Weak and Strong Convergence Theorems of an Implicit Iteration Process for a Countable Family of Nonexpansive Mappings

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Volume 2008, Article ID 732193,18pages doi:10.1155/2008/732193

Research Article

Weak and Strong Convergence Theorems of

an Implicit Iteration Process for a Countable

Family of Nonexpansive Mappings

Kittikorn Nakprasit,1_{Weerayuth Nilsrakoo,}2_{and Satit Saejung}1

1_{Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand} 2_{Department of Mathematics, Statistics and Computer, Ubon Rajathanee University,}

Ubon Ratchathani 34190, Thailand

Correspondence should be addressed to Satit Saejung,[email protected]

Received 22 July 2008; Accepted 18 November 2008

Recommended by Anthony Lau

Using the implicit iteration and the hybrid method in mathematical programming, we prove weak and strong convergence theorems for finding common fixed points of a countable family of nonexpansive mappings in a real Hilbert space. Our results include many convergence theorems by Xu and Ori2001and Zhang and Su2007as special cases. We also apply our method to find a common element to the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem. Finally, we propose an iteration to obtain convergence theorems for a continuous monotone mapping.

Copyrightq2008 Kittikorn Nakprasit et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

LetHbe a real Hilbert space with inner product·,·and norm·,and letCbe a nonempty subset ofH. A mappingT :C → His said to benonexpansiveif

Tx−Ty ≤ x−y ∀x, y∈C. 1.1

We denote byFTthe set of all fixed points ofT. IfCis bounded closed convex andT is a nonexpansive mapping ofCinto itself, thenFTis nonemptysee1. We writexn → x

xn x, resp.if {xn} converges strongly weakly, resp. to x. There are many methods

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x1α1x01−α1T1x1,

x2α2x11−α2T2x2, ..

.

xNαNxN−1

1−αNTNxN,

xN1αN1xN1−αN1

T1xN1

.. .

1.2

where{αn}is a sequence in0,1. The iteration above can be written in the following compact form:

xnαnxn−1

1−αnTnxn, n≥1, 1.3

whereTn≡TnmodN,here the mod Nfunction takes values in{1,2, . . . , N}.They proved that

this process converges weakly to a common fixed point of{T_i}N_i₁. Recently, to obtain a strong convergence theorem, Zhang and Su 3 modify iteration processes 1.3 by the implicit hybrid method for a finite family of nonexpansive mappings{Ti}N_i₁: an initial pointx0∈C,

x0∈Cis arbitrary,

ynαnxn1−αnTnzn,

znβnyn1−βnTnyn,

Cnz∈C:yn−z≤xn−z,

Qnz∈C:xn−z, x0−xn≥0,

xn1PCn∩Qnx0, n0,1,2, . . . ,

1.4

whereTn≡TnmodN, {αn}and{βn}are real sequences in0,1withαn<1.

In this paper, we establish weak and strong convergence theorems for finding common fixed points of a countable family of nonexpansive mappings in a real Hilbert space. Our results include many convergence theorems by 2, Theorems 2 and 3, Theorems 2.4 as special cases. The new iteration introduced in this paper is applied to find a common element to the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem. We also propose an iteration to obtain convergence theorems for a continuous monotone mapping.

2. Preliminaries

LetHbe a real Hilbert space. Then,

x−y2_x2₋_y2₋₂_x₋_{y, y}_, _2.1

_λx 1−λy2λx2 ₁₋_λ_y2₋_λ₁₋_λ_x₋_y2 _2.2

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1Opial’s condition4, that is, for any sequence{xn}withxn x, the inequality

lim inf

n→ ∞ xn−x<lim infn→ ∞ xn−y 2.3

holds for everyy∈Hwithy /x.

2The Kadec-Klee property 1, that is, for any sequence {xn} with xn x and xn → xtogether impliesxn−x → 0.

LetCbe a nonempty closed convex subset ofH. Then, for anyx∈H, there exists the nearest pointPCxinCsuch that

_x₋_P_C_x_≤_x₋_y_∀_y_∈_C. 2.4

Such a mapping, PC is called the metric projection of H onto C. We know that PC is

nonexpansive. Furthermore, forx∈Handz∈C,

zPCx iﬀx−z, z−y ≥0∀y∈C. 2.5

Lemma 2.1see5, Lemma 1. Suppose that{an}and{bn}are two sequences of nonnegative real numbers such that

an1≤anbn ∀n≥1, 2.6

and ∞_n₁bn<∞, thenlimn→ ∞anexists. In particular, iflim infn→ ∞an0, thenlimn→ ∞an0.

