Mở rộng và ứng dụng bổ đề Gronwall - Bellman

u(t)S u(t)exp[( [a(s) +b(s)Jds, 'r7tEQ. (2.2) b. Ne'up :;z!:1 thi I u(t)sexp[ rta(s)ds]{uq(to)+qrtb(s)exp[-qr~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 Chungminhb6d~b6trQ.Xem[5].(0) 2.2.DinhIy 2.1. Chou(t)la hamlientl:lCtrenQ. Gid si/:art),bet),cp(t)la cachamlien Luijn vanthlJcsi loan h{Jc Mil nganh..1.01.01 14 Mil ri)ngvaringd(tngBd d~Gronwall-Bellman HoangThanhLong tf:lC,khongam 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 ChungminhdfnhIf 2.1.Di;it vet) =2 rt<p(s)u(s)ds,'v'tEQ.Jto (2.6) Lffy d~ohamhaivii cua(2.6),apdl;lngbfftd~ngthlicCauchyvakiit hqpvoi (2.4),taduQc: 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 Thayvao(2.4)vaIffydin, taduqc(2.5).(0) (2.8) 2.3.Dfnh If 2.2. Cha u(t), b(t), <P(t) la cac hamlien tf:lC,khongam tren£2 0 ::;p :;z!:1, a la cac hangso: Gid sit b(t) la melthamkhonggidmva 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 +qr<p(s)bP(s)dsj,q l7tE[to,tp), b(to) to (2.10) trangdo tp=SUp{tEQI[~Jq +q r cp(s)bl'(s)ds>OJ vaq =1- p. b(to) Jto Luqn vanth{lcsl loanh(JC Mil nganh : 1.01.01 15 MlIri)ngvaungd~tngBifd€ Gronwall-Bellman HoangThanhLong Chungminh dinh Iy 2.2.E>~tvet)la v€ philicua(2.9).Khi do,taco: 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 dlJngb6d~b6trq,taduqc: b'( ) b ' ( ) I t S t s r - vet)s;exp[r -ds]{aq +q rqJ(s)b(s)exp[-qr -dr]ds}q (2.12) Jto b(s) Jto Jto her) . I V~y vet)S;betH[~]q +q rtlp(s)bP(s)ds}4(D) beta) Jto . 2.4.DinhIy 2.3. Chou(t),fer),qi.t)lacachamlient¥c,khongamtrenQ. 0~p <1la m(jthlingso: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) trangdoM =Sup{f(t)E IRI tED}. ChungminhdinhIy 2.3.E>~t vet)=rt lp(s)uP(s)ds. Jto (2.15) La'ydC;lOhamhaiv€ va chuyrAngu(t)S;f(t) +v(t), tathuduqc: v'et)s;qJ(t)[f(t)+v(t)]P. v'(t)s;lp(t)[M+v(t)]P. (2.16) Lufjn van th{lCsi loan h(Jc Mil nganh : 1.01.01 16 MiJri)ngvaungd(tngBddi Gronwall-Bellman HoangThanhLong Chiahaiv€ cua(2.16)cho[M +v(t)]P,tadtiQc: viet) [M +v(t)]P~lp(t). (2.17) Lgy tichphanhaiv€ tutod€n thai v€ bgtd~ngthuc,tadtiQc: [M +v(t)]q~Mq+q rtlp(s)ds. Jto (2.18) Suyra 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 trenkhongc~ntinhdondi~u,khavi cuahamf. Tuy nhienta c~nphaitinhSup{f(t)EIRI tEn}. 