Fixed Point Theory and Applications Volume 2010, Article ID 407651,26pages doi:10.1155/2010/407651

*Research Article*

**Approximation of Common Fixed**

**Points of a Countable Family of Relatively**

**Nonexpansive Mappings**

**Daruni Boonchari**

**1**

_{and Satit Saejung}

_{and Satit Saejung}

**2**

*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

Copyrightq2010 D. Boonchari and S. Saejung. 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.

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 obtain several convergence theorems announced by many authors but also prove them under weaker assumptions. 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 also discussed.

**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{*xn*}is a sequence in*E*, we denote strong convergence of{*xn*}to*x*∈*E*by*xn* → *x*and

weak convergence by*xn* * 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

{*xn*}in*C*such that*xn* * p*and*xn*−*Txn* → 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 if*xy/*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 *xy/*2 *<* 1−*δ* for

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

lim

*t*→0

*xtx* − *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 fixed point of a certain mapping or a common fixed point of a family of mappings. This paper deals with 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 of maximal monotone operatorssee2–4.

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

*φx, yx*2_{−}_{2}_{x, Jy}_{}* _{y}*2

_{∀}

_{x, y}_{∈}

_{E,}_{}

_{1.2}

_{}

where*J*is the normalized duality mapping from*E*to the dual space*E*∗given by the following
relation:

*x, Jx* *x*2_{}_{}_{Jx}_{}2_{.}_{}_{1.3}_{}

We know that if *E*is smooth, strictly convex, and reflexive, then the duality mapping*J* is
single-valued, one-to-one, and onto. The duality mapping*J*is said to be weakly sequentially
continuous if*xn x*implies that*Jxn Jx*∗ see5for more details.

Following Matsushita and Takahashi6, a mapping*T* :*C* → *E*is said to be relatively
nonexpansive if the following conditions are satisfied:

R1*FT*is nonempty;

R2*φu, Tx*≤*φu, x*for all*u*∈*FT*,*x*∈*C*;

R3*FT* *FT*.

If *T* satisfiesR1and R2, then*T* is called relatively quasi-nonexpansive 7. Obviously,
relative nonexpansiveness implies relative quasi-nonexpansiveness but the converse is
not true. Relatively quasi-nonexpansive mappings are sometimes called hemirelatively
nonexpansive mappings. But we do prefer the former name because in a Hilbert space setting,
relatively quasi-nonexpansive mappings are nothing but quasi-nonexpansive.

In2, Alber introduced the generalized projectionΠ*C* from*E*onto*C*as follows:

Π*Cx* arg min
*y*∈*C* *φ*

*y, x* ∀*x*∈*E.* _{}_{1.4}_{}

If*E*is a Hilbert space, then*φy, x* *y*−*x*2 andΠ*C* becomes the metric projection of *E*

onto*C*. Alber’s generalized projection is an example of relatively nonexpansive mappings.
For more example, see1,8.

Recently, a problem of finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a relatively nonexpansive mapping is studied 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 the corresponding convergence theorems under the weaker assumptions. More precisely, many restrictions 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**see7, Lemma 2*.*5. *LetCbe a nonempty closed convex subset of a strictly convex*
*and smooth Banach spaceEand letTbe a relatively quasi-nonexpansive mapping fromCinto itself.*
*ThenFTis closed and convex.*

**Lemma 1.2**see17, Proposition 5. *LetCbe a nonempty closed convex subset of a smooth, strictly*
*convex, and reflexive Banach spaceE. Then*

*φx,*Π*Cyφ*Π*Cy, y*≤*φx, y* 1.5

*for allx*∈*Candy*∈*E.*

**Lemma 1.3**see17. *LetEbe a smooth and uniformly convex Banach space and letr >*0*. Then*
*there exists a strictly increasing, continuous, and convex functionh*:0*,*2*r* → R*such thath*0 0

