Hiệu chỉnh bất đẳng thức biến phân trong không gian Banach

Phgm Thanh Hilu Tgp chf KHOA HQC & CONG NGHE 126(12): 87-92

HIEU CHINH BAT HANG THlTC BIEN PHAN

TRONG KHONG GIAN BANACH

Phgm Thanh Hieu

Trudng Dai hgc Ndng Ldm - DH Thdi Nguyen

TOMTAT

Bat ^ng thffc bien phan ndi chung la mgt bdi toan dat khong chinh theo nghTa nghiem cua bdi toan

khong phu thugc lien tyc vdo dff ki^n ban diu. Do do vific xdy dyng cac phuang phap giai on djnh

cho ldp bai toan nay la mdt vifc cin thilt.Trong bai bao ndy chung toi de xuit phuang pl ap hifu

chinh cho bat ding thffc bien phan vdi tap chap nhan dugc Id tap dilm bit ddng chung .;ua nffa

nhdm khong giSn tren khong gian Banach dya tren phuang phap hi6u chinh ndi tilng Browder￾Tikhonov.

Ttr khda:6af ddng thue bien phdn, bdi todn dgt khdng chinh, phuang phdp hliu chinh, lap diem

bdl ddng, niea nhom khdng gidn.

GIOI THIEU

Cho E la khdng gian Banach vdi khdng gian

ddi ngau ky hieu la £". Ta dung ky hieu ||. ||

de chi chuan cffa ca E^va £", Ta viet (s^.x')

dl thay cho gia trj cffa phiem ham x' ^ E'

tgi dilm x E E, tuc la (x. x') = x' (x).

Khdng gian Banach E dugc ggi la ldi dlu niu

vdi mgi s £ (0,2], cac dilm A, V € E sao

cho \\x II < 1, lly II < 1 vd Ih- - y\^ > s

thi tdn tai o = 5(f) > 0 sao cho

ll(.t- + y)/2ii<l-^ .

Anh xg J: E -^ 2^ dugc ggi Id dnh xg ddi

ngau chuan tac ciia E neu

iW = {r£r:t.,.v) = |W!M, |k1 = |:il, JEE),

Anh xg ddi nglu chuin tic /la don tri (ky bieu

bdi ;•) va lien tuc dlu tren cdc tap bj chgn cffa

E niu E la khdng gian Banach tron dlu theo

nghTa md dun tron cffa E xac dinh bdi

Pf(t) = sup{i(|lr-y||-h-.v|-l>M = l,W;St},

thda man lim^_Q.-^j^ = 0. Ta cd

/(Ax) = A/(A-) vdi mgi i > 0, V.v e E va

/(—x) = —fix), Vx e E. Trong trudng

hgp E = H, vdi H la khdng gian Hilbert, thi

/ = /, dnh xa ddng nhit cua M.

Cho C la tap con khac rdng Ioi ddng b-ong E.

Anh xg T: C -* -C dugc ggi Id dnh xa khdng

gian tren C neu

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\\Tix) - T{y) II < \x - y k Vx. y S C.

Ky hieu Fix(T) la tap diem bat ddng cua dnh

xa khdng gian T, tffc la

Fix(r) = {,T e C: Tx = x).

Mdt hg cac anh xa [T(t)'. t > 0] tff C vdo C

dugc ggi la nua nhdm khdng gian tren C neu

no thda mdn cdc tinh chat sau:

(1) vdi mdi t > 0, T(t) Id mdt anh xg

khdng gian tren C,

(2)T(0)x = x, VxeC, (3)T(s + r) = r(s) 0 7(t)Vs, r > o, (4) vdi mdi A: £ C, anh xa i (. J.lidi tff

(0,OT ) vdoCldlidntyc.

Ky hieu ^=M-.oFix(T(0)l a tap dilm bit

ddng chung cffa nffa nhdm khdng gian

[Tit): t > 0}. Ta biit ring (J i= 0n6u C la

tap bi chgn (xem [3]).

Anh xa A: E -^ E lin lugt dugc ggi la }}-J￾don dieu manh val-lien tuc Lipschitz neu A

thda man cdc diiu ki?n tuong ffng sau:

{A(x)-A(y).j(x - y)) > vl^ - .^l "•

va(l.l)|UC0--4(y)li<L|k-y||, vdi mgi x.y £ E, f(x — y) Ej(x - y), vd

t), L la cac hdng so cd dinh duong. Trong

(1.1), neu 0 < L < 1, thi -4 dugc ggi la dnh

xg co va niu L= 1 thi j4dugc ggi la anh xa

khdng gian.

Anh xa<?c: E —^ C dugc ggi la phep sunny co

rut khdng gian tff E\enC neu (^c thda man cac

dieu ki^n:

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