2.5.Dfnh ly 2.4(Xem [3]). Chau(t),a(t),b(t) la cachamlientl;lc,khongamtrenD, p 20, c la cachangso'saacha u(t)~c+ rl[a(s)u(s)+b(s)uP(s)]ds,'rItEil. Jlo (2.20) Khi dotuythepp, taco cackit quasau: a.Niu 0 sp <1thi I u(t)~exp[(a(s)ds]{cq+q( b(s)exp[-q 1:a(r)dr]dsii (2.21) litED, trangdo q =1 -p. b.Niu p =1thi u(t)~cexp[([a(s)+b(s)]ds], 'rItEQ (2.22) Llli)n vanth{lcsf loan h{Jc Mil nganh : 1.01.01 17 Milri}ngvaungd1!ngBfldi Gronwall-Bellman HoangThanhLong c.Ntup >1thz 1 u(t)::; c{exp[q 1:a(s)ds]+c-'1q1:b(s)exp[qra(r )dr]ds}q(2.23) 1 I VtE[to,tp),t =Sup{tEQlexp[q rta(s)ds]/q{-qrtb(s)ds]}q>c}. fJ Jto Jto ChungminhdinhIf 2.4.Tachungminhr5rangnhusau: D~tvet)lavfiphaicua(2.20).Taco: v'(t)::;a(t)v(t)+b(t)vP(t). (2.24) a.Nfiu 0 ::;p <1,thl apdl;mgb6d~b6trQ,taduQc(2.21). b.Nfiup =1,tu(2.20),taco: u(t)::;c+ ([a(s) +b(s)]u(s)ds. Ap dlJngdinh1y1.2,taduQc(2.21). c.Nfiup > 1,apdlJngb6d~b6trQcho(2.24),taduQc(2.23).(0) 2.6.Dinh If 2.5. Chou(t),a(t),b(t) la cachamlientl;lc,khongamtrenQ, c ;:::0,p ;:::0, 0 ::;q ::;1 la cachangso:p ;:::q.Ntu u(t) ::;c + rta(s)uP(s)ds + rt b(s)14'1(s)ds, titED. Jto Jto (2.25) Khi dotuytheop, tacocackefquasau: a.Ntu p =1 thz u(t)::;exp[rta(s)ds] Jto 1 (Cl-'1 +(1-q) rtb(s)exp[(q-1) r~a(r)dr]dsp-'1 (2.26) Jto Jto b.Ntu p <1thz Lu{jnvanth{Jcsi loan h(Jc Mii nganh: 1.01.01 18 MlJrl)ngvaungdljngBddffGronwall-Bellman HoangThanhLong I-p I u(t) So[ZI-q(t)+(1-p) rta(s)dst-p, Jto (2.27) trongdoZ(t)=Sup{K(s) I s E[ta,t]}. c.Ne'up > 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~ trongcaebatdcingthactrenK (t)=c1-q +(1- q) rtb(s)ds.Jto (2.29) Chungminhdjnh Iy 2.5.£)~tvet)Ia vfiphiiicua(2.25).Khi do,taco: 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~nvg(t)sangtnii vaIffytkh phanhaivfi tll todfint, tadU<;lc: vI-get)Socl-q +(1- q) rt[b(s)+a(s)vp-q(s)]ds. Jto (2.31) £)~tyet)=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,tucIa p =1,tll (2.32),taco: yet)SoK(t) +(1- q) (a(s)y(s)ds. (2.33) Ap d1;1ngdinhIy 1.5,tadu<;lc: Lluln vanth{lcsi loanh{JC Mii nganh: 1.01.01 19 MlIri)ngvallngd~tngBli di Gronwall-Bellman HoangThanhLong yet)~cl-qexp[(I- q) rta(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,tucla p < 1,apdl;lngd~nhly 2.7trongtruonghQpd~c bi~tvao (2.32),ta duQc: I-p I-q yet)~[Zl-q(t) +(1- p) rta(s)ds]I-P,Jto (2.36) I-p I hay net) ~[ZI-q (t) +(1- p) rta(s)ds]I-PJto (2.37) c.N€u 8 > 1,tucla p >1,apdl;lngd~nhly 2.10vao(2.32),taduQc: 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) rta(s)K1=4(s)ds]1=P.