*and*

*hx*−*y*≤*φx, y* 1.6

*for allx, y*∈*Br* {*z*∈*E*:*z* ≤*r*}*.*

**Lemma 1.4**see17, Proposition 2. *LetEbe a smooth and uniformly convex Banach space and let*

{*xn*}*and*{*yn*}*be sequences ofEsuch that either*{*xn*}*or*{*yn*}*is bounded. If*lim*n*→ ∞*φxn, yn* 0*,*
*then*lim*n*→ ∞*xn*−*yn*0*.*

**Lemma 1.5**see2. *LetCbe a nonempty closed convex subset of a smooth, strictly convex, and*
*reflexive Banach spaceE, letx*∈*E,and letz*∈*C. Then*

*z* Π*Cx*⇐⇒*y*−*z, Jx*−*Jz*≤0*,* ∀*y*∈*C.* 1.7

**Lemma 1.6**see18. *LetEbe a uniformly convex Banach space and letr >*0*. Then there exists a*
*strictly increasing, continuous, and convex functiong* :0*,*2*r* → R*such thatg*0 0*and*

_{tx}_{1}_{−}_{t}_{}* _{y}*2

_{≤}

_{t}_{}

_{x}_{}2

_{1}

_{−}

_{t}_{}

*2*

_{y}_{−}

_{t}_{}

1−*tgx*−*y* 1.8

*for allx, y*∈*Br* *andt*∈0*,*1*.*

**Lemma 1.7.** *Let LetCbe a closed convex subset of a smooth Banach spaceE. LetT* *be a relatively*
*quasi-nonexpansive mapping fromEintoEand let*{*Si*}*Ni*1*be a family of relatively quasi-nonexpansive*

*mappings fromCinto itself such thatFT*∩*N _{i}*

_{}

_{1}

*FSi/*∅

*. The mappingU*:

*C*→

*Eis defined by*

*UxTJ*−1*N*

*i*1

*ωiαiJx* 1−*αiJSix* 1.9

*for allx* ∈ *Cand* {*ωi*}*,*{*αi*} ⊂ 0*,*1*,i* 1*,*2*, . . . , N* *such thatNi*1*ωi* 1*. Ifx* ∈ *Cand* *z* ∈

*FT*∩*N*

*i*1*FSi, then*

*φz, Ux*≤*φz, x.* 1.10

*Proof.* The proof of this lemma can be extracted from that ofLemma 1.8; so it is omitted.

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

**Lemma 1.8.** *LetC* *be a closed convex subset of a uniformly smooth Banach space* *E. Let* *r >* 0*.*
*Then, there exists a strictly increasing, continuous, and convex functiong*∗:0*,*6*r* → 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*{*Si*}*Ni*1 :*C* → *Csuch thatFT*∩
*N*

*i*1*FSi/*∅*,*

*N*

*i*1

*ωiαi*1−*αig*∗*Jz*−*JSiz*≤*φu, z*−*φu, Uz* 1.11

*for allz*∈*C*∩*Br* *andu*∈*FT*∩*Ni*1*FSi*∩*Br, where*

*UxTJ*−1*N*

*i*1

*ωiαiJx* 1−*αiJSix* 1.12

*x*∈*Cand*{*ωi*}*,*{*αi*} ⊂0*,*1*,i*1*,*2*, . . . , Nsuch thatNi*1*ωi* 1*.*

*Proof.* Let *r >* 0. From Lemma 1.6and *E*∗ is uniformly convex, then there exists a strictly
increasing, continuous, and convex function*g*∗:0*,*6*r* → Rsuch that*g*∗0 0 and

* _{tx}*∗

_{1}

_{−}

_{t}_{}

*∗2*

_{y}_{≤}

_{t}_{}

_{x}_{∗}

_{}2

_{1}

_{−}

_{t}_{}

*∗2*

_{y}_{−}

_{t}_{}

1−*tg*∗*x*∗−*y*∗ 1.13

for all*x*∗*, y*∗ ∈ {*z*∗ ∈ *E*∗ : *z*∗ ≤ 3*r*}and *t* ∈ 0*,*1. Let*T* : *E* → *E*and {*Si*}*Ni*1 : *C* → *C*
be relatively quasi-nonexpansive for all *i* 1*,*2*, . . . , N* such that*FT*∩*N _{i}*

