Mở rộng và ứng dụng bổ đề Gronwall -BellMan 8

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

MiJ ri)ng va ung d~tngBd d€ Gronwall-Bellman Hoang Thanh Long

CHUaNG 3

MOT SO MO RONG D~NG HAM EXPONENT

TruDc lien chung Wi xin neu mQt b6 d~ b6 trQ duQCsa dl;1ngde chung minh cac dinh 19 trong su6t chudng llCIY

3.1 B6 d~ b6 trq.

Cho u(t) fa ham duang, khd vi tren Q, aCt),b(t) fa cac ham lien tl;lc

tren Q Ntu

u'(t) ~ a(t)u(t) + b(t)u2(t), \7tEQ, (3.1)

thi

it

is

u(t) ~ exp[ a(s)ds]{ b(s)exp[ a(r )dr jds;-l ,(3.2)

'v'tE [to ,tp), tp = Sup {t E Q I u(to) r0b( s)exp[ r0a(r )dr ids < I}.

Chung minh b6 d~ Suy ra tu b6 d~ b6 trQtrong truong hQpP =2.(0)

3.2 Djnh Iy 3.1.

Cho u( t) fa ham so' lien tl;lc,khong am tren Q, a > 0, b > 0 fa cac hling so: Ntu

u(t)~a+bexp[ r u(s)dsj, \7tEQ,

thi

u(t) ~ a + bexp[a(t - to)]{a - b(exp[a(t - to)]-I)} -I,

\7tE[to,a-), a- = Sup{t EQla > b(exp[a(t - to)]-I)}.

(3.4)

MiJ rf)ng va zing d~tngBtl di Gronwall-Bellman Hoang Thanh Long

Chung minh djnh ly 3.1 D~t

vet) =exp[ rt u(s)ds]

Ta co:

v'(t) ::;av(t) + bv2(t)

Ap dvng b6 d~ b6 trq VaG(3.6), ta du'qc:

(3.6)

v(t)::; exp[a(t - to)]{a - b(exp[a(t - to)] -1) rl

Tli (3.3) va (3.5), ta du'qc:

u(t)::; a + bexp[a(t - to)]{a - b(exp[a(t - to)] -1) rl (0)

(3.7)

Dinh 1y 3.2, dinh 1y 3.3 sail day chung minh tu'dng tV'dinh 1y 3.1

3.3 Djnh ly 3.2.

Cho u(t), art), b(t) fa cac ham s5fien tl;lc,kh6ng am tren Q Ne'u

u(t)::;a(t)+b(t)exp[ f' u(s)ds], litED,

the

u(t) ::;a(t)+b(t)exp[ J~rt a(s)ds]{1- J~rt b(s)exp[ rva(r)dr]dsF1 ,(3.9)J~

'v'tE[to,a),a=Sup{tEQII- rtb(s)exp[ rsa(r)dr]ds>O}.

Jto Jto

Chung minh djnh ly 3.2 D~t

Ta co:

Ap dvng b6 d~ b6 trq va thay vet) vaG (3.8), ta du'qc (3.9).(0)

Chu y: NSu aCt)va bet) 1a cac ham h~ng thl chung ta co kSt qua cua

MiJ ri)ng va lcng d(lng Bd d€ Gronwall-Bellman Hoang Thanh Long

dinh 1y 3.1

3.4 Djnh Iy 3.3.

Cho u(t), art), b(t), k(t) la cac ham so'lien tl:lc,khong am tren Q, rp(t,s)

lien tl;lc tren to s s st stl Niu

u(t) ~ a(t) + b(t )exp{ r'[k( s) + <p(t,s)u(s )Jds), \it EQ,

thi

u(t) ~ art) + b(t)exp{ r'[k( s) + <p(s,s)a( s)]ds)

J,o

(l- f. b(s)<p(s,s)exp{ C'f [k(r) + <p(r,r)a(r)]dr)ds;-l,

\it E [to,G),G = Sup{t E Q IJ,or'b( s)<p(s,s)exp[ J,oc'~ [k(r) + <p(r,r)a(r )]dr Jds < I)

Chung minh djnh Iy 3.3 f)~t

vet) =exp{ rt [k(s) + <p(t,s)u(s)]ds}.

Ta co:

u(t) ~ aCt)+ b(t)v(t)

Tu (3.14) va (3.15), ta thu du'cJc:

(3.15)

Ap d\lng b6 d~ b6 trcJva thay vet) vao (3.15), ta du'cJC(3.13).(0)

Chu y:dinh 1y 3.2 du'cJcsuy ra tu dinh 1y 3.3 trong tru'ong hcJp cac

ham sO' k(t) =0 va (p(t,s) = 1.

3.5 Djnh Iy 3.4.

Cho u(t) la ham S(]'lien tl:lc,khong am tren Q a > 0, p :::0 la cac hiing s6 Niu

Mil ri)ng va ung dl.WgBiJ di Gronwall-Bellman Hoang Thanh Long

u(t)~a+exp[ r uP(s)ds], 'vItEO,

thl

u(t)~(a+l)[l- p(a+1jP(t-to)] P, 'vItE[to,(J), a = Sup(tEOlp(a+ljP(t-to)<l}.

(3.18)

Chung minh dinh Iy 3.4 £)~t

vet) = exp[ft uP (s)ds].

v'et) =uP(t)v(t)

(3.21) vdi R(t)=[a+v(t)]P

Tli (3.20), ta suy fa:

Dodo

R(t)exp[-p it R(s)ds]~(a+l)P

La'y tich phan hai vfi cua ba't d~ng thuc tren, ta dU<;1c:

1

exp[ r R(s)ds]~[l-p(a+l)P(t-to)] P

Tli (3.22) va (3.24), ta suy fa:

vet) + a ~ (a + 1)[1- pea + l)P(t - to)] P

Tli (3.25), ta dU<;1c(3.18).(0)

(3.25)