(0) Jto 2.7.DinhIy 2.6(Xem[2]). Cia situ(t),b(t),K(t,s),h(t,s,o-)la cachamlientf:lc,khongamtrang to:;;():;;s :;;t :;;tJ saGcha u(t)~a+ rtb(s)uP(s)ds+rt r~K(s,1:)uP(1:)d1:ds Jto Jto Jto +rt r~rth(s,1:,a)uP(a)dad1:ds,VtEQ, Jto Jto Jto (2.39) trangdo0 <a la mQthangso'va0 :;;p ::;z!: 1 thi Luqnvanth{lcsi loan h(Jc Mii nganh: 1.01.01 20 M1Jri)ngvaungd(tngBddi Gronwall-Bellman HoangThanhLong 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 titE{to,tp), tp=SUp{tEQIaq+qf [b(s)+ rl' K( s,T)dT+ r r'h(s,T,cr)dTdcr]ds> OJ.to Jto Jto Jto Chungminhdjnhly 2.6.Xem[2].(0) 2.8.Bjnh ly 2.7(Xem[2]). Chou(t),b(t),K(t,s),h(t,s,a) la cachamlientl;lc,khongamtrongto::; , .? ? a::;s ::;t ::;tj va gzasa u(t)~a(t)+ rtb(s)uP(s)ds+rt r~K(S,T)UP(T)dTds Jto Jto Jro +rtr r'h(s,T,a)uP(a)dadTds, 'r/tEQ, Jto Jto Jto (2.41) trangdoa(t)20 la mqthamso'lientl;lc,khonggidmtrenQva O::;p;z:Jla ml)th!:ingso:Taco: ] 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 Chungminbdjnbly 2.7.Xem[2].(0) £)~tIi ={(tl,h,...,ti)EIRI I a ~ ti ~ ~t1~t ~~},i =1, ,n. 2.9.Bjnb ly 2.8(Xem[2]). Chou(t),b(t)la cachamlientl;lc,khongamtrongJ =[a,p] va u(t)~b(t)[a+IK](t,tj)uP(tl)dtl r rl rtl1I +...+Ja(Ja ...(Ja- Kn(t,tl'...,tn)uP(tn)dtn)...)dt]],'r/tEJ,(2.43) Lllqn van th{lcsFloan h(JC Mil nganh : 1.01.01 21 MiJ rl)ngvalingd~tngBd di Gronwall-Bellman HoangThanhLong trang doa >0va0S'p :/=1la cachangso:Ki (t,t],..,Ula hamso'lientl;lc, khongam trangJi WYii =1,...,n.Gid sit a:i ton tc;zi,khongam va lien tl;lc trangJi wJi i =1,...,n.Khi do,taco: I u(t)~b(t)[alJ +q((R[bPJ(s) +Q[bPJ(s))dsjq, WE[a,fJ]), (2.44) trangdoq=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 IlaKQ[w](t)= ---1-(t,ti)W(ti)dtia at ~II ill il;-I aK,+L.. ( ...( 1-(t,tp...,t)w(t)dt)...)dtp (2.46), a a a at 1 11=2 wJi mQihamlientl;lcw(t) trangJ. Chungminhdfnhly 2.8.Xem[2].(0) Dinh19saudayIii h~quacuadinh192.8. 2.10.Dfnh ly 2.9(Xem [2]). Cha u(t)la hamlientl;lc,khongamtrenJ =[a,fJ].Gidsit u(t)~a+ (K(t,s)uP(s)ds+ ({h(t,s,i:)uP(r)di:ds, WEJ, (2.47) trangdoa >0 va0S'p :/=1 la hangso:'K(t,s)vah(t,s,r) la cachamlien khA A ,. < < < <fJ, aK ,ah ~ . khA A I .Atuc, ang amvala - r - s - t - ,. -va - tantal, ang am, len tUG. at at' . Lt«fn 1'0'1tlt(lCsf todn It(lC Mti nganlt ..IJllJJI 22 MlJr(mgvaungdl.lngBddi Gronwall-Bellman HoangThanhLong , . < < < <j3. Kh ' d ' , vO'l a - 'f - S - t -. 1 0, ta co: I u(t)::=;;[ail + qL (R(s)+Q(s))dsjq,b1E[a,fJJ), (2.48) WJiq =1- p vafJJ =Sup{tE J Iail+qL (R(s)+Q(s))ds>O}, R(t)=K(t,t)+ Lh(t,t,1:)d1:, (2.49) It aK It IsahQ(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]). Chou(t),b(t)la caehamlientl:le,khongamtren12K(t,s),h(t,s,(J) la caehamlien tl:le,khongamtrong 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) trangdoa(t) ;:0 la melthamso'lientl:le,khonggidmtrenQ, 1 <p la hang so:Ta co: J It -~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 minhdjnh ly 2.10.Xem [2].