_{}

_{1}

*FSi/*∅. For

*z*∈*C*∩*Br* and*u*∈*FT*∩*Ni*1*FSi*∩*Br*. It follows that

and hence*Siz* ≤3*r*. Consequently, for*i*1*,*2*, . . . , N*,

*αiJz* 1−*αiJSiz*2≤*αiJz*2 1−*αiJSiz*2−*αi*1−*αig*∗*Jz*−*JSiz.* 1.15

Then

*φu, Uz*≤*φ*

*u, J*−1*N*

*i*1

*ωiαiJz* 1−*αiJSiz*

*u*2_{−}_{2}

*u,* *N*

*i*1

*ωiαiJz* 1−*αiJSiz*

_{}*N*

*i*1

*ωiαiJz* 1−*αiJSiz*

2

≤ *N*

*i*1

*ωi*

*u*2_{−}_{2}_{}_{u, α}

*iJz* 1−*αiJSiz* *αiJz* 1−*αiJSiz*2

≤ *N*

*i*1

*ωi*

*u*2_{−}_{2}_{}_{u, α}

*iJz* 1−*αiJSiz* *αiJz*2 1−*αiJSiz*2

−*αi*1−*αig*∗*Jz*−*JSiz*

*N*

*i*1

*ωiαiφu, z* 1−*αiφu, Siz*−*αi*1−*αig*∗*Jz*−*JSiz*

≤*φu, z*− *N*

*i*1

*ωiαi*1−*αig*∗*Jz*−*JSiz.*

1.16

Thus

*N*

*i*1

*ωiαi*1−*αig*∗*Jz*−*JSiz*≤*φu, z*−*φu, Uz.* 1.17

**Lemma 1.9.** *Let* *C* *be a closed convex subset of a uniformly smooth and strictly convex Banach*
*spaceE. LetT* *be a relatively quasi-nonexpansive mapping fromEintoEand let*{*Si*}*Ni*1 *be a family*

*of relatively quasi-nonexpansive mappings fromC* *into itself such thatFT*∩*N _{i}*

_{}

_{1}

*FSi/*∅

*. The*

*mappingU*:

*C*→

*Eis defined by*

*UxTJ*−1*N*

*i*1

*ωiαiJx* 1−*αiJSix* 1.18

*for allx*∈*Cand*{*ωi*}*,*{*αi*} ⊂0*,*1*,i*1*,*2*, . . . , Nsuch thatNi*1*ωi*1*. Then, the following hold:*

1*FU* *FT*∩*N _{i}*

_{}

_{1}

*FSi,*

*Proof.* 1Clearly,*FT*∩*N _{i}*

_{}

_{1}

*FSi*⊂

*FU*. We want to show the reverse inclusion. Let

*z*∈

*FU*and*u*∈*FT*∩*N _{i}*

_{}

_{1}

*FSi*. Choose

*r* :max{*u,z,S*1*z,S*2*z, . . . ,Smz*}*.* 1.19

FromLemma 1.8, we have

*N*

*i*1

*ωiαi*1−*αig*∗*Jz*−*JSiz* 0*.* 1.20

From*ωiαi*1−*αi>*0 for all*i*1*,*2*, . . . , N*and by the properties of*g*∗, we have

*JzJSiz* 1.21

for all*i*1*,*2*, . . . , N*. From*J*is one to one, we have

*zSiz* 1.22

for all*i*1*,*2*, . . . , N*. Consider

*zUzTJ*−1*N*

*i*1

*ωiαiJz* 1−*αiJSiz* *Tz.* 1.23

Thus*z*∈*FT*∩*N _{i}*

_{}

_{1}

*FSi*.

2It follows directly from the above discussion.