Lemma 2.2see6, Lemma 2.2. Suppose that{a_n}and{b_n}are two sequences of nonnegative real numbers such that ∞_n₁an∞and ∞n1anbn<∞. Then,lim infn→ ∞bn0.

Lemma 2.3see7, Lemma 3.2. LetCbe a nonempty closed convex subset of a real Hilbert space

H. Let{x_n}be a sequence inHsuch that

_x_n₁₋_y_≤_x_n₋_y _∀_y_∈_{C, n}_∈_N_. 2.7

Then, the sequence{PCxn}converges strongly to somez∈C.

To deal with a family of mappings, the following conditions are introduced. LetCbe a subset of a Banach space, let{Tn}andTbe families of mappings ofCwith ∞_n₁FTn

FT/∅, whereFTis the set of all common fixed points of all mappings inT.

a{T_n}is said to satisfy the AKTT-condition8if for each bounded subsetBofC,

∞

n1

supTn1z−Tnz:z∈B<∞. 2.8

b{T_n}is said to satisfy the NST-conditionIwithT9if for each bounded sequence

{zn}inC,

lim

n→ ∞zn−Tnzn0 implies limn→ ∞zn−Tzn0 ∀T ∈ T. 2.9

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c{Tn}is said to satisfy the NST-conditionII 9if for each bounded sequence{zn} inC,

lim

n→ ∞zn1−Tnzn0 implies limn→ ∞zn−Tmzn0 ∀m∈N. 2.10

Inspired by conditions above, we introduce the following one.

d{T_n}is said to satisfy the NST∗-condition withTif for each bounded sequence{z_n} inC,

lim

n→ ∞zn−Tnzn0, nlim→ ∞zn−zn10 2.11

imply that lim_n_{→ ∞}z_n−Tz_n 0 for allT ∈ T. In particular, ifT {T}, then we simply say that{Tn}satisfies the NST∗-condition withT.

Remark 2.4. iIf {Tn}satisfies the NST-conditionIwith T, then {Tn} satisfies the NST∗ -condition withT.

iiIf{Tn}satisfies the NST-conditionII, then{Tn}satisfies the NST∗-condition with

{Tn}.

Lemma 2.5 see8, Lemma 3.2. LetCbe a nonempty closed subset of a Banach space, and let {Tn}be a family of mappings ofCinto itself which satisfies the AKTT-condition, then there exists a mappingT :C → Csuch that

Tx lim

n→ ∞Tnx ∀x∈C, 2.12

andlimn→ ∞sup{Tz−Tnz:z∈B}0for each bounded subsetBofC.

Lemma 2.6. LetCbe a nonempty closed subset of a Banach space, and let{Tn}be a family of mappings ofCinto itself which satisfies AKTT-condition and∞_n₁FT_n_/∅. LetTbe the mapping fromCinto itself defined byTz limn→ ∞Tnzfor allz ∈ Cand suppose thatFT ∞n1FTn. Then,{Tn}

satisfies the NST-condition (I) withT. This implies that{Tn}satisfies the NST∗-condition withT. Proof. Let {z_n} be a bounded sequence in Csuch that lim_n_{→ ∞}z_n−T_nz_n 0.We apply

Lemma 2.5to get

_z_n₋_Tz_n_≤_z_n₋_T_n_z_n_T_n_z_n₋_Tz_n

≤zn−TnznsupTnz−Tz:z∈zn−→0.

2.13

Hence, we obtain that{Tn}satisfies the NST-conditionIwithT. This completes the proof.

Lemma 2.7. Let C be a nonempty subset of a Banach space, and let {T_n}N_n₁ be a finite family of nonexpansive mappings of Cinto itself with a common fixed point. Then,{Tn} satisfies NST∗ -condition withT{T1, T2, . . . , TN}, whereTn≡TnmodN.

Proof. Let{z_n}be a bounded sequence inCsuch that lim

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Obviously, it is easy to see that limn→ ∞zni−zn0 for eachi1,2, . . . , N. Consequently,

_z_n₋_T_n_i_z_n_≤_z_n₋_z_n_i_z_n_i₋_T_n_i_z_n_i_T_n_i_z_n_i₋_T_n_i_z_n

≤2zn−znizni−Tnizni−→0 asn−→ ∞.

