(u) va khong gidm tren

IR+,-d 0::; K(t,s) la ham lien tl;lctren ox.ova co dqLOham rieng the a t la ham khong am, lien tl;lc;

e f'(t){ 1 I} S 0, t En, v6'i 77la ham lien tl;lc,khong am tren

g(7J(t))

.ova G dtnh ngh'ia nhu:(2.87).

Khi do, ta co:

u(t) S G-1 (G( f(to)) + f(t) - f(to) + Lcprs)dsJ, bt E[ta,a), (2.89)

a =Sup(t E.o IG( f(to)) + JtJf'( s) + cprs)Jds E Dom(G- )}

Chung minh dinh ly 2.18 Ta chung minh r6 rang nhli sail:

f)~t vet) la v6 phiii cua (2.88), Iffy d(;loham hai v6, ta dli<;1c:

Tli (2.92) va di~u ki~n e, ta dli<;1C:

Lllfjn van th{lc Sf loan h(JC Mil nganh : 1.01.01

MlJ ri)ng va ling dl;l11g Bd de'Gronwall-Bellman Hoang Thanh Long

Xo= sup{xE[a,bJI{G(u(a))- (/ cp(S)dsjEDom(G- )j.

Chung minh dinh Iy 2.19 Xem [3].(0)

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

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) f K(t,s)ds} J, (2.98)

V'tE[c,d') va G(u)= r ~~ ' (E>O, u>O),

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;lC tren [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:

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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)

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

l OA

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

the v6i tEl, ta co:

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

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.

Chung minh djnh ly 2.23 Suy fa tu dinh 1y2.22.(0)

Lllq,n van thlJc sl loan h(JC Mil nganh : 1.01.01