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Volume 2011, Article ID 671754, 24 pagesdoi:10.1155/2011/671754 Research Article New Iterative Approximation Methods for a Countable Family of Nonexpansive Mappings in Banach Spaces 1 De

Volume 2011, Article ID 671754, 24 pages

doi:10.1155/2011/671754

Research Article

New Iterative Approximation Methods for

a Countable Family of Nonexpansive Mappings in Banach Spaces

1 Department of Mathematics, School of Science and Technology, Phayao University,

Phayao 56000, Thailand

2 Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

3 Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

Correspondence should be addressed to Rabian Wangkeeree,rabianw@nu.ac.th

Received 5 October 2010; Accepted 13 November 2010

Academic Editor: Qamrul Hasan Ansari

Copyrightq 2011 K Nammanee and R Wangkeeree This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited

We introduce new general iterative approximation methods for finding a common fixed point of acountable family of nonexpansive mappings Strong convergence theorems are established in theframework of reflexive Banach spaces which admit a weakly continuous duality mapping Finally,

we apply our results to solve the the equilibrium problems and the problem of finding a zero of

an accretive operator The results presented in this paper mainly improve on the correspondingresults reported by many others

1 Introduction

In recent years, the existence of common fixed points for a finite family of nonexpansivemappings has been considered by many authorssee 1 4 and the references therein Thewell-known convex feasibility problem reduces to finding a point in the intersection of thefixed point sets of a family of nonexpansive mappingssee 5,6 The problem of finding

an optimal point that minimizes a given cost function over the common set of fixed points

of a family of nonexpansive mappings is of wide interdisciplinary interest and practicalimportance see 2, 7 A simple algorithmic solution to the problem of minimizing aquadratic function over the common set of fixed points of a family of nonexpansive mappings

is of extreme value in many applications including set theoretic signal estimationsee 7,8

Let E be a normed linear space Recall that a mapping T : E → E is called nonexpansive

if

Tx − Ty  ≤ x − y, ∀x,y ∈ E. 1.1

We use FT to denote the set of fixed points of T, that is, FT  {x ∈ E : Tx  x} A self mapping f : E → E is a contraction on E if there exists a constant α ∈ 0, 1 such that

f x − f

y  ≤ αx − y, ∀x,y ∈ E. 1.2

One classical way to study nonexpansive mappings is to use contractions toapproximate a nonexpansive mapping9 11 More precisely, take t ∈ 0, 1 and define a contraction T t : E → E by

T t x  tu  1 − tTx, ∀x ∈ E, 1.3

where u ∈ E is a fixed point Banach’s contraction mapping principle guarantees that T thas a

unique fixed point x t in E It is unclear, in general, what is the behavior of x t as t → 0, even if

T has a fixed point However, in the case of T having a fixed point, Browder 9 proved that

if E is a Hilbert space, then {x t } converges strongly to a fixed point of T Reich 10 extended

Browder’s result to the setting of Banach spaces and proved that if E is a uniformly smooth

Banach space, then{x t } converges strongly to a fixed point of T and the limit defines the

unique sunny nonexpansive retraction from E onto FT Xu 11 proved Reich’s resultshold in reflexive Banach spaces which have a weakly continuous duality mapping

The iterative methods for nonexpansive mappings have recently been applied to solveconvex minimization problems; see, for example,12–14 and the references therein Let H be

a real Hilbert space, whose inner product and norm are denoted by·, · and ·, respectively Let A be a strongly positive bounded linear operator on H; that is, there is a constant γ > 0

with property

Ax, x ≥ γx2

A typical problem is to minimize a quadratic function over the set of the fixed points of a

nonexpansive mapping on a real Hilbert space H

min

x∈F T

1

2Ax, x − x, b, 1.5

where b is a given point in H In 2003, Xu 13 proved that the sequence {x n} defined by the

iterative method below, with the initial guess x0∈ H chosen arbitrarily

x n1  I − α n A Tx n  α n u, n ≥ 0 1.6

converges strongly to the unique solution of the minimization problem 1.5 providedthe sequence {α n} satisfies certain conditions Using the viscosity approximation method,Moudafi15 introduced the following iterative process for nonexpansive mappings see 16for further developments in both Hilbert and Banach spaces Let f be a contraction on H