(0) 2.12.Djnh ly 2.11(Xem [2]). Chou(t),b(t),K(t,s),a(t)la caehamlientl:le,khongamtrongto::;s::; Luijnvanth{lcsi loanh(Jc Mil nganh : 1.01.01 23 MlJ r{}ngvaungd~tngBd d~Gronwall-Bellman HoangThanhLong t :::::tj.Ne'u u(t) S;a(t){a +it b(s)uP(s)ds+it is K (s,T)UP('r:)dTds),t7t6.0,(2.54)to to to trangdoa :? 0,P :? 1facaehlingso:thitaco: I u(t) s;aa(t)[1- rarit 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 ChungminhdinhIy 2.11.Tachungminhrorangnhusail:D~t vet) = rtb(s)uP(s)ds+ rt rsK(s, T)uP('r:)d'T:ds. Jto Jto Jto (2.57) La'ydqohamhaivS (2.57),tadu<;jc: v'(t)=b(t)uP(t)+ rtK(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),tasuyfa: vet)+as;aexp[rtR(s)ds].Jto (2.60) La'ylUythuahaivSva nhanr vaohaivSba'td£ngthuc,tadu<;jc: fRet)S;rB(t)af (t)afexp[r rtR(s)ds]. Jto (2.61) NhanhaivS(2.61)vdi exp[-r r R(s)ds]vala'ytkhphanhaivStutoJto Lllqn vanth[JcSfloan h{Jc Mii nganh : 1.01.01 24 MlJ rQngvalingd1!ngBdd~Gronwall-Bellman HoangThanhLong dSnt, taduQc: 1- exp[-r rtR(s)ds]:::;far rtB(s)crP(s)ds. Jto Jto (2.62) Vdi to:::;t:::;PP'tu(2.62),tasuyfa: 1 exp[rtR(s)ds]:::;[1- far rtB(s)crP(s)dsf~.J~ J~ (2.63) Thayvao(2.60),taduQc: 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 rB(s)crP(s)ds]r .(0) Jto 2.13.DfnhIy 2.12(Xem[2]). Cha u(t) ;;:0, art) ;;:0, b(t)> 0, la cac hamlien tl;lctrangJ =[a, 13], Gid sit aCt)la mothamtangtrangJ, 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) trangdop > 1 la m(Jthiingso:Ki (t,tj,...,ti)la hamso'lien tl;lc,khongam trangJi wJi i =1,...,n,va a:i tbnt(li,khongamva lien tl;lctrangJi wJi i = 1,...,n.Khi do, ta co: 1 u(t):::;a(t)[l- rr[a(s)T(R[bP](s)+Q[bP](s))dsJ--;., ME[a ,131),(2.66) a b(S) trangdo r =p -1, Llli)n vanth(lcsf loan hf)C Mil nganh : 1.01.01 25 M1Jri)ngvaungd(tngB6di Gronwall-Bellman HoangThanhLong /31 =Sup{tE J I r rlaCs)T(R[b" J(s) +Q[b"J(s))ds<1j, a b(S) R[wJ(t) =Ki(t,t)w(t)+L K2(t,t,t2)W(t2)dt2 n II 12 I; I+~ (r ...(r- K(t,t,t 2,...,t.)w(t.)dt.)...)dt2,(2.67)L... a Ja Ja t t I I i=3 IlaKQ[wJ(t)= --L(t,ti)W(ti)dtia at ~ I I III 1/;-1 aK.+L... ( ...( ---1-(t,tl,...,t)w(t)dtJ..)dtl' (2.68)i=2 a a a at vciimqi hamlien tZ:tcw(t) trang1. Chungminhdjnhly 2.12.Xem[2].(0) Dinh 1ysau1ah~quacuadinh1y2.12. 