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Volume 2010, Article ID 407651, 26 pagesdoi:10.1155/2010/407651 Research Article Approximation of Common Fixed Points of a Countable Family of Relatively Nonexpansive Mappings Daruni Boo

Volume 2010, Article ID 407651, 26 pages

doi:10.1155/2010/407651

Research Article

Approximation of Common Fixed

Points of a Countable Family of Relatively

Nonexpansive Mappings

Daruni Boonchari1 and Satit Saejung2

1 Department of Mathematics, Mahasarakham University, Maha Sarakham 44150, Thailand

2 Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand

Correspondence should be addressed to Satit Saejung,saejung@kku.ac.th

Received 22 June 2009; Revised 20 October 2009; Accepted 21 November 2009

Academic Editor: Tomonari Suzuki

Copyrightq 2010 D Boonchari and S Saejung This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited

We introduce two general iterative schemes for finding a common fixed point of a countable family

of relatively nonexpansive mappings in a Banach space Under suitable setting, we not only obtainseveral convergence theorems announced by many authors but also prove them under weakerassumptions Applications to the problem of finding a common element of the fixed point set

of a relatively nonexpansive mapping and the solution set of an equilibrium problem are alsodiscussed

1 Introduction and Preliminaries

Let C be a nonempty subset of a Banach space E, and let T be a mapping from C into itself.

When{x n } is a sequence in E, we denote strong convergence of {x n } to x ∈ E by x n → x and weak convergence by x n  x We also denote the weak∗convergence of a sequence{x∗

n} to

x∗in the dual E∗by x∗n  x∗ ∗ A point p ∈ C is an asymptotic fixed point of T if there exists {x n } in C such that x n  p and x n − Tx n → 0 We denote FT and FT by the set of fixed points and of asymptotic fixed points of T, respectively A Banach space E is said to be strictly

convex ifx  y/2 < 1 for x, y ∈ SE  {z ∈ E : z  1} and x / y It is also said to be uniformly convex if for each  ∈ 0, 2, there exists δ > 0 such that x  y/2 < 1 − δ for

x, y ∈ SE and x − y ≥  The space E is said to be smooth if the limit

lim

t → 0

x  tx − x

exists for all x, y ∈ SE It is also said to be uniformly smooth if the limit exists uniformly in

x, y ∈ SE.

Many problems in nonlinear analysis can be formulated as a problem of finding a fixedpoint of a certain mapping or a common fixed point of a family of mappings This paper dealswith a class of nonlinear mappings, so-called relatively nonexpansive mappings introduced

by Matsushita and Takahashi1 This type of mappings is closely related to the resolvent ofmaximal monotone operatorssee 2 4

Let E be a smooth, strictly convex and reflexive Banach space and let C be a nonempty closed convex subset of E Throughout this paper, we denote by φ the function defined by

Following Matsushita and Takahashi6, a mapping T : C → E is said to be relatively

nonexpansive if the following conditions are satisfied:

In2, Alber introduced the generalized projection ΠC from E onto C as follows:

For more example, see1,8

In 2004, Masushita and Takahashi 1,6 also proved weak and strong convergencetheorems for finding a fixed point of a single relatively nonexpansive mapping Severaliterative methods, as a generalization of1,6, for finding a common fixed point of the family

of relatively nonexpansive mappings have been further studied in7,9 14

Recently, a problem of finding a common element of the set of solutions of anequilibrium problem and the set of fixed points of a relatively nonexpansive mapping isstudied by Takahashi and Zembayashi in15,16 The purpose of this paper is to introduce

a new iterative scheme which unifies several ones studied by many authors and to deduce thecorresponding convergence theorems under the weaker assumptions More precisely, manyrestrictions as were the case in other papers are dropped away

First, we start with some preliminaries which will be used throughout the paper

Lemma 1.1 see 7, Lemma 2.5 Let C be a nonempty closed convex subset of a strictly convex

and smooth Banach space E and let T be a relatively quasi-nonexpansive mapping from C into itself Then FT is closed and convex.

Lemma 1.2 see 17, Proposition 5 Let C be a nonempty closed convex subset of a smooth, strictly

convex, and reflexive Banach space E Then

for all x ∈ C and y ∈ E.

Lemma 1.3 see 17 Let E be a smooth and uniformly convex Banach space and let r > 0 Then

there exists a strictly increasing, continuous, and convex function h : 0, 2r → R such that h0  0 and

h x − y ≤ φx,y 1.6

for all x, y ∈ B r  {z ∈ E : z ≤ r}.

