Weak convergence theorems for an infinite family of nonexpansive mappings and equilibrium problems (tt)

‘Weak Convergence Theorems for an Infinite Family .... ‘Weak Convergence Theorems for an Infinite Family ... and compute the next iteration point by Mann-type fixed points, To investiga

14 P.N Anh, L B Long, N V Quy and L Q Thuy

1 Introduction

Let # be areal [lilbert space with inner product (., -) and norm ||-|| Let

C be a closed convex subset of a real Hilbert space Ho and Pr be the

Frojection of 7 onto C When {x} is a sequence in H, then x” > =

(resp x° — ¥} will denole strong (esp weak) convergence of the sequence

{&"} to ¥ asa > Amapping $:C > C is said to be nonexpansive if

||S(x)- #(y

x~y|, À.yĂeC

Fad) is denoted by the set of fixed points of 8 Let f:CxC +R bea

‘bifunction such that f(x, x)= 0 forall xc C, We consider the equilibrium problems in the sense of Blum and Oettli [8| which are presented as follows:

Find x" c C suchthat f(x", y} 2 0 forall yc C EP(f,C) The set of solutions of ZP(/, C) is denoted by Sof, C) The bifunction f

is called strongly monotone on C with B > 0 if

Se v)+ ZŒ, z) < =B|x= ylỄ Vx, ye monotone on Cif

pseuslomanotone om Cif

Lipschite-type continuous on C with constants cy > 0 and cy > 0 if

Fle, y+ fly 2) 2 fle zì—a[|x—+ ?2— |» V8 zCC,

In this paper, we are interested in the problem of Ñnding ä conunon element of the solntion set of the equilibrium problems ZP(f, () and the set

of Bxcd points ne Fe(Sy ) of an infinite fanily of uoncxpansive mappings

‘Weak Convergence Theorems for an Infinite Family 115

{Se}, namely’

where uke bifmction f and the mappings Sy (k= 1,2 ) salisly he following, conditions

AL fis Lipschitz-type continuous on C,

À2 fis pseudomonotone an C,

A3 fis weakly continuous on C,

AA, Sy is nonexpansive on C for all & > 1, m Fix(S i Sol f, C)

#O,

AS YY suplll S164) — S,lx) |:

A

of CL

D} <2 for any bounded subsct D

An important special case of problem (1.1) is that fix, y)—

Ure), ¥

common element of the solution set of variational inequalities and the set of fixed points of an infinite family of nonexpansive mappings (see [6, 14, tế,

17, 22])

x), where 2°; C —> ?⁄ and this problem is reduced (o finding a

Motivated by the viscosity method in [16] and the approximation methad

in [7] via an duuprovernent sel of extragradient methods in [2-4], we introduce anew ileration algorilinu for finding, a common element of problem (1.1) Al cach main iteration m, we only solve two strongly convex prograns with ạ pseudomonotone ‘aud Lipschit:-iype continuous bifunclion, We show thai all

of Uie iterative scquences generated by this algonillun converge weakly to the conuuon clement in areal Hilbert space

This paper is organized us fullows: Scetion 2 recalls some concepts in equilibritan problems and fixed point probleras thai will be used in the sequel

116 Ð.N Anh, L B Long, N V Quy and L Q Thuy

and an iterative algorithm for solving problem (1.1) Section 3 investigates the convergence of the algorithms presented in Section 2 as the main results

of our paper

2 Preliminaries

In 1953, Mann [13] introduced a well-known classical iteration method

lu approximate a ñxed poinl of a nonexpansive mappmg SCC ina teal Hilbert space 7¢ This iteration is defined as

xf# = axŸ —(1—d„)8@Ÿ), VÉ >0,

where C 1s nonemply closed convex subsel of 7 and {q;}© [0, 1] Then 4x") converges weakly to x" c Fax(S)

For finding a conunon fixed point of an infinite family of noncxpansive

mappings {Š„}, Aoyama el a [7] introduced an iterative sequence {x*} of

C defined by +? = C and

where C is anonemply closed convex subsel of a real Lilbert space, {u,.}

Jo, 1] and (2, Fads, }# 2 The anthors proved that the sequence {x*}

converges slrongly to x” tm HWS;

