Một số phương pháp lặp tìm điểm bất động chung của một họ hữu hạn các ánh xạ giả co chặt = Some interation methods for common fixed points of a finite family of strictly pseudocontractive mappings

\guyen Dire Fiing Tap chi KHOA HOC & c6\ G NGI IE 90(02): 129- 1(0

SOME ITERATION METHODS EOR COMMON EIXED POINTS OF A EINITE

EAMILY OE STRICTLY PSEUDOCONTRACTIA E MAPPINGS

Nguyen Due Lang

Thainguyen Universtly. University of Sciences

Abstract. In this paper, we introduce new implicit and explicil iteration methods based

oil the Kiasnoselskii-Matin iteration method and a coiitraetioii for hiidiiig a ioiiiiuoii fixed

point of a finite lainilv of strictly psoiidocontractivc^ self-mappings ol a clo.sod eonvox subset

in real Hilbert spaces. An extension to the problem of convex optimization is showed.

Key words' Nonexpansive inappiiig, fixed points, variational inequalities

2000 Malhrmatir.s Subjoet Cla.ssifiratioii I7II17. 471100.

1, Introduction and preliminaries

kel C be a nonempty closed and convex subset of a real Hilbert space H with inner product (.. >

and norm ||.|| iind lot T be a 7-strictly pseudocontactive and self-miipping of C , i.e.,

\\T.r - Tyf = \\x - y\\'' + ^||(/ - T)x - (y - Ty)\\''

and •/• : C • C , rospoelively, for all x.y e C . where -. is a fixed niimbrr in [0, I). When ^, = 0. 7 is

called nonexpansive Denote the sel of fixed points of T by Fix(T). i.e.. Fi.r{ / ) := {.;• 6 C • .;• = 7".;}.

iuid the projection of ., e // onto C hy Pc-(x). .Note, that in „ Biinack sapce E.T is ,. -, -sliiellv

pseudoeiailaelivo, if

{'Ex - Ty..i(x - y)) < \\x - y\f - 3\\(I - T)x - (y - •Ty)\\'

u'lieie j(x y) € .J(x - y), and J is the normalized duality map])iiig of E. i.e., J E -^ E' and

salisfies the condition (.;', J(x)) = ||.'r|p for all x £ E.

ket {/';},1i.l < N < K., he a A' -,-slrictly iiseudocontactivo and self-iiiiippiiigs T, of C In

this paper, we assuiue that nfliAJ(Tj) ^ 0 and introduce some new ileralion methods for finding an

element p" S n;'ljFi.i:(T,).

The rkiss of strielly pseudocontractive mappings has been studied intensivciy hy several aulliors

(see for example [1]-[18| and references therein). Clearly this class of mappings includes the ckuss of

iionexiiaiisivo mappings.

In order lo study the fixed poiiil problem tor a nonexpansive seli-mapping T of a closed convex

subset C in a real Ililhert space, one recent way is to construct the iterative scheme, the sociilled

viscosity itiM'alioii luothod:

•I'l^i = Aj./(.rt) + (1 - \c-'Tx^.,k > 0, (1,1;

proposed fir.stly hy Moudafi [19|. whole Ay. S (0.1) and / is a contraction o! C with constant o S [0,1),

Xu (20| proved that the sequence {xt,,} generaled by (1.1) converges strongly to a fixed point p"

of T . which is the unique solution of the following variational inequality:

(F(p-).p- -p ) < 0 VpeF;.r("A). (1'21

where F = I - f In 200C, relaled to a certain optimization problem. Marino and Xu (21] introdiieed

Ihe lollovvuig general iterative .scheme for the fixed point problem of a nonexpansive mapping:

X,., = X,u.-f(x,) + (1 - \cA)Tx,. A- > 0, (1.:!

where .4 is a strongly positive bounded linear operator. A^ G (0.1) ;uid uj > 0. fliey proved that

the soqiienee {.rt} .nenerated by (1.3) converges strongly to the unique solution of the vaiialional

iiieiiuality (1.2) with A = .4 - ..,•/ Further, algorithm (1.3) was exlended in 2009 hv Clio el ,d. '!)] lo

the class ,if A-..iiietly pseucloeont rac live mappings

"K-niail, iii;ii>t'iiiliKl.inR'-'n0'2"\'.ilKHi.c()ni

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