Lemma 1.4 see 17, Proposition 2 Let E be a smooth and uniformly convex Banach space and let

{x n } and {y n } be sequences of E such that either {x n } or {y n } is bounded If lim n → ∞ φx n , y n   0,

then lim n → ∞ x n − y n   0.

Lemma 1.5 see 2 Let C be a nonempty closed convex subset of a smooth, strictly convex, and

reflexive Banach space E, let x ∈ E, and let z ∈ C Then

Lemma 1.6 see 18 Let E be a uniformly convex Banach space and let r > 0 Then there exists a

strictly increasing, continuous, and convex function g : 0, 2r → R such that g0  0 and

tx  1 − ty2

≤ tx2 1 − t y2− t1 − tgx − y 1.8

for all x, y ∈ B r and t ∈ 0, 1.

We next prove the following three lemmas which are very useful for our main results

Lemma 1.7 Let Let C be a closed convex subset of a smooth Banach space E Let T be a relatively

quasi-nonexpansive mapping from E into E and let {S i}N

i1 be a family of relatively quasi-nonexpansive mappings from C into itself such that FT ∩N i1 FS i  / ∅ The mapping U : C → E is defined by

If E has a stronger assumption, we have the following lemma.

Lemma 1.8 Let C be a closed convex subset of a uniformly smooth Banach space E Let r > 0.

Then, there exists a strictly increasing, continuous, and convex function g∗:0, 6r → R such that

g∗0  0 and for each relatively quasi-nonexpansive mapping T : E → E and each finite family of

relatively quasi-nonexpansive mappings {S i}N i1 : C → C such that FT ∩N i1 FS i  / ∅,

N

i1

ω i α i 1 − α i g∗Jz − JS i z ≤ φu, z − φu, Uz 1.11

for all z ∈ C ∩ B r and u ∈ FT ∩N

i1 FS i  ∩ B r , where

Ux  TJ−1 N

i1

x ∈ C and {ω i }, {α i } ⊂ 0, 1, i  1, 2, , N such that N i1 ω i  1.

increasing, continuous, and convex function g∗:0, 6r → R such that g∗0  0 and

tx∗ 1 − ty∗2

≤ tx∗2 1 − t y∗2

− t1 − tg∗x∗− y∗ 1.13

for all x∗, y∗ ∈ {z∗ ∈ E∗ : z∗ ≤ 3r} and t ∈ 0, 1 Let T : E → E and {S i}N i1 : C → C

be relatively quasi-nonexpansive for all i  1, 2, , N such that FT ∩N i1 FS i  / ∅ For

i1 FS i  ∩ B r It follows that

u − S i z2≤ φu, S i z ≤ φu, z ≤ u  z2≤ 2r2 1.14

and henceS i z ≤ 3r Consequently, for i  1, 2, , N,

α i Jz  1 − α i JS i z2≤ α i Jz2 1 − α i JS i z2− α i 1 − α i g∗Jz − JS i z. 1.15Then

ω i α i 1 − α i g∗Jz − JS i z ≤ φu, z − φu, Uz. 1.17

Lemma 1.9 Let C be a closed convex subset of a uniformly smooth and strictly convex Banach

space E Let T be a relatively quasi-nonexpansive mapping from E into E and let {S i}N i1 be a family

of relatively quasi-nonexpansive mappings from C into itself such that FT ∩N i1 FS i  / ∅ The

Ux  TJ−1 N

i1

for all x ∈ C and {ω i }, {α i } ⊂ 0, 1, i  1, 2, , N such that N

i1 ω i  1 Then, the following hold:

1 FU  FT ∩N i1 FS i ,

2 U is relatively quasi-nonexpansive.

Proof 1 Clearly, FT ∩N i1 FS i  ⊂ FU We want to show the reverse inclusion Let z ∈

2 It follows directly from the above discussion

2 Weak Convergence Theorem

Theorem 2.1 Let C be a nonempty closed convex subset of a uniformly smooth and uniformly convex

Banach space E Let {T n}∞

n1 : E → C be a family of relatively quasi-nonexpansive mappings and let {S i}N i1 : C → C be a family of relatively quasi-nonexpansive mappings such that F :∞

ω n,i α n,i Jx n  1 − α n,i JS i x n 2.1

for any n ∈ N, {ω n,i }, {α n,i } ⊂ 0, 1 for all n ∈ N, i  1, 2, , N such that N i1 ω n,i  1 for all

n ∈ N Then {Π F x n } converges strongly to z ∈ F, where Π F is the generalized projection of C onto F.