2.14.Djnh ly 2.13(Xem [2]). Cha u(t)2:0,aCt)2:0,bet)>0, la cachamlientZ:tctrangJ =[a,/3J. Gid sit aCt)la mothamtang trang 1.Ne'u bet) . u(t) s,a(t)+b(t)L K (s)un(s)ds,'rItEJ, (2.69) trangdol <n la mQtso'nguyen,thztaco: I u(t)S,aCt)[1- (n-1) L K( s)b(s)an-l(s)ds/-n, [;ItE[ a ,/3J),(2.70) /31=Sup{tEJI(n-1) LK(s)b(s)an-l(s)ds<lj. Chung minhdjnh ly 2.13. Suy fa tll dinh 1y2.12khi P =n, K1(t,tl)=K(t), Kj =0, i ~2.(0) Lllqn vanth{lcSl loan h(Jc Mil nganh ..1.01.01 26 Mi'Jrf)ngvaungdz.mgBd di Gronwall-Bellman HoangThanhLong 2.15.Dinh If 2.14. Chau(t),art),b(t,s)la cachamlientf:tC,khongamtrangto: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, Jto Jto Jto (2.71) trangdoc ~0,p la cachlingso:q =1 - p. Khi do tilyrheap, taco cackef ? qua sau: a.Ntup <1the u(t)s,exp[r rsb(s,r)drds] JtoJto 1 (cq +qra(s)exp[-qr~rrb(r,h)dhdr]ds;7i. (2.72) Jto Jto Jto b. Ntu p =1 the u(t)s,cexp[rt(a(s)+ r~b(s,r)dr)ds],WED. Jto Jto (2.73) c.Ntup >1vathemdduki~n 1 I C <{exp[q rto+h rto+h b(s,r)drds]/q (-q rto+h a(s)ds]}q.J~ J~ J~ (2.74) wJi h > 0 naGdo, theWYito:5:t:5:to+ h, ta co: I u(t) S,c{exp[q( 1:b(s,r)drds] +c-qq( a(s)exp[qf1:b(r,h)dhdr]ds)q. (2.75) ChungminhdinhIf 2.14.Tuongtvchungminhdinh1y2.4. Di\ltvet)1Av€ phaicua(2.71).Khi do,taco: v'et)s,a(t)vP(t)+vet)rtbet,'t)d't. Jto (2.76) Ap dl:mgb6d~b6trQ,taduQca,b, c.(D) Lllljn vanth(lcsl loanh()c Mil nganh: 1.01.01 27 Mli ri)ng vaungdljng Btl di Gronwall-Bellman HoangThanhLong 2.16.Dfnh Iy 2.15. Chau(t),art),b(t,s)lacachamlientl;lc,khongamtrangtossststj , .? ? vagzasa u(t)sc+ fl a(s)u(s)ds+r r~b(S,1)U]J('t)d1ds,btEQ, Jlo Jlo Jlo (2.77) trangdo 0 :::;c,p lacachangsa,q=1- p.Khi dotuytheap, tacocackit ? qua sau: a.Niu p < 1 thi u(t)sexp[ fl a(s)ds] Jlo I {cq +q fl (r~b(s,r)dr)exp[-q CSa(r)dr]dsii. Jlo Jlo Jlo (2.78) b.Niup =1thi u(t)scexp[fl (a(s)+ r~b(s,r)drjds],btEf2 Jlo Jlo (2.79) c.Niu p > 1va themdi~uki~n I I C <(exp[q flo+h a(s)dsJ) -q{-q flo+hflo+hb(s,r )drds] if. Jlo Jlo Jlo (2.80) wJi h >0 newdothiwJi to.st .sto+h, taco: u(t)scexp[ fl a(s)ds] Jlo I (1+c-qqfl ( fsb(s,r)dr)exp[-q fsa(r)drjdsjq. Jlo Jlo Jlo (2.81) 2.17.Dfnh Iy 2.16(Bihari, Xem [3]). Chau(t), qX.t) la cachamlientl;lc,khongamtrenQ vathoaman u(t)sM + fl cp(s)g(u(s))ds,btEQ, Jlo (2.82) Lu{jnvanthlJcsi loan h(JC Mil nganh..1.01.01 28 Mli rf)ngvau'ngd~tngBdde'Gronwall-Bellman HoangThanhLong trang doM la hangso'khongamvag: IR+-f (0, ro)la hamtang, lien tl;lC. Khi do, ta co: u(t):5,G-1(G(M) + (cp(s)ds), VtED, (2.83) WJiG: (0,ro)~IR chab(ji: [u da G(u)=J, -, (c>0,u>0). E g(a) (2.84) ChungminhdinhIy 2.16.Xem[3].