Recently, Yao el al, [20] inueduced au iterutive scheme for Gnding common clement of te set of solutions of problem (1.1)

we

*—xbi>0, weed,

0t, xi+ Lee yh, ¥

Te

et opagtet ys Best 1 yy),

where g:C > ís contractive and Wy, is W-mapping of {8,} Under mild

‘Weak Convergence Theorems for an Infinite Family 7

aesumplions on parameters, the authors proved thal the sequences {x"} and

{)"} converge strongly to x”, where

at — Pr

nancy, fe)

Py Fly

Methods for volving problem (1.1) have been well developed by many roscarchers (sce [5, 9-11, 18-21]) ‘These methods require solving approximation cqutlibrium problems with strongly monotone or monotone and Lipschits-type continuous bifuniclions

In our scheme, the main steps are to solve two strongly convex problems

1

l — mg in| 4z 7P, +)+t>|y-## Poy ec}

|e = axg:min{ AS" vt |e fips cÌ

and compute the next iteration point by Mann-type fixed points,

To investigale Lhe convergence of this scheme, we 1

(echnical Jerumas which val be used in the sequel

call the Dollowing,

Lemma 2.1 (See [I]} Let C be a nonempty closed convex subset of a real Hilbert space H Let [Cx CR be a pseudomonvtone, Lipschitz-

type continuous Vifunction For each xc, let fix.) be convex and

subdifferentiable on C Suppose that the sequences {x}, {y*}, {(È} am

generated by scheme (2.1) and x" = Sol f, C) Then

FP <i ah a" PG ange) xy P

(L 2yedpeY AP whee Lemma 2.2 [17] Let 1 be a real Hilbert space, {3,} be a sequence of real numbers such that {8,}c |o, B|< (0,1), © > 0 and two sequences

118 P.N, Anh, L.B Long, N V Quy and L Q Thuy

tx}, fy") of 1 such that

im sup] x* |e,

ae

Tim sup|| yŸ

kao

Than

Lemma 2.3 [7] Jet C he a nonempty clased convex subset of a Banach

apace Let {S} be a sequence of nonexpansive mappings of C into itself and

Sis @ mapping of C into itself such that

[Shem seule) 8469 [sxe Che,

tal

| (x) dim Sg(x) Ve eC

Then lun supl|Sv(x) #(x)|:x< C}=0

eae

Lemma 2.4 (See [12

@ nonempiy closed comes subset C of a real Iiflbert space H if Fie(S}

Assume that 8 ís œ nonexpansive selfmapping of

#, then I S is demiclosed: that is, whenever {x*} is a sequence in C weakly converging to some ¥ © C and the sequence {(1 - S)(x")} strongly

converging fo some y, il follows that UZ — 8)( 3 Here J is the identity

operator of TÍ

Lemma 2.5 (See [17]) Zet C be a nonempty closed comes subsel of a

real Hilbert space H Suppose thal, for all u € C, the sequence {x"} satisfies

Then the sequence (Pra (x* )} converges strongly to some x © C

‘Weak Convergence Theorems for an Infinite Family uy

3 Convergence Results Now, we prove (he main convergence theorem,

Theorem 3.1 Suppose that Assumptions Al-A5 are satisfied, x° = C

and bo positive sequences {h,}, {ay} satisfy the folowiny restrictions:

fox} [ed] 0.05

;

fig) <a, Bl for some œ 6 ca +} where E = max{2ey,

Then the sequences {x*}, {v} and {PP} generated by (2.1) and (2.2) converge weakly to the same patat x” © (2, Fil) SoK S.C), where