2.15

This implies that limn→ ∞zn−Tmzn0 for eachm1,2, . . . , N. This completes the proof.

Remark 2.8. There are families of mappings{Tn}andTsuch that

1{Tn}satisfies the NST∗-condition withT;

2{Tn}fails the NST-conditionIwithTand the NST-conditionII.

The following example shows that the NST∗-condition withTis strictly weaker than NST-conditionIwithTand the NST-conditionII.

Example 2.9. LetH:R2_and_C_: ₀_,₁_×₀_,₁_{. Define}_{T1, T2}_:_C _→ _C_{as follows:}

T1x, y x,1−y, T2x, y 1−x, y 2.16

for allx, y∈C. Hence,T1andT2are nonexpansive mappings with

FT1∩FT2

0,1×

1 2

∩

1 2

×0,1

1 2,

1 2

/

∅. 2.17

LetTnTnmod 2. ByLemma 2.7, we have{Tn}satisfies NST∗-condition with{T1, T2}.

a{T_n}fails the NST-condition Iwith T {T1, T2}. In fact, letz2_n₋1 1,1/2and z2n 1/2,1for alln∈N. Then,z2n−1 ∈FT2n−1 FT1andz2n∈FT2n FT2.

In particular,zn−Tnzn ≡0. Clearly,

_z_n₋_T1z_n0, z_n−T2z_n0. 2.18

Hence,{Tn}fails the NST-conditionIwith{T1, T2}.

b{Tn} fails the NST-condition II. To this end, let z4n−3 1/4,1/4, z4n−2

1/4,3/4, z4n−1 3/4,3/4, andz4n 3/4,1/4for all n ∈ N. Then,zn1 − Tnzn ≡0. But,

_z_n₋_T1z_n_0, _z_n₋_T2z_n₀_. _2.19

Hence,{Tn}fails the NST-conditionII.

Lemma 2.10see10. LetCbe a nonempty closed convex subset of a strictly convex Banach space,

SandTbe two nonexpansive mappings ofCinto itself with a common fixed point, and0< β <1. Let

Ube a mapping defined by

UTβI 1−βS, 2.20

whereIis the identity mapping. Then,Uis a nonexpansive mapping fromCinto itself andFU

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Lemma 2.11. LetCbe a nonempty closed convex subset of a strictly convex Banach space. Let{Tn} andTbe two families of nonexpansive mappings fromCinto itself such that∞_n₁FT_n FT_/∅, and suppose that{Tn} satisfies the NST∗-condition withT. Let {Un} be a family of nonexpansive mappings fromCinto itself defined by

UnTnβnI1−βnTn 2.21

for alln ∈N, whereI is the identity mapping, and{βn}is a sequence ina,1for somea ∈0,1. Then,{U_n}satisfies the NST∗-condition withT.

Proof. ByLemma 2.10, we haveFUn FTnfor alln∈Nand so,

∞

n1

FUnFT/∅. 2.22

Let{z_n}be a bounded sequence inCsuch that

lim

n→ ∞zn−Unzn0, nlim→ ∞zn1−zn0. 2.23

Since

_z_n₋_T_n_z_n_≤_z_n₋_U_n_z_n_T_n

βnzn1−βnTnzn−Tnzn ≤zn−Unzn1−βnzn−Tnzn

≤zn−Unzn 1−azn−Tnzn,

2.24

it follows that

_z_n₋_T_n_z_n_≤ 1

azn−Unzn−→0. 2.25

Since{T_n}satisfies the NST∗-condition withT, we have

lim

n→ ∞zn−Tzn0 ∀T ∈ T. 2.26

Hence, we obtain that{U_n}satisfies the NST∗-condition withT. This completes the proof.