(0) 2.18.Binh Iy 2.17(Xem[3]). Cha u(t), rp(t) la cac hamlien tl;lc,khongam tren1 =[to,ro).Gid sa f( t) la hamkhdvi, khongam; g la hamlien tl;lc,duang,khonggidmva thoa man u(t) :5,f (t) + (cp( s)g(u(s))ds, titE1. (2.85) Ne'uf'(t){ 1 -1}:5,0,tE1,WJi 17lahamlientl;lc,khongam,thi g(17(t)) taco: u(t):5,G-1(G(f(to))+r [f'(s) +cp(s)]ds}, VtE[to,a), Jlo (2.86) trangdo a= Sup(tE[to,ro)1G(f(to)) + ([f'(s)+CP(s)]ds EDam(G-1)}, [u da G(u)=J, -, (u>0,c> 0). E g(a) (2.87) ChungminhdinhIy2.17.Xem[3].(0) 2.19.Binh Iy 2.18.(Xem[3]) Gid sau(t)la hamlientl;lc,khongamtrenQ saDcha Luijn vanth(lcSfloanh{Jc Mil nganh..1.01.01 29 Mll ri)ngvaungd(tngBii di Gronwall-Bellman HoangThanhLong f ~(t) u(t) sf(t)+ K(t,s)g(u(s))ds, btEn, to (2.88) v6'icachamf(t),fjJ(t),K(t,s),g(u)thoamancacddu ki~nsau: a.f(t) la hamkhdvi,khongamvakhonggidmtren.0; b. fjJ(t)la hamkhdvi,khonggidmtren.ova fjJ(t)::;t,fjJ(ta)=ta; c. 0 <g(u)vakhonggidmtrenIR+,- d.0::;K(t,s)la hamlientl;lctren.ox.ovacodqLOhamriengthea t la hamkhongam,lientl;lc; e.f'(t){ 1 I}S 0,tEn, v6'i77lahamlientl;lc,khongamtren g(7J(t) .ova G dtnhngh'ianhu:(2.87). Khi do,taco: u(t) S G-1(G( f(to)) +f(t) - f(to) + L cprs)dsJ, btE[ta,a), (2.89) trongdo cp(t)=K(t,~(t))~'(t)+ r . ~(t) 3K(t,s) Jt 3 ds, 0 t (2.90) rt 1 a =Sup(tE.o IG(f(to)) + JtJf'( s) +cprs)Jds E Dom(G- )}. Chung minhdinh ly 2.18.Ta chungminhr6rangnhli sail: f)~tvet)la v6phiiicua(2.88),Iffyd(;lohamhaiv6,tadli<;1c: i~(t) 3K(t s)viet)=f'(t) +~'(t)K(t,~(t))g[u(~(t))]+ ' g(u(s))ds.(2.91)to at Suyra viet) f'(t) g(v( I)) :0;g(v(t)) +~'(t)K(t,W)) + r;«) 8K(t,s)Jto at ds. (2.92) Tli (2.92)va di~uki~ne, tadli<;1C: Lllfjn van th{lcSf loan h(JC Mil nganh: 1.01.01 30 MlJ ri)ngvalingdl;l11gBdde'Gronwall-Bellman HoangThanhLong v'(t) g(v(t)) S;f'(t) +(p(t). (2.93) La'ytichphanhaiv€ va d6ibi€n, tadU'<;5C: G(v(t)) S;G(v(to))+ I: [t'CS)+cp(s)]ds. u(t) S;G-1{G(f(to))+f(t) - f(to) + I: (p(s)ds}.(D) (2.94) 2.20.Dinh Iy 2.19(Xem [3]). Chou(t),qi,t),f(t) la cachamlientl:lC,khongamtren[a,bJ, g(u)la hamlientl:lC,duong,khonggidmwJi u >O.GidsawJi m6iYE[a,xJ, taco: u(y)S;f(x)+ f;cp(s)g(u(s))ds. (2.95) Khi do,taco: U(X)2G-I{G(u(a)- fcp(s)dsj, t7XE[a,xo), (2.96) trongdoG durJcdinhnghfab(ji(2.84)va Ix IXo=sup{xE[a,bJI{G(u(a))- (/ cp(S)dsjEDom(G-)j. ChungminhdinhIy2.19.Xem[3].(0) 2.21.DinhIy 2.20(Xem[3]). Cho u(t),art),b(t)la cachamduClng,lientl:lCtren[c,dJ; k(t,s)la ham khongam,lientl:lCwJi C::;s ::;t ::;d; 0 <f(u) la hamlientl:lC,tangngc;it; 0 0) la hamlientl:lC,khonggidm. 