Starting with an arbitrary initial x0∈ H, define a sequence {x n} recursively by

x n1  1 − σ n Tx n  σ n f x n , n ≥ 0, 1.7

where {σ n } is a sequence in 0, 1 It is proved 15, 16 that under certain appropriateconditions imposed on{σ n }, the sequence {x n} generated by 1.7 strongly converges to the

unique solution x∗in C of the variational inequality

where A is a strongly positive bounded linear operator on H They proved that if the

sequence{α n} of parameters satisfies the following conditions:

C1 limn → ∞ α n 0,

C2∞n1 α n ∞,

C3∞

n1 |α n1 − α n | < ∞,

then the sequence{x n} generated by 1.9 converges strongly to the unique solution x∗in H

of the variational inequality

On the other hand, in order to find a fixed point of nonexpansive mapping T, Halpern

18 was the first who introduced the following iteration scheme which was referred to as

Halpern iteration in a Hilbert space: x, x0∈ C, {α n } ⊂ 0, 1,

x n1  α n x  1 − α n Tx n , n ≥ 0. 1.11

He pointed out that the control conditionsC1 limn → ∞ α n  0 and C2∞

n1 α n  ∞ arenecessary for the convergence of the iteration scheme1.11 to a fixed point of T Furthermore,

the modified version of Halpern iteration was investigated widely by many mathematicians.Recently, for the sequence of nonexpansive mappings{T n}∞

n1with some special conditions,Aoyama et al 1 studied the strong convergence of the following modified version of

Halpern iteration for x0, x ∈ C:

x n1  α n x  1 − α n T n x n , n ≥ 0, 1.12

where C is a nonempty closed convex subset of a uniformly convex Banach space E

whose norm is uniformly G´ateaux differentiable, {αn } is a sequence in 0, 1 satisfying C1

limn → ∞ α n 0, C2∞n1 α n ∞, and either C3∞n1 |α n − α n1 | < ∞ or C3

 α n ∈ 0, 1 for all n ∈ N and lim n → ∞ α n /α n1  1 Very recently, Song and Zheng 19 also introduced theconception of the conditionB on a countable family of nonexpansive mappings and proved

strong convergence theorems of the modified Halpern iteration1.12 and the sequence {y n}defined by

in a reflexive Banach space E with a weakly continuous duality mapping and in a reflexive

strictly convex Banach space with a uniformly G´ateaux differentiable norm

Other investigations of approximating common fixed points for a countable family ofnonexpansive mappings can be found in1,20–24 and many results not cited here

In a Banach space E having a weakly continuous duality mapping J ϕ with a gauge

function ϕ, an operator A is said to be strongly positive 25 if there exists a constant γ > 0

with the property

Banach space E which admits a weakly continuous duality mapping J ϕ with gauge ϕ:

where A is strongly positive defined by 1.15, {T n : E → E} is a countable family

of nonexpansive mappings, and f is an α-contraction We will prove in Section 3 that ifthe sequence {α n} of parameters satisfies the appropriate conditions, then the sequences

{x n }, {z n }, and {y n } converge strongly to the unique solution x of the variational inequality

Finally, we apply our results to solve the the equilibrium problems and the problem of finding

a zero of an accretive operator

2 Preliminaries

Throughout this paper, let E be a real Banach space, and E∗be its dual space We write x n  x

resp., x n ∗x  to indicate that the sequence {x n} weakly resp., weak∗ converges to x; as usual x n → x will symbolize strong convergence Let U  {x ∈ E : x  1} A Banach space

E is said to uniformly convex if, for any  ∈ 0, 2, there exists δ > 0 such that, for any x, y ∈ U,

x − y ≥  implies x  y/2 ≤ 1 − δ It is known that a uniformly convex Banach space is

reflexive and strictly convexsee also 26 A Banach space E is said to be smooth if the limit

limt → 0 x  ty − x/t exists for all x, y ∈ U It is also said to be uniformly smooth if the limit is attained uniformly for x, y ∈ U.