3. Weak convergence theorems

Lemma 3.1. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Let{T_n}be a family of nonexpansive mappings fromCinto itself with a common fixed point. Let{xn}be a sequence inC defined byx0∈Cand

xnαnxn−1

1−α_nT_nx_n 3.1

for alln∈N, where{αn}is a sequence in0,1. Then,

ilimn→ ∞xn−pexists for eachp∈∞n1FTn;

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Proof. Observe that ifCis a nonempty closed convex subset of a real Hilbert space Hand

T : C → C is a nonexpansive mapping, then for everyu ∈ C, α ∈ 0,1, the mapping

SSα,T:C → Cdefined by

Sxαu 1−αTx x∈C 3.2

is a1−α-contraction, that is, for allx, y∈C,

Sx−Sy 1−αTx−Ty ≤1−αx−y. 3.3

Consequently,Shas a unique fixed pointx∗∈C. Thus, there exists a uniquex∗∈C, that is,

x∗_αu ₁₋_α_Tx∗_. _3.4

This implies that the implicit iteration scheme 3.1 is well defined. To seei, we letp ∈ _∞

n1FTn. It follows from2.2that

_x_n₋_p2 _α

nxn−1−p

1−αnTnxn−p2

αnxn−1−p2

1−αnTnxn−p2−αn1−αnxn−1−Tnxn2

≤αnxn−1−p2

1−α_nx_n−p2−α_n1−α_nx_n₋1−Tnxn2.

3.5

Sinceαn>0, we have

_x_n₋_p2_≤_x

n−1−p2−

1−αnxn−1−Tnxn2. 3.6

In particular,

_x_n₋_p_≤_x_n₋₁₋_p_. 3.7

So, limn→ ∞xn−pexists. Furthermore, from3.6, we have

1−αnxn−1−Tnxn2≤xn−1−p2−xn−p2. 3.8

Summing from 1 tomand tending to infinity form, we haveii. This completes the proof.

Theorem 3.2. LetCbe a nonempty closed convex subset of a real Hilbert space H. Let {Tn}and T be two families of nonexpansive mappings fromCinto itself such that∞_n₁FT_n FT_/∅, and suppose that{T_n}satisfies the NST∗-condition withT. Then, the sequence{x_n}inCdefined by

3.1, where{αn}is a sequence in0, bfor someb∈0,1, converges weakly tow∈FT. Moreover,

limn→ ∞PFTxnw.

Proof. It follows fromLemma 3.1ithat{x_n}is bounded. ByLemma 3.1iiandα_n ≤ b, we have

∞

n1

_x_n₋₁₋_T_n_x_n2 _<_∞_.

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It follows that limn→ ∞xn−1−Tnxn0.From3.1, we immediately have

lim

n→ ∞xn−Tnxnnlim→ ∞αnxn−1−Tnxn0, 3.10

and so,

lim

n→ ∞xn−xn−10. 3.11

Since{Tn}satisfies the NST∗-condition withT, we have

lim

n→ ∞xn−Txn0 ∀T ∈ T. 3.12

We now extract a subsequence{xni}of{xn}such thatxni w. So, by the demiclosedness

principle,w ∈FT. To prove thatxn w, suppose that there exists another subsequence {xmj}of{xn}such thatxmj w/w. So, we havew ∈FT. It follows fromLemma 3.1i

and Opial’s condition that

lim

n→ ∞xn−wilim→ ∞xni−w<ilim→ ∞xni−w

lim

j→ ∞xmj−w

_< _lim

j→ ∞xmj−w

lim

n→ ∞xn−w,

3.13

arriving at a contradiction. Hence,xn w ∈ FT.Finally, we prove that limn→ ∞zn w,

wherez_n P_F_Tx_n for eachn ∈ N. By3.7andLemma 2.3, there isw0 ∈ FTsuch that

zn → w0.FromznPFTxnandw∈FT, we have

xn−zn, zn−w≥0 ∀n∈N. 3.14

It follows fromzn → w0andxn wthat

w−w0, w0−w≥0, 3.15

and thenw0w.This completes the proof.

UsingTheorem 3.2andLemma 2.7, we have the following result.

Corollary 3.3see2, Theorem 2. LetCbe a nonempty closed convex subset of a real Hilbert spaceH,and let{Tn}N_n₁be a finite family of nonexpansive mappings ofCinto itself with a common fixed point. Then, the sequence{x_n}inCdefined by1.3, where{α_n}is a sequence in0, bfor some

b∈0,1, converges weakly towlim_n_{→ ∞}PN

n1FTnxn.

In the presence of the stronger condition than NST∗-condition withT, we are able to weaken the restriction on{αn}.