Proof Let u∈∞n1 FT n ∩N i1 FS i Put

U n  T n J−1 N

i1

FromLemma 1.7, we have

Therefore limn → ∞ φu, x n  exists This implies that {φu, x n }, {x n } and {S i x n} are bounded

This implies that limn → ∞ φy n , x n  exists This together with the boundedness of {x n} gives

r : sup n∈N y n  < ∞ UsingLemma 1.3, there exists a strictly increasing, continuous, and

convex function h : 0, 2r → R such that h0  0 and

h y n − y nm  ≤ φy n , y nm

≤ φy n , x n

− φy nm , x nm

. 2.7

Since{φy n , x n } is a convergent sequence, it follows from the properties of g that {y n} is a

Cauchy sequence Since F is closed, there exists z ∈ F such that y n → z.

We first establish weak convergence theorem for finding a common fixed point of

a countable family of relatively quasi-nonexpansive mappings Recall that, for a family ofmappings{T n}∞n1 : C → E with∞n1 FT n  / ∅, we say that {T n} satisfies the NST-condition

19 if for each bounded sequence {z n } in C,

Theorem 2.2 Let C be a nonempty closed convex subset of a uniformly smooth and uniformly

convex Banach space E Let {T n}∞

n1 : E → C be a family of relatively quasi-nonexpansive mappings

satisfying NST-condition and let {S i}N i1 : C → C be a family of relatively nonexpansive mappings

such that F :∞n1 FT n ∩N i1 FS i  / ∅ and suppose that

ω n,i α n,i Jx n  1 − α n,i JS i x n 2.10

for any n ∈ N, {ω n,i }, {α n,i } ⊂ 0, 1 for all n ∈ N, i  1, 2, , N such that N

i1 ω n,i  1 for all

n ∈ N, lim inf n → ∞ ω n,i α n,i 1 − α n,i  > 0 for all i  1, 2, , N If J is weakly sequentially continuous,

then {x n } converges weakly to z ∈ F, where z  lim n → ∞ΠF x n

Proof Let u ∈ F FromTheorem 2.1, limn → ∞ φu, x n  exists and hence {x n } and {S i x n} are

bounded for all i  1, 2, , N Let

r  sup

n∈N {x n , S1x n , S2x n , , S N x n }. 2.11

By Lemma 1.8, there exists a strictly increasing, continuous, and convex function g∗ :

0, 2r → R such that g∗0  0 and

N

i1

ω n,i α n,i 1 − α n,i g∗Jx n − JS i x n  ≤ φu, x n  − φu, x n1 . 2.12

In particular, for all i  1, 2, , N,

ω n,i α n,i 1 − α n,i g∗Jx n − JS i x n  ≤ φu, x n  − φu, x n1 . 2.13Hence,

∞

n1

ω n,i α n,i 1 − α n,i g∗Jx n − JS i x n  < ∞ 2.14

for all i  1, 2, , N Since lim inf n → ∞ ω n,i α n,i 1 − α n,i  > 0 for all i  1, 2 , N and the properties of g, we have

lim

n → ∞ Jx n − JS i x n  0 2.15

for all i  1, 2 , N Since J−1is uniformly norm-to-norm continuous on bounded sets, wehave

lim

n → ∞ x n − S i x n  0 2.16

for all i  1, 2 , N Since {x n } is bounded, there exists a subsequence {x n k } of {x n} such that

x n k  z ∈ C Since S i is relatively nonexpansive, z ∈ FS i   FS i  for all i  1, 2 , N.

We show that z∈∞n1 FT n Let

y n  J−1 N

i1

ω n,i α n,i Jx n  1 − α n,i JS i x n . 2.17

We note from2.15 that

It follows fromLemma 1.4that limn → ∞ T n y n − y n  0 From 2.19 and x n k  z, we have

y n k  z Since {T n } satisfies NST-condition, we have z ∈∞n1 FT n  Hence z ∈ F.