By a gauge function ϕ, we mean a continuous strictly increasing function ϕ : 0, ∞ →

0, ∞ such that ϕ0  0 and ϕt → ∞ as t → ∞ Let E∗be the dual space of E The duality mapping J ϕ : E → 2 E∗associated to a gauge function ϕ is defined by

ϕx/xJx for all x / 0 see 27 Browder 27 initiated the study of certain classes

of nonlinear operators by means of the duality mapping J ϕ Following Browder27, we say

that a Banach space E has a weakly continuous duality mapping if there exists a gauge ϕ for which the duality mapping J ϕ x is single valued and continuous from the weak topology

to the weak∗ topology, that is, for any{x n } with x n  x, the sequence {J ϕ x n} convergesweakly∗to J ϕ x It is known that l phas a weakly continuous duality mapping with a gauge

function ϕt  t p−1 for all 1 < p < ∞ Set

where ∂ denotes the subdifferential in the sense of convex analysis.

Now, we collect some useful lemmas for proving the convergence result of this paper.The first part of the next lemma is an immediate consequence of the subdifferentialinequality and the proof of the second part can be found in28

Lemma 2.1 see 28 Assume that a Banach space E has a weakly continuous duality mapping J ϕ with gauge ϕ.

i For all x, y ∈ E, the following inequality holds:

Φx  y  ≤ Φx  y,J ϕ



x  y

In particular, for all x, y ∈ E,

x  y2≤ x2 2y, J

x  y

ii Assume that a sequence {x n } in E converges weakly to a point x ∈ E,

then the following identity holds:

where {α n }, {b n }, {c n } satisfying the restrictions

i∞n1 α n  ∞; ii∞n1 b n < ∞; iii lim sup n → ∞ c n ≤ 0.

Lemma 2.5 see 1, Lemma 3.2 Suppose that {Tn } satisfies AKTT-condition, then, for each y ∈

C, {T n y} converses strongly to a point in C Moreover, let the mapping T be defined by

Ty  lim

Then, for each bounded subset B of C, lim n → ∞supz∈B Tz − T n z  0.

The next valuable lemma was proved by Wangkeeree et al.25 Here, we present theproof for the sake of completeness

Lemma 2.6 Assume that a Banach space E has a weakly continuous duality mapping J ϕ with gauge

ϕ Let A be a strongly positive bounded linear operator on E with coefficient γ > 0 and 0 < ρ ≤ ϕ1A−1, then I − ρA ≤ ϕ11 − ργ.

Proof From 1.15, we obtain that A  sup x≤1 |Ax, J ϕ x| Now, for any x ∈ E with

x  1, we see that



I − ρA

x, J ϕ x ϕ1 − ρAx, J ϕ x≥ ϕ1 − ρA ≥ 0. 2.10

That is, I − ρA is positive It follows that

I − ρA   supI − ρAx,J ϕ x: x ∈ E, x  1

with coefficient γ > 0 and 0 < γ < γϕ1/α Define the mapping St : E → E by

Remark 2.7 We note that l pspace has a weakly continuous duality mapping with a gauge

function ϕt  t p−1 for all 1 < p < ∞ This shows that ϕ is invariant on 0, 1.

Lemma 2.8 see 25, Lemma 3.3 Let E be a reflexive Banach space which admits a weakly

continuous duality mapping J ϕ with gauge ϕ such that ϕ is invariant on 0, 1 Let T : E → E

be a nonexpansive mapping with FT /  ∅, f an α-contraction, and A a strongly positive bounded linear operator with coefficient γ > 0 and 0 < γ < γϕ1/α Then, the net {x t } defined by 2.14

converges strongly as t → 0 to a fixed point x of T which solves the variational inequality

3 Main Results

We now state and prove the main theorems of this section

Theorem 3.1 Let E be a reflexive Banach space which admits a weakly continuous duality mapping

J ϕ with gauge ϕ such that ϕ is invariant on 0, 1 Let {T n : E → E}∞n0 be a countable family of nonexpansive mappings satisfying F :∞

n0 FT n  / ∅ Let f be an α-contraction and A a strongly positive bounded linear operator with coefficient γ > 0 and 0 < γ < γϕ1/α Let the sequence {x n}

be generated by1.16, where {α n } is a sequence in 0, 1 satisfying the following conditions:

Suppose that {T n } satisfies the AKTT-condition Let T be a mapping of E into itself defined by Tz 

limn → ∞ T n z for all z ∈ E, and suppose that FT ∞

n0 FT n  Then, {x n } converges strongly to x which solves the variational inequality