Theorem 3.4. Let Cbe a nonempty closed convex subset of a real Hilbert space H, and let{Tn} be a family of nonexpansive mappings of C into itself which satisfies the AKTT-condition and

_∞

n1FTn/∅. LetTbe the mapping fromCinto itself defined byTzlimn→ ∞Tnzfor allz∈C,and

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Proof. By Lemmas2.2and3.1iiand _n∞₁1−αn ∞, we have

lim inf

n→ ∞ xn−1−Tnxn0, 3.16

and hence,

lim inf

n→ ∞ xn−Tnxnlim infn→ ∞ αnxn−1−Tnxn0. 3.17

Next, we prove that the limit limn→ ∞xn−Tnxnexists. Since{xn}is bounded, it follows from

AKTT-condition that

∞

n1

supTnz−Tn−1z:z∈xn<∞. 3.18

Notice that

_x_n₋_x_n₋₁

1−α_nx_n₋1−Tnxn

≤1−αnxn−1−Tn−1xn−1Tn−1xn−1−Tn−1xnTn−1xn−Tnxn

≤1−αnxn−1−Tn−1xn−1

1−αnxn−1−xn

1−α_nsupT_nz−T_n₋1z:z∈x_n,

3.19

so we have

αnxn−xn−1≤

1−αnxn−1−Tn−1xn−1

1−αnsupTnz−Tn−1z:z∈xn.

3.20

It follows that

_x_n₋_T_n_x_n αn

1−αn

_x_n₋_x_n₋₁

≤xn−1−Tn−1xn−1supTnz−Tn−1z:z∈xn.

3.21

ByLemma 2.1and3.18, we have lim_n_{→ ∞}x_n−T_nx_nexists. Thus, we have

lim

n→ ∞xn−Tnxn0. 3.22

From the definition ofT, we haveT is nonexpansive. ByLemma 2.6, we have{Tn}satisfies the NST∗-condition withT. As in the proof ofTheorem 3.2,{xn}converges weakly to w

lim_n_{→ ∞}P_F_Tx_n.

Remark 3.5. Since the NST∗-condition is implied by the AKTT-condition,Theorem 3.4still holds under the same condition of{αn}as inTheorem 3.2.

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Corollary 3.6. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Let{αn}be a sequence in0,1with ∞_n₁1−α_n ∞. Let{βk_n}be a family of positive real numbers with indices

n, k∈Nwithk≤nsuch that

i n_k₁βk_n1for everyn∈N;

iilimn→ ∞βkn>0for everyk∈N;

iii ∞_n₁ n_k₁|βk_n₁−βk_n|<∞.

Let {Sk} be a family of nonexpansive mappings from C into itself with a common fixed point. Then, the sequence {x_n} in C defined by 3.1, where T_n ≡ n_k₁βk_nS_k, converges weakly to

wlimn→ ∞P∞

k1FSkxn.

4. Strong convergence theorems

We next use the hybrid method from mathematical programming to obtain several strong convergence theorems.

Theorem 4.1. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Let{T_n}andT be two families of nonexpansive mappings fromCinto itself such that∞_n₁FTn FT/∅,and suppose that{Tn}satisfies the NST∗-condition withT. Let{xn}be a sequence inCdefined as follows:

x0∈Cis arbitrary,

ynαnxn1−αnTnyn,

xn1PCn∩Qnx0, n0,1,2, . . . ,

4.1

where{αn}is a sequence in0, bfor someb∈0,1. Then,{xn}converges strongly toPFTx0. Proof. We first prove thatC_n andQ_n are closed and convex for eachn ∈N∪ {0}. From the definitions ofCnandQn, it is obvious thatCnis closed andQnis closed and convex for each

n∈N∪ {0}. We prove thatCnis convex. Sinceyn−z ≤ xn−zis equivalent to

_y_n₋_x_n2

2yn−xn, xn−z≤0, 4.2

by2.1it follows thatCnis convex. Next, we show that

FT⊂Cn ∀n∈N∪ {0}. 4.3

Letp∈FTandn∈N∪ {0}.Since

_y_n₋_p_α_n_x_n

1−αnTnyn−p ≤αnxn−p1−αnTnyn−p ≤αnxn−p1−αnyn−p,

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it follows that

_y_n₋_p_≤_x_n₋_p_, _4.5

and hence,p∈Cn. Therefore, we obtain4.3. Now, we show that

FT⊂Qn ∀n∈N∪ {0}. 4.6

We prove this by induction. Forn0, we haveFT⊂CQ0. Suppose thatFT⊂Q_n. Then, ∅/FT⊂Cn∩Qnand there exists a unique elementxn1 ∈Cn∩Qnsuch thatxn1PCn∩Qnx0.