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) taco: Lllqn vanth{lcsf loan h(JC Mil nganh: 1.01.01 31 MiJ rl}ngvaungd(tngBll di Gronwall-Bellman HoangThanhLong 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{rE[c,dJ IG[A(r)J +B(r) fK(r,s)ds::; G[ f(oo)]}. Chung minh djnh IS'2.20.Ta coth~chungminhr6 rangnhusail: E>~tvet)la vii phaicua(2.97). Voi m6iTE[C,d],xetc::;t::;T::; d.Ta co: v(t)::;A(T) +B(T) fK(T,s)g(u(s))ds. (2.100) Liy d~ohamhaivii, taduQc: v'(t)::;B(T)K(T, t)g(u(t)), ::;B(T)K(T, t)g[CI(v(t))]. (2.101) Chuy~nvii, liy tichphanhaivii tuc diint va d6ibiiin, taduQc: G(v(t))::;G(v(c))+B(T) fK(T,s)ds. (2.102) Suyra v(t)::; G-I {G(A(T)) +B(T) fK(T,s)ds}. (2.103) Cho t=T valiy hamnguQc,taduQc(2.98).(0) 2.22.Djnh IS'2.21(Xem [3]). Cha u(t)la hamduong,lient1;lCtren[c, d]. Gid silvdi u >0, Y(u) la hamtangng(it;vdiu >0,g(u)la hamlient1;lC,duangvakh6nggidm.Ne'u Y(u(t)) ::;f(t) +fcp(s)g(u(s))ds,t7tE[C,dJ, (2.104) trangdof(t), rp(t) thoamandi~uki~ntrangdjnhly 2.17,thEtaco: Lllfjn van th(lc si loan h(JC Mil nganh ..1.01.01 32 MiJ ri}ngvazingd(tngBli di Gronwall-Bellman HoangThanhLong 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.Tlidngtlj chungminhdinh1:92.17.(D) 2.23.Djnh Iy 2.22(Xem [3]). Cho u(t),fer),F(t,s)thoamancacdiJu ki~n: a. u(t),f( t), F( t,s) la cac hamdU:rJnglien tl;lctren IR+va s ::;t. h aF(t,s)I , h ' khA A l OA. a am ongam, zentuc. at 0 c. g(u)la hamdU:rJng,lientl;lc,kh6nggidmtren(0,ro). d.h(z) >0,kh6nggidmvalientl;lctren(0,ro). Ntu u(t)~fer) +h(f~F(t,s)g(u( s))ds), (2.107) thev6i tEl, taco: u(t)~f(t) +h(G-1[G(f~F(t,s)g(u(s))ds)+ f~rjJ(s)ds]),(2.108) trongdo 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 G(oo)}. Chung minhdjnh Iy 2.22.Xem [3].(D) Lllqn vanthlJ-cS1loan h(}c Mil nganh..1.01.01 33 MlJ r~ngvaungdlJngBdd€ Gronwall-Bellman HoangThanhLong Dinh 1ysau1ah~quacuadinh1y2.22. 2.24.Djnh ly 2.23.(Xem [3]). Chau(t),f(t), F(t) thoamancacdi~uki?n: a. u(t),f(t),F(t) la cachamdu(JJ1glientl;lctren(0,co)vas :::;t. b.g(u)la hamdu(JJ1g,lientl;lc,khonggiamtren(0,co). c. h(z)>0,khonggiamvalientl;lctren(0,co). Ne'u u(t)~f(t)+h[ S;F(s)g(u(s))dsJ, Vt6(0,CO), (2.111) thiwJi t61,taco: u(t)~f(t)+h(G-1[G( S;F(s)g(u(s))ds)+S;F(s)dsJ), (2.112) trangdoG djnhnghzanhu(2.109)va 1={tE (0,00)I G(S;F(s)g(u(s))ds)+S;F(s)ds~G(00n. Chungminhdjnhly 2.23.Suyfatudinh1y2.22.(0) Lllq,n van thlJc sl loan h(JC Mil nganh: 1.01.01 ._.