Let z n ΠF x n FromLemma 1.5and z ∈ F, we have

Moreover, since J is monotone,

We next apply our result for finding a common element of a fixed point set of

a relatively nonexpansive mapping and the solution set of an equilibrium problem Thisproblem is extensively studied in11, 14–16 Let C be a subset of a Banach space E and let f : C × C → R be a bifunction The equilibrium problem for a bifunction f is to find x ∈ C such that fx, y ≥ 0 for all y ∈ C The set of solutions above is denoted by EPf, that is

x ∈ EPf

iff fx, y

≥ 0 ∀y ∈ C. 2.25

To solve the equilibrium problem, we usually assume that a bifunction f satisfies the

following conditionsC is closed and convex:

A1 fx, x  0 for all x ∈ C;

A2 f is monotone, that is, fx, y  fy, x ≤ 0, for all x, y ∈ C;

A3 for all x, y, z ∈ C, lim sup t↓0 ftz  1 − tx, y ≤ fx, y;

A4 for all x ∈ C, fx, · is convex and lower semicontinuous.

The following lemma gives a characterization of a solution of an equilibrium problem

Lemma 2.3 Let C be a nonempty closed convex subset of a Banach space E Let f be a bifunction from

C × C → R satisfying (A1)–(A4) Suppose that p ∈ C Then p ∈ EPf if and only if fy, p ≤ 0 for all y ∈ C.

Proof Let p ∈ EPf, then fp, y ≥ 0 for all y ∈ C From A2, we get that fy, p ≤ −fp, y ≤

So f x t , y ≥ 0 for all t ∈ 0, 1 From A3, we have

Takahashi and Zembayashi proved the following important result

Lemma 2.4 see 15, Lemma 2.8 Let C be a nonempty closed convex subset of a uniformly smooth,

strictly convex and reflexive Banach space E Let f be a bifunction from C × C → R satisfying (A1)– (A4) For r > 0 and x ∈ E, define a mapping T r : E → C as follows:

4 EPf is closed and convex.

We now deduce Takahashi and Zembayashi’s recent result fromTheorem 2.2

Corollary 2.5 see 15, Theorem 4.1 Let C be a nonempty closed convex subset of a uniformly

smooth and uniformly convex Banach space E Let f be a bifunction from C × C to R satisfying (A1)– (A4) and let S be a relatively nonexpansive mapping from C into itself such that FS ∩ EPf / ∅ Let the sequence {x n } be generated by u1 ∈ E,

for every n ∈ N, {α n } ⊂ 0, 1 satisfying lim inf n → ∞ α n 1 − α n  > 0 and {r n } ⊂ a, ∞ for some

a > 0 If J is weakly sequentially continuous, then {x n } converges weakly to z ∈ Π FS∩EPf , where

z  lim n → ∞ΠFS∩EPf x n

Proof Put T n ≡ T r n where T r n is defined by Lemma 2.4 Then ∞

To see that {T n } satisfies NST-condition, let {z n } be a bounded sequence in C such that

limn → ∞ z n − T n z n   0 and p ∈ ω w {z n } Suppose that there exists a subsequence {z n k} of

{z n } such that z n k  p Then T n k z n k  p ∈ C Since J is uniformly continuous on bounded

sets and r n k ≥ a, we have

EPf ∩ FS, where z  limn → ∞ΠEPf∩FSx n

Using the same proof as above, we have the following result

Corollary 2.6 see 11, Theorem 3.5 Let C be a nonempty and closed convex subset of a uniformly

convex and uniformly smooth Banach space E Let f be a bifunction from C × C to R satisfies (A1)–

EP f / ∅ Let the sequence {x n } be generated by the following manner:

b lim infn → ∞ α n β n > 0, lim inf n → ∞ α n γ n > 0;

c {r n } ⊂ a, ∞ for some a > 0.

If J is weakly sequentially continuous, then {x n } converges weakly to z ∈ F, where z  lim n → ∞ΠF x n

The following result also follows fromTheorem 2.2

Corollary 2.7 see 9, Theorem 5.3 Let E be a uniformly smooth and uniformly convex Banach

space and let C be a nonempty closed convex subset of E Let {S i}N i1 be a finite family of relatively nonexpansive mappings from C into itself such that F  N i1 FS i  is a nonempty and let {α n,i :

lim infn → ∞ α n,i 1 − α n,i  > 0 and lim inf n → ∞ ω n,i > 0 for all i ∈ {1, 2, , N} and N

i1 ω n,i  1 for

all n ∈ N Let U n be a sequence of mappings defined by

U n x  Π C J−1 N

i1

for all x ∈ C and let the sequence {x n } be generated by x1 x ∈ C and

x n1  U n x n n  1, 2, . 2.39

Then the following hold:

1 the sequence {x n } is bounded and each weak subsequential limit of {x n } belongs to

N

i1 FS i ;

2 if the duality mapping J from E into E∗ is weakly sequentially continuous, then {x n}

converges weakly to the strong limit of{ΠF x n }.