Then,

xn1−z, x0−xn1

≥0 4.7

for eachz∈C_n∩Q_n. In particular,

xn1−p, x0−xn1

≥0 4.8

for eachp∈FT. It follows thatFT⊂Q_n1,and hence4.6holds. Therefore,

FT⊂Cn∩Qn ∀n∈N∪ {0}. 4.9

This implies that{xn}is well defined. It follows from the definition ofQn thatxn PQnx0,

that is,

_x_n₋_x0_≤_z₋_x0 _∀_z_∈_Q_n _{and all} _n_∈_N_{∪ {}₀_}_. _4.10

In particular,

_x_n₋_x0_≤_z₋_x0 _∀_z_∈_F_Tand all n∈N∪ {0}. 4.11

On the other hand, fromx_n1PCn∩Qnx0∈Qn, we have

_x_n₋_x0_≤_x_n₁₋_x0 _∀_n_∈_N_{∪ {}0}. 4.12

Therefore,{xn−x0}is nondecreasing and bounded. So, limn→ ∞xn−x0exists. This implies

that{xn}is bounded. Sincexn1 PCn∩Qnx0∈Qn, we have

xn−xn1, x0−xn≥0. 4.13

It follows from2.1that

_x_n₁₋_x_n2_x

n1−x0

−xn−x02

xn1−x02−xn−x02−2xn1−xn, xn−x0 ≤xn1−x02−xn−x02

4.14

for alln∈N∪ {0}.This implies that

lim

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Sincexn1∈Cn, we have

_y_n₋_x_n_≤_y_n₋_x_n₁_x_n₋_x_n₁

≤2xn−xn1−→0.

4.16

It follows fromαn≤b <1 that

_x_n₋_T_n_x_n_≤_x_n₋_T_n_y_n_T_n_y_n₋_T_n_x_n

≤xn−Tnynyn−xn

1

1−αn

_y_n₋_x_n_y_n₋_x_n

≤ 1

1−byn−xnyn−xn−→0.

4.17

Since{Tn}satisfies the NST∗-condition withT, we have

lim

n→ ∞xn−Txn0 ∀T ∈ T. 4.18

Finally, we show thatxn → w, wherew PFTx0. Since{xn} is bounded, let{xnk} be a

subsequence of{xn}such thatxnk w. SinceI−T is demiclosed and by using4.18, we

havew∈FT. By4.11, we have

_x_n₋_x0_≤_w₋_x0_. _4.19

It follows fromwPFTx0and the lower semicontinuity of the norm that

_w₋_x0_≤_w₋_x0_≤_{lim inf}

k→ ∞ xnk−x0≤lim sup_k_{→ ∞} xnk −x0≤w−x0. 4.20

Thus, we obtain that lim_k_{→ ∞}x_nk−x0w−x0w−x0. Using the Kadec-Klee property ofH, we obtain that limk→ ∞xnk w w.Since{xnk}is an arbitrary subsequence of{xn},

we can conclude that the whole sequence{xn}converges strongly toPFTx0.

UsingTheorem 4.1and Lemmas2.7and2.11, we have the following result.

Corollary 4.2see3, Theorem 2.4. LetCbe a nonempty closed convex subset of a real Hilbert spaceH,and let{T_n}N_n₁be a finite family of nonexpansive mappings ofCinto itself with a common fixed point. Then, the sequence{xn}inCdefined by1.4, where{αn}is a sequence in0, afor some

a∈0,1,and{βn}is a sequence inb,1for someb∈0,1, converges strongly toPN

n1FTnx0. 5. Applications

5.1. Equilibrium problems

LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Letfbe a bifunction of

C×CintoR, whereRis the set of real numbers. The equilibrium problem forf:C×C → R is to findx∈Csuch that

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The set of solutions of 5.1 is denoted by EPf. Numerous problems in physics, optimization, and economics are reduced to find a solution of 5.1. Some methods have been proposed to solve the equilibrium problem11–17. In 2005, Combettes and Hirstoaga

12introduced an iterative scheme of finding the best approximation to the initial data when EPfis nonempty, and they also proved a strong convergence theorem.

For solving the equilibrium problem, let us assume that the bifunctionf satisfies the following conditionssee11.