HE, Pop meegmnenty e9 }

The proof of this theorem is divided into several steps

In Steps | and 2, we will consider weak clusters of {x*} It follows Lom T.emma 2.1 that

and hence there exists

pen

8a, CC), & 29,

61)

k

‘The sequence {x} is bounded and there cxsis a subsequence ‡x 2}

converges weakly to ¥ as j — ,

Step 1 Claim that x € 1 Fats, }

Proof of Step 1 By Iemma 2.1 and (3.1), we have

(=a) tt - 26g) * - »* Ễ s 4— đj@Œ Daye] xf - x P

effoe Pope oe B

Oa kom,

120 P.N Anh, L B Long, N V Quy and L Q Thuy

lim |xỶ yŸ ||=0

ame oF ll

By the siztilar way, also

lim | yy #

fon

lim |

Since x” € làm Eix(S )(1 Sal(Ƒ, C], Lemma 2.1 and (3.1), we have

J##2) v<|J# #J<l## #I

and henec

lim sup] Se

row

ied

Using (3.1) and + aya + — ay 5, 2"), we have

Firm || u(x — x7 )4 (1 — ay Spf) — 2") | — Tim | xT

abit g(x — 2°) + (1 ony Meee") — 2°) | — J |

By Lemma 2.2, we have

bso

Af follows from (3.1) and 3.3) Ubal the vequence f*} is bounded By

Assumption AS, we have

a1) Spx): PB <0

‘Weak Convergence Theorems for an Infinite Family 121 Let S be a mapping of C into itself defined by

forall x ¢ C and suppose thal ##(Š) (YP

2.3, (3.3) and (3.4), we obtain

Jix(S,) ‘Then, using Lemma

| #&#)—zŸ |

<||#G#)- #0#)||+| #@#)— s0#)|+| se0#)— xŸ |

<||xF —# |+ epl|[S()— 8z(3)||: z s #È}†~ | sy(Ÿ)—

— 0ásŠ cm

k

‘Then, by Lemma 24 and the sequence x J} cơnverges weakly lo X, we have x © Fie(S), ic,

Step2 When x <x as j › s, we show thất # Sel(ƒ, C)

Pruol of Step 2 Sinci

problem

iz the unique solulion of (he strongly convex

„ {1

mà ly xF Ê 1/6 y)ryc

we have

0c0(A,/6*, yet] y— x P (9+ Nel)

This follows thal

122 P.N Anh, L B Long, N V Quy and L Q Thuy

where w & 62,f(x", »”) má 7 © Ne(y"), By the definition of the norrual

cone ÄV 2, we imply that

On the other tund, since f(x®, -) is subdifferentiable on C, by the well known Moreau-Rockafellar theorem, there exists wc Ga f(x" y”) such thar

ey h

Combining this wilh (3.6), we have

relist’ feet, o> OF

Hence

ag Uf”, 9)— 688, shay» Oh —

Then, nsing {Á„} C |a, b| = (a ty G

continuity of f we have

This means that x £ #øi(7, Œ)

Step 3 Claim that the sequences {x}, {yŸ} and {1°} converge weakly

to Lhe same point x", where

fim Pree mg naili.c *)

Proof of Step 3 It follows from Steps 2 and 3 that for every weak cluster

point x of the sequence {x"} swisfies x € (Y", Fils; )A Sof C) We show that {x"} converges weakly to x Now, we aseume that {x”*} is an

‘Weak Convergence Theorems for an Infinite Family 123

another subsequence of {x”} such that x" — & as & —x co Then

Se Sollf, CH FS), where # is defined by (3.5) We will show that ¢= ¥ If ¥ + ¥, then from (3.1) and the Opiel condition, it follows that

c= Em | xŸ

= timing] x - x | "

< liming] x“! — £ |

joe

= lim |

toa

= liming] x"* £|

ben

< liminf| x" - © koe

— lint] sẼ

ber

=e

‘This ix a contraction Thns, we have X = & Tt implies

x _ x € Sollf.C)[) Fix(S) as k +0

It follows fiom (3.2) and (3.3) that

TA sx a k ow

Setting

Pris cy ray"

)-124 Ð.N Anh, L B Long, N V Quy and L Q Thuy

‘Then, from x ¢ §ø(ƒ, C){1 Ph{§), tt imphies

Tây Lemma 2.5 and Step 1, the sequence {=} convernes sirongly lo 5© Soi f, C)N #ix(S) Hence, we have

z>0,

s0, we have = = 5, This shows that

PP mGi)si0,eS )

4 Applications

Let Cbe a nanempty closed convex subset of areal Hilbert space 7 { and

F be a function fiom C into 1 In this section, we consider the vaiadional inequality problemi which is presented as follows:

Kind x* © C such thal (F(x"), x2") 20 forall xe FICK, C}

Lot fC xC — R be defined by ffs, y) = F(a) » — x} Then problem EPCS, C) can be writen in V2(F, C) The set of solutions of H7(F, C) is

denoted by Sof(¥, C) Recall that the fimction F ig called strongly monotone

on C with B > 0 if

(F()- Fy) e- y)> Blx-y Pe pec,

monotone on Cif

ŒG)- FỤ) xo y)> 0 Wx, pec preudomonctone on Cif

Lipschit continuous on C with constanis £ > 0 1F

|F(a)- Fly}| <i|x-y], Ys, yee

‘Weak Convergence Theorenis for an Infinite Family 125 Since

yh eae nin| hy Fs, viet yx Five ct

- angi ig

= Protx’ AF"), using, (2.1), (2.2) and ‘Theorem 3.1, we obtain the following, convergence

theorem for finding a common clement of the set of fixed points of an

infinite family of nonexpansive mappings {S,} and the solntion set of

problem W7(F, €)

Theorem 4.1 Let C he a nonempty closed comex subset of a rat Tlithert space ‘H Let F be a function from C to H such that Fis psexdomonotore and L-Lipschitz continuous on C For each i-}, , 8,

CC is nonexpansive such that [Y2, Fie(S,}[ Solr, C) + and

Le SuPlll Seals) Sy(x)| sD} < «for any bounded subset D of C

Ifpositive sequences {az} and {0y} sat the following restrictions:

far} < [e, d] ~ (0, 1),

/

far} < [eB] for some a, bc [0, 1)

XE

then the sequences {x"}, {y*} and {t"} generated by

cá PGŸ),

*

¥

Pe ến B= Prelx! — ke FP),

= = ayxŸ + (I— dự J8; 0),

converge weakly ta the same paint 4” € [YŸ, Fix(S,) 1 Soll, C), where

txt

= dim Prom mcs inser, CV

126 Ð.N Anh, L B Long, N V Quy and L Q Thuy

Acknowledgement

The work is supported by the Vietuam National Foundation for Science

Technology Development (NAFOSTED)

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References

PLN Amb, A hybrid extragrashentt method extended to fixed pont problems and equilibrinm problems, Optimization, 2011, DOI: 10.1080/02331934.2011.607497 PON, Auk, A logarithmic quadratic regulsrization method for solving pscudo-

monotone equilibrium problems, Acta Math Viemam 34 (2009), 183-200

PN Anh Án LQP regularization method for equilibrium problems on polyhedral, Vietnam J Math 36 (2008), 209-228

P.N Ảnh and J, K Kim, Outer approximation algorithms for parudomenotone equilibrinm probtems, Comput, Math Appl 61 (2011), 2588-2595,

P.M Aub, J K Kim and J M Nam, Strong convergence of an extragradient method for equilibrium problems ond fixed point problems, J Korean Math Soe (2011), accepted

P, N Anh and D X Son, A new iterative scheme for pseudomenotone eqnifibrinm problems and a fmite family of pseudocontractions, J Appl Math Inform 29 (2411), 1129-1191

K Aoyama, ¥ Kimura, W Tokahashi and M Toyoda, Approximation of common fixed points of a conntahle amily of novexpansive anappings in a Ranach space, Nonlinear Anal 67 (2007), 2450-2360,

E Blum and W Cettli, From optimization and variational inequality to

iam problems, Math, Stndent 63 (1994), 127-149

equi

L ©, Ceng, § Schaible and J C Yao, Implicit iteration scheme with perturbed mapping for equilibrinm yroblems amd fixed point problems of finitely many nonexpancive mappings, J Optim Theory Appl 139 (2008), 483-418

L ΠCeng, A Petrusel, C Lee and M M, Wong, Two extragradicnt approximation methods for variational inequalities and fixed point problems of strict psendo-contractions, Taiwanese J Math 13 (2909), 607-632

R Chen, X Shem and 8 Cui, Wenk and strong convergence theorems for equilibrium problems and conntable strict pseudocontrastions mappings in Hilbert space, J Inequal, Appl 2010, Att, ID 474813, 11 pọ doi:10.1155/2010/474813