**2. Weak Convergence Theorem**

**Theorem 2.1.** *LetCbe a nonempty closed convex subset of a uniformly smooth and uniformly convex*
*Banach spaceE. Let*{*Tn*}∞*n*1:*E* → *Cbe a family of relatively quasi-nonexpansive mappings and let*

{*Si*}*Ni*1:*C* → *Cbe a family of relatively quasi-nonexpansive mappings such thatF*:
_{∞}

*n*1*FTn*∩
*N*

*i*1*FSi/*∅*. Let the sequence*{*xn*}*be generated byx*1∈*C,*

*xn*1 *TnJ*−1
*N*

*i*1

*ωn,iαn,iJxn* 1−*αn,iJSixn* 2.1

*for anyn* ∈ N*,*{*ωn,i*}*,*{*αn,i*} ⊂ 0*,*1*for alln* ∈N*,i* 1*,*2*, . . . , N* *such thatNi*1*ωn,i* 1*for all*

*Proof.* Let*u*∈∞_{n}_{}_{1}*FTn*∩*Ni*1*FSi*. Put

*Un* *TnJ*−1
*N*

*i*1

*ωn,iαn,iJ* 1−*αn,iJSi.* 2.2

FromLemma 1.7, we have

*φu, xn*1 *φu, Unxn*≤*φu, xn.* 2.3

Therefore lim*n*→ ∞*φu, xn*exists. This implies that{*φu, xn*}*,*{*xn*}and{*Sixn*}are bounded

for all*i*1*,*2*, . . . , N*.

Let*yn*≡Π*Fxn*. From2.3and*m*∈N, we have

*φyn, xnm*≤*φyn, xn.* 2.4

Consequently,

*φyn, ynmφynm, xnm*≤*φyn, xnm*≤*φyn, xn.* 2.5

In particular,

*φyn*1*, xn*1

≤*φyn, xn.* 2.6

This implies that lim*n*→ ∞*φyn, xn*exists. This together with the boundedness of{*xn*}gives

*r* : sup_{n}_{∈}_{N}*yn* *<* ∞. UsingLemma 1.3, there exists a strictly increasing, continuous, and

convex function*h*:0*,*2*r* → Rsuch that*h*0 0 and

*hyn*−*ynm*≤*φyn, ynm*≤*φyn, xn*−*φynm, xnm.* 2.7

Since{*φyn, xn*}is a convergent sequence, it follows from the properties of*g*that{*yn*}is a

Cauchy sequence. Since*F*is closed, there exists*z*∈*F*such that*yn* → *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 of
mappings{*Tn*}∞*n*1 :*C* → *E*with

_{∞}

*n*1*FTn/*∅, we say that{*Tn*}satisfies the NST-condition

19if for each bounded sequence{*zn*}in*C*,

lim

*n*→ ∞*zn*−*Tnzn*0 implies*ωw*{*zn*} ⊂
∞

*n*1

*FTn,* 2.8

**Theorem 2.2.** *Let* *C* *be a nonempty closed convex subset of a uniformly smooth and uniformly*
*convex Banach spaceE. Let*{*Tn*}∞*n*1 :*E* → *Cbe a family of relatively quasi-nonexpansive mappings*

*satisfying NST-condition and let*{*Si*}*Ni*1 : *C* → *Cbe a family of relatively nonexpansive mappings*

*such thatF*:∞_{n}_{}_{1}*FTn*∩*Ni*1*FSi/*∅*and suppose that*

*φu, Tnx* *φTnx, x*≤*φu, x* 2.9

*for allu*∈∞_{n}_{}_{1}*FTn,n*∈N*andx*∈*E. Let the sequence*{*xn*}*be generated byx*1∈*C,*

*xn*1 *TnJ*−1
*N*

*i*1

*ωn,iαn,iJxn* 1−*αn,iJSixn* 2.10

*for anyn* ∈ N*,*{*ωn,i*}*,*{*αn,i*} ⊂ 0*,*1*for alln* ∈N*,i* 1*,*2*, . . . , N* *such thatNi*1*ωn,i* 1*for all*

*n*∈N*,*lim inf*n*→ ∞*ωn,iαn,i*1−*αn,i>*0*for alli*1*,*2*, . . . , N. IfJis weakly sequentially continuous,*
*then*{*xn*}*converges weakly toz*∈*F, wherez*lim*n*→ ∞Π*Fxn.*

*Proof.* Let*u* ∈ *F*. FromTheorem 2.1, lim*n*→ ∞*φu, xn*exists and hence{*xn*}and {*Sixn*}are

bounded for all*i*1*,*2*, . . . , N*. Let

*r*sup

*n*∈N{*xn,S*1*xn,S*2*xn, . . . ,SNxn*}*.* 2.11

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

0*,*2*r* → Rsuch that*g*∗0 0 and

*N*

*i*1

*ωn,iαn,i*1−*αn,ig*∗*Jxn*−*JSixn*≤*φu, xn*−*φu, xn*1*.* 2.12

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

*ωn,iαn,i*1−*αn,ig*∗*Jxn*−*JSixn*≤*φu, xn*−*φu, xn*1*.* 2.13

Hence,

∞

*n*1

*ωn,iαn,i*1−*αn,ig*∗*Jxn*−*JSixn<*∞ 2.14

for all *i* 1*,*2*, . . . , N*. Since lim inf*n*→ ∞*ωn,iαn,i*1−*αn,i* *>* 0 for all*i* 1*,*2*. . . , N* and the

properties of*g*, we have

lim

for all*i* 1*,*2*. . . , N*. Since*J*−1 _{is uniformly norm-to-norm continuous on bounded sets, we}
have

lim

*n*→ ∞*xn*−*Sixn*0 2.16

for all*i*1*,*2*. . . , N*. Since{*xn*}is bounded, there exists a subsequence{*xnk*}of{*xn*}such that

*xnk* * z*∈*C*. Since*Si*is relatively nonexpansive,*z*∈*FSi* *FSi*for all*i*1*,*2*. . . , N*.

We show that*z*∈∞_{n}_{}_{1}*FTn*. Let

*yn* *J*−1
*N*

*i*1

*ωn,iαn,iJxn* 1−*αn,iJSixn.* 2.17

We note from2.15that

*N*

*i*1

*ωn,iαn,iJxn* 1−*αn,iJSixn*−*Jxn*

≤

*N*

*i*1

*ωn,i*1−*αn,iJSixn*−*Jxn* −→0*.* 2.18

Since*J*−1_{is uniformly norm-to-norm continuous on bounded sets, it follows that}

lim

*n*→ ∞*yn*−*xnn*lim→ ∞
*J*−1

_{N}

*i*1

*ωn,iαn,iJxn* 1−*αn,iJSixn*

−*J*−1_{Jx}

*n*

0*.* 2.19

Moreover, by2.9and the existence of lim*n*→ ∞*φu, xn*, we have

*φTnyn, yn*≤*φu, yn*−*φu, Tnyn*

*φ*

*u, J*−1*N*

*i*1

*ωn,iαn,iJxn* 1−*αn,iJSixn*

−*φu, xn*1

≤*φu, xn*−*φu, xn*1−→0*.*

2.20

It follows fromLemma 1.4that lim*n*→ ∞*Tnyn*−*yn* 0. From2.19and*xnk* * z*, we have

*ynk* * z*. Since{*Tn*}satisfies NST-condition, we have*z*∈

_{∞}

*n*1*FTn*. Hence*z*∈*F*.
Let*zn* Π*Fxn*. FromLemma 1.5and*z*∈*F*, we have

*znk* −*z, Jxnk*−*Jznk* ≥0*.* 2.21

FromTheorem 2.1, we know that*zn* → *z*∈*F*. Since*J*is weakly sequentially continuous, we

have

Moreover, since*J*is monotone,

*z*_{−}_{z, Jz}_{−}_{Jz}_{≤}_{0}_{.}_{}_{2.23}_{}

Then

*z*_{−}_{z, Jz}_{−}_{Jz}_{}_{0}_{.}_{}_{2.24}_{}

Since*E* is strictly convex, *z* *z*. This implies that *ωw*{*xn*} {*z*} and hence *xn* * z*

lim*n*→ ∞Π*Fxn*.

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. This
problem is extensively studied in11, 14–16. Let*C* be a subset of a Banach space *E*and
let*f*:*C*×*C* → Rbe 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 EP*f*, that is

*x*∈EP*f* iﬀ*fx, y*≥0 ∀*y*∈*C.* 2.25

To solve the equilibrium problem, we usually assume that a bifunction *f* satisfies the
following conditions*C*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*;

A3for all*x, y, z*∈*C*, lim sup_{t}_{↓}_{0}*ftz* 1−*tx, y*≤*fx, y*;

A4for 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.** *LetCbe a nonempty closed convex subset of a Banach spaceE. Letfbe a bifunction from*

*C*×*C* → R*satisfying (A1)–(A4). Suppose thatp*∈*C. Thenp*∈*EPfif and only iffy, p*≤0*for*
*ally*∈*C.*

*Proof.* Let*p*∈EP*f*, then*fp, y*≥0 for all*y*∈*C*. FromA2, we get that*fy, p*≤ −*fp, y*≤
0 for all*y*∈*C*.

Conversely, assume that*fy, p*≤0 for all*y*∈*C*. For any*y*∈*C*, let

*xtty* 1−*tp,* for*t*∈0*,*1*.* 2.26

Then*fxt, p*≤0 and hence

So*fxt, y*≥0 for all*t*∈0*,*1. FromA3, we have

0≤lim sup

*t*↓0

*fty* 1−*tp, y*≤*fp, y* ∀*y*∈*C.* _{}_{2.28}_{}

Hence*p*∈EP*f*.

Takahashi and Zembayashi proved the following important result.

**Lemma 2.4**see15, Lemma 2*.*8. *LetCbe a nonempty closed convex subset of a uniformly smooth,*
*strictly convex and reflexive Banach spaceE. Letfbe a bifunction fromC*×*C* → R*satisfying (A1)–*
*(A4). Forr >*0*andx*∈*E, define a mappingTr* :*E* → *Cas follows:*

*Trx*

*z*∈*C*:*fz, y*1

*r*

*y*−*z, Jz*−*Jx*≥0∀*y*∈*C*

2.29

*for allx*∈*E. Then, the following hold:*

1*Tr* *is single-valued;*

2*Tr* *is a firmly nonexpansive-type mapping [20], that is, for allx, y*∈*E*

*Trx*−*Try, JTrx*−*JTry*≤*Trx*−*Try, Jx*−*Jy*; 2.30

3*FTr* *EPf;*

4*EPfis closed and convex.*

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

**Corollary 2.5**see15, Theorem 4*.*1. *LetCbe a nonempty closed convex subset of a uniformly*
*smooth and uniformly convex Banach spaceE. Letfbe a bifunction fromC*×*Cto*R*satisfying (A1)–*
*(A4) and letSbe a relatively nonexpansive mapping fromCinto itself such thatFS*∩*EPf/*∅*.*
*Let the sequence*{*xn*}*be generated byu*1 ∈*E,*

*xn*∈*Csuch thatfxn, y _{r}*1

*n*

*y*−*xn, Jxn*−*Jun*≥0 ∀*y*∈*C,*

*un*1 *J*−1*αnJxn* 1−*αnJSxn*

2.31

*for everyn* ∈ N*,*{*αn*} ⊂ 0*,*1*satisfying*lim inf*n*→ ∞*αn*1−*αn* *>* 0*and* {*rn*} ⊂ *a,*∞*for some*

*a >*0*. IfJ* *is weakly sequentially continuous, then*{*xn*}*converges weakly toz* ∈Π*FS*∩*EPf, where*