A1fx, x 0 for allx∈C;

A2fis monotone, that is,fx, y fy, x≤0 for anyx, y∈C;

A3fis upper-hemicontinuous, that is, for eachx, y, z∈C,

lim sup

t→0 f

tz 1−tx, y≤fx, y; 5.2

A4fx,·is convex and lower semicontinuous for eachx∈C.

The following lemma is shown in11, Corollary 1and12, Lemma 2.12.

Lemma 5.1. LetCbe a nonempty closed convex subset of a real Hilbert spaceH, letfbe a bifunction fromC×CintoRsatisfies (A1)–(A4), and letr >0andx∈H. Then, there exists a uniquex∗∈C such that

fx∗_{, y} 1

r

y−x∗_{, x}∗₋_x_≥₀ _∀_y_∈_C. _5.3

Moreover, letTr be a mapping ofHintoCdefined by

Trx x∗ ∀x∈H. 5.4

Then, the following conditions hold:

iTr is firmly nonexpansive, that is, for anyx, y∈H,

_T_r_x₋_T_r_y2_≤_x₋_y2₋_T

rx−x−Try−y2; 5.5

iiFTr EPf;

iiiEPfis closed and convex.

Lemma 5.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let S be a nonexpansive mapping of Cinto H, and letT be a firmly nonexpansive mapping fromH into C such thatFS∩FT/∅. Then,STis a nonexpansive mapping fromHinto itself and

FST FS∩FT. 5.6

Proof. SinceTis firmly nonexpansive, there exists a nonexpansive mappingUsuch thatT

1/2IUandFU FT. As in the proof ofLemma 2.10, the conclusion holds.

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Theorem 5.3. Let C be a nonempty closed convex subset of a real Hilbert space H. Let f be a bifunction fromC×CintoRsatisfying (A1)–(A4), and letSbe a nonexpansive mapping ofCintoH such thatFS∩EPf_/∅. Let{xn}and{un}be two sequences generated byx0∈Hand

fun, y _r1 n

y−un, un−xn≥0 ∀y∈C,

xnαnxn−1

1−αnSun

5.7

for alln ∈ N, where{αn}is a sequence in0,1with ∞_n₁1−αn ∞,and {rn}is a sequence in0,∞withlim infn→ ∞rn > 0and ∞n1|rn1 −rn| < ∞. Then,{xn}converges weakly tow ∈ FS∩EPf. Moreover,lim_n_{→ ∞}P_F_S_∩EPfxnw.

Proof. It is noted that the iteration scheme is well defined. As in the proof of14, Theorem 16, it follows from lim infn→ ∞rn>0 and ∞n1|rn1−rn|<∞that

∞

n1

supTrn1z−Trnz:z∈B

<∞ 5.8

for any bounded subsetBofH. Moreover, byLemma 2.5, the mappingT defined by

Tx lim

n→ ∞Trnx ∀x∈H 5.9

satisfies

FT ∞

n1

FTrnEPf. 5.10

It is easy to see thatT is a firmly nonexpansive mapping ofHintoC. WriteTn ≡STrn then,

byLemma 5.2, we haveTnis a nonexpansive mapping fromHinto itself, and

FTnFSTrnFS∩FTrnFS∩EPf FST 5.11

for alln∈Nand so,

∞

n1

FTn FST FS∩EPf. 5.12

SinceSis nonexpansive,5.8and5.9, we have

∞

n1

supT_n1z−T_nz:z∈B<∞ 5.13

for any bounded subsetBofH,and

STxS

lim

n→ ∞Trnx

lim

n→ ∞STrnxnlim→ ∞Tnx ∀x∈H. 5.14

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Similarly, we have the following strong convergence theorem. We safely suppress the proof.

Theorem 5.4. Let C be a nonempty closed convex subset of a real Hilbert space H. Let f be a bifunction fromC×CintoRsatisfying (A1)–(A4), and letSbe a nonexpansive mapping ofCintoH such thatFS∩EPf_/∅. Let{x_n}and{u_n}be two sequences generated byx0∈Hand

fun, y_r1 n

y−un, un−yn≥0 ∀y∈C,

ynαnxn−1

1−αnSun,

xn1PCn∩Qnx0, n0,1,2, . . . ,

5.15

where {α_n} is a sequence in 0, a for some a ∈ 0,1, and {r_n} is a sequence in 0,∞ with

lim infn→ ∞rn>0and n∞1|rn1−rn|<∞. Then,{xn}converges strongly toPFS∩EPfx0.

5.2. Convergence theorem for monotone mappings

LetHbe a real Hilbert space, andCbe a nonempty closed convex subset ofH. LetA:C → H be a mapping. The classical variational inequality is to findx∈Csuch that

Ax, y−x ≥0 ∀y∈C. 5.16

The set of solutions of classical variational inequality is denoted by VIPC, A. The variational inequality has been extensively studied in the literaturessee7,18–23and the references therein. We recall that a mappingA:C → His said to be

amonotone if

Au−Av, u−v ≥0 ∀u, v∈C; 5.17

bα-inverse-strongly monotone if there exists a constantα >0 such that

Au−Av, u−v ≥αAu−Av2 _∀_{u, v}_∈_C_; _5.18

cr-strongly monotone if there exists a constantr >0 such that

Au−Av, u−v ≥ru−v2 _∀_{u, v}_∈_C_; _5.19

drelaxedγ, r-cocoercive if there exist constantsγ,r >0 such that

Au−Av, u−v ≥ −γAu−Av2_r_u₋_v2 _∀_{u, v}_∈_C_; _5.20

eμ-Lipschitzian if there exists a constantμ >0 such that

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Remark 5.5. 1Everyα-inverse-strongly monotone mapping is monotone and 1/α -Lipschit-zian.

2Everyr-strongly monotone is monotone.

3 Every relaxed γ, r-cocoercive and μ-Lipschitzian mapping with γμ2 _≤ _r _is

monotone.

Lemma 5.6. Let C be a nonempty closed convex subset of a real Hilbert space H. Let A be a continuous monotone mapping ofCintoH. Define a bifunctionf :C×C → Ras follows:

fx, y Ax, y−x ∀x, y∈C. 5.22

Then,

i[14, Lemma 19]fsatisfies (A1)–(A4) and VIPC, A EPf;

ii[14, Lemma 20] Ifx∈H, u∈C,andr >0, then

fu, y 1_ry−u, u−x ≥0 ∀y∈C⇐⇒uPCx−rAu. 5.23

UsingTheorem 5.3andLemma 5.6, we haveTheorem 5.7.

Theorem 5.7. Let C be a nonempty closed convex subset of a real Hilbert space H. Let A be a continuous monotone mapping ofC,and letS be a nonexpansive mapping of Cinto H such that

FS∩VIPC, A/∅. Let{xn}and{un}be sequences generated byx0 ∈Hand

unPCxn−rnAun,

xnαnxn−1

1−αnSun

5.24

for alln ∈ N, where{αn}is a sequence in0,1with ∞n11−αn ∞,and {rn}is a sequence

in0,∞withlim inf_n_{→ ∞}r_n > 0and ∞_n₁|r_n1 −rn| < ∞. Then,{xn}converges weakly tow ∈

FS∩VIPC, A. Moreover,limn→ ∞PFS∩VIPC,Axnw.

UsingTheorem 5.4andLemma 5.6, we also haveTheorem 5.8.

Theorem 5.8. Let C be a nonempty closed convex subset of a real Hilbert space H. Let A be a continuous monotone mapping ofC,and letS be a nonexpansive mapping of Cinto H such that

FS∩VIPC, A_/∅. Let{x_n}and{u_n}be sequences generated byx0 ∈Hand

unPCyn−rnAun,

ynαnxn−1

1−α_nSu_n,

xn1PCn∩Qnx0, n0,1,2, . . . ,

5.25

where {αn} is a sequence in 0, a for some a ∈ 0,1, and {rn} is a sequence in 0,∞ with

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Remark 5.9. 1 By Remark 5.5, we obtain a strong convergence theorem for α -inverse-strongly monotone mappings,r-strongly monotone and continuous mappings and relaxed

γ, r-cocoercive andμ-Lipschitzian mappings withγμ2_≤_r_.

2Some weak and strong convergence theorems for monotone Lipschitzian mappings were established by several authors 7, 18–23. However, there is a monotone continuous mapping which is not Lipschitziansee14, Remark 23. Therefore, Theorems5.7and5.8

provide a new convergence theorem for a wider class of mappings.

Acknowledgments

The authors would like to thank the referee for comments which improve the manuscript. Satit Saejung was supported by the Commission on Higher Education and the Thailand Research Fund under Grant MRG5180146.

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