Proof Applying Lemma 2.8, there exists a point x ∈ FT which solves the variational

inequality3.1 Next, we observe that {x n } is bounded Indeed, pick any p ∈ F to obtain

for all n ≥ 1, where M is a constant satisfying M ≥ sup n≥1 γfT n−1 x n−1  − AT n−1 x n−1 Putting

μ n  T n x n−1 − T n−1 x n−1   |α n − α n−1 |M From AKTT-condition and C3, we have

Let{x n k } be a subsequence of {x n} such that

If follows from reflexivity of E and the boundedness of a sequence {x n k} that there exists

{x n ki } which is a subsequence of {x n k } converging weakly to w ∈ E as i → ∞ Since J ϕ isweakly continuous, we have byLemma 2.1that

Next, we show that x n → x as n → ∞ In fact, since Φt  t

0ϕτdτ, for all t ≥ 0, and

ϕ : 0, ∞ → 0, ∞ is a gauge function, then for 1 ≥ k ≥ 0, ϕkx ≤ ϕx and

It follows that from conditionC1, limn → ∞ x n1 − x n  0 and 3.9 that

ApplyingLemma 2.2 to3.20, we conclude that Φx n1 − x → 0 as n → ∞; that is,

x n → x as n → ∞ This completes the proof.

Setting γ  1, A ≡ I, where I is the identity mapping and fx  x for all x ∈ E in

Theorem 3.1, we have the following result

Corollary 3.2 Let E be a reflexive Banach space which admits a weakly continuous duality mapping

J ϕ with gauge ϕ Suppose that {T n : E → E} is a countable family of nonexpansive mappings satisfying F : ∞

n0 FT n  / ∅ Assume that {x n } is defined by, for x0, x ∈ E,

Suppose that {T n } satisfies the AKTT-condition Let T be a mapping of E into itself defined by Tz 

limn → ∞ T n z for all z ∈ E, and suppose that FT  ∩∞n0 FT n , then {x n } converges strongly to x of

F which solves the variational inequality

Theorem 3.3 Let E be a reflexive Banach space which admits a weakly continuous duality mapping

J ϕ with gauge ϕ such that ϕ is invariant on 0, 1 Let {T n : E → E}∞n0 be a countable family of nonexpansive mappings satisfying F :∞

n0 FT n  / ∅ Let f be an α-contraction and A a strongly positive bounded linear operator with coefficient γ > 0 and 0 < γ < γϕ1/α Let the sequence {z n}

be generated by1.17, where {α n } is a sequence in 0, 1 satisfying the following conditions:

Suppose that {T n } satisfies the AKTT-condition Let T be a mapping of E into itself defined by Tz 

limn → ∞ T n z for all z ∈ E, and suppose that FT  ∞

n0 FT n , then {z n } converges strongly to x which solves the variational inequality3.1.

Proof Let {x n } be the sequence given by x0 z0and

x n1  α n γf T n x n   I − α n A T n x n , n ≥ 0. 3.25FormTheorem 3.1, x n → x We claim that z n → x ApplyingLemma 2.6, we estimate

Theorem 3.4 Let E be a reflexive Banach space which admits a weakly continuous duality mapping

J ϕ with gauge ϕ such that ϕ is invariant on 0, 1 Let {T n : E → E}∞n0 be a countable family of nonexpansive mappings satisfying F :∞

n0 FT n  / ∅ Let f be an α-contraction and A a strongly positive bounded linear operator with coefficient γ > 0 and 0 < γ < γϕ1/α Let the sequence {y n}

be generated by1.17, where {α n } is sequence in 0, 1 satisfying the following conditions:

Suppose that {T n } satisfies the AKTT-condition Let T be a mapping of E into itself defined by Tz 

limn → ∞ T n z for all z ∈ E, and suppose that FT ∞

n0 FT n , then {y n } converges strongly to x which solves the variational inequality3.1.

Proof Let the sequences {z n } and {β n} be given by

z n  α n γf

y n



 I − α n A y n , β n  α n1 ∀n ∈ N. 3.27

Other investigations of approximating common fixed points for a countable family ofnonexpansive mappings can be found in 1,20–24... Browder27, we say

that a Banach space E has a weakly continuous duality mapping if there exists a gauge ϕ for which the duality mapping J ϕ x is single valued and continuous... theconception of the conditionB on a countable family of nonexpansive mappings and proved