*Proof.* Put *Tn* ≡ *Trn* where *Trn* is defined by Lemma 2.4. Then

_{∞}

*n*1*FTn* EP*f*. By
reindexing the sequences {*xn*} and {*un*} of this iteration, we can apply Theorem 2.2 by

showing that the family {*Tn*} satisfies the condition 2.9and NST-condition. It is proved

in15, Lemma 2*.*9that

*φu, Tnx* *φTnx, x*≤*φu, x* ∀*x*∈*E, u*∈

∞

*n*1

*FTn.* 2.32

To see that {*Tn*} satisfies NST-condition, let {*zn*} be a bounded sequence in *C* such that

lim*n*→ ∞*zn*−*Tnzn* 0 and*p* ∈ *ωw*{*zn*}. Suppose that there exists a subsequence{*znk*} of
{*zn*}such that*znk* * p*. Then*Tnkznk* * p* ∈*C*. Since*J* is uniformly continuous on bounded

sets and*rnk* ≥*a*, we have

lim

*k*→ ∞
1

*rnk*

*Jznk*−*JTnkznk*0*.* 2.33

From the definition of*Tr _{nk}*, we have

*fTnkznk, y*

* _{r}*1

*nk*

*y*−*Tnkznk, JTnkznk* −*Jznk*

≥0 ∀*y*∈*C.* 2.34

Since

*fy, Tnkznk*

≤ −*fTnkznk, y*

≤ 1

*rnk*

*y*−*Tnkznk, JTnkznk*−*Jznk*

≤ * _{r}*1

*nk*

_{y}_{−}_{T}_{n}_{k}_{z}_{n}_{k}_{}_{JT}_{n}_{k}_{z}_{n}_{k}_{−}_{Jz}_{n}_{k}_{}

2.35

and*f*is lower semicontinuous and convex in the second variable, we have

*fy, p*≤lim inf

*k*→ ∞ *f*

*y, Tnkznk*

≤0*.* _{}_{2.36}_{}

Thus*fy, p*≤0 for all*y*∈*C*. FromLemma 2.3, we have*p*∈EP*f*. Then{*Tn*}satisfies the

NST-condition. FromTheorem 2.2where*N*1,{*xn*}converges weakly to*z*∈*FTn*∩*FS*

EP*f*∩*FS*, where*z*lim*n*→ ∞ΠEP*f*∩*FSxn*.

**Corollary 2.6**see11, Theorem 3*.*5. *LetCbe a nonempty and closed convex subset of a uniformly*
*convex and uniformly smooth Banach spaceE. Letf* *be a bifunction fromC*×*Cto*R*satisfies (A1)–*
*(A4) and letT, S*:*C* → *Cbe two relatively nonexpansive mappings such thatF* :*FT*∩*FS*∩

*EPf/*∅*. Let the sequence*{*xn*}*be generated by the following manner:*

*xn*∈*Csuch thatfxn, y _{r}*1

*n*

*y*−*xn, Jxn*−*Jun*≥0 ∀*y*∈*C,*

*un*1*J*−1

*αnJxnβnJTxnγnJSxn* ∀*n*≥1*.*

2.37

*Assume that*{*αn*}*,*{*βn*}*,and*{*γn*}*are three sequences in*0*,*1*satisfying the following restrictions:*

a*αnβnγn*1*;*

blim inf*n*→ ∞*αnβn>*0*,*lim inf*n*→ ∞*αnγn>*0*;*

c{*rn*} ⊂*a,*∞*for somea >*0*.*

*IfJis weakly sequentially continuous, then*{*xn*}*converges weakly toz*∈*F, wherez*lim*n*→ ∞Π*Fxn.*

The following result also follows fromTheorem 2.2.

**Corollary 2.7**see9, Theorem 5*.*3. *LetEbe a uniformly smooth and uniformly convex Banach*
*space and letCbe a nonempty closed convex subset ofE. Let*{*Si*}*Ni*1 *be a finite family of relatively*

*nonexpansive mappings fromCinto itself such thatF* *N _{i}*

_{}

_{1}

*FSiis a nonempty and let*{

*αn,i*:

*n, i* ∈ N*,* 1 ≤ *i*≤ *N*} ⊂ 0*,*1*and*{*ωn,i* : *n, i* ∈ N*,* 1 ≤ *i* ≤ *N*} ⊂ 0*,*1*be sequences such that*

lim inf*n*→ ∞*αn,i*1−*αn,i>*0*and*lim inf*n*→ ∞*ωn,i>*0*for alli*∈ {1*,*2*, . . . , N*}*andNi*1*ωn,i*1*for*

*alln*∈N*. LetUnbe a sequence of mappings defined by*

*Unx* Π*CJ*−1
*N*

*i*1

*ωn,iαn,iJx* 1−*αn,iJSix* 2.38

*for allx*∈*Cand let the sequence*{*xn*}*be generated byx*1*x*∈*Cand*

*xn*1*Unxn* *n*1*,*2*, . . ..* 2.39

*Then the following hold:*

1*the sequence*_{} {*xn*} *is bounded and each weak subsequential limit of* {*xn*} *belongs to*
*N*

*i*1*FSi;*

2*if the duality mapping* *J* *from* *E* *into* *E*∗ *is weakly sequentially continuous, then* {*xn*}
*converges weakly to the strong limit of*{Π*Fxn*}*.*

*Proof.* Since Π*C* is relatively nonexpansive, the family {Π*C*} satisfies the NST-condition.

Moreover,*F*Π*C* *C*and

*φx,*Π*Cyφ*Π*Cy, y*≤*φx, y* ∀*y*∈*E, x*∈*C.* 2.40

**3. Strong Convergence Theorem**

In this section, we prove strong convergence of an iterative sequence generated by the hybrid method in mathematical programming. We start with the following useful common tools.

**Lemma 3.1.** *LetCbe a nonempty closed convex subset of a uniformly smooth and uniformly convex*
*Banach space* *E. Let* {*Tn*}∞*n*1 : *E* → *E* *and* {*Si*}*Ni*1 : *C* → *C* *be families of relatively *

*quasi-nonexpansive mappings such thatF*:∞_{n}_{}_{1}*FTn*∩*Ni*1*FSi/*∅*, and*

*φu, Tnx* *φTnx, x*≤*φu, x* 3.1

*for allu*∈∞_{n}_{}_{1}*FTn,n*∈N*andx*∈*E. Let*{*xn*} ⊂*Cbe such that*{*xn*}*and*{*Sixn*}*are bounded for*
*alli*1*,*2*, . . . , N, and*

*ynJ*−1
*N*

*i*1

*ωn,iαn,iJxn* 1−*αn,iJSixn,*

*unTnyn,*

3.2

*where*{*ωn,i*}*,*{*αn,i*} ⊂ 0*,*1*for alln* ∈N*andi* 1*,*2*, . . . , N* *satisfyNi*1*ωn,i* 1*for alln* ∈ N*,*
lim inf*n*→ ∞*ωn,i*1−*αn,i>*0*for alli*1*,*2*, . . . , Nand*lim*n*→ ∞*xn*−*un*0*. Then the following*
*statements hold:*

1lim*n*→ ∞*φu, xn*−*φu, un* 0*for allu*∈*C,*

2lim*n*→ ∞*un*−*yn*0*,*

3*ωw*{*xn*}*ωw*{*yn*}*,*

4*if*lim*n*→ ∞*xn*1−*xn*0*, then*lim*n*→ ∞*xn*−*Sixn*0*for alli*1*,*2*, . . . , N,*

5*ifxn* → *z, thenun* → *zandyn* → *z.*

*Proof.* 1 Since lim*n*→ ∞*xn* − *un* 0 and *J* is uniformly norm-to-norm continuous on

bounded sets,

lim

*n*→ ∞*Jxn*−*Jun*0*.* 3.3

We note here that{*un*}is also bounded. For any*u*∈*C*, we have

*φu, xn*−*φu, unxn*2− *un*2−2*u, Jun*−*Jxn*

≤*xn*2− *u*22|*u, Jun*−*Jxn* |

≤ *xn*−*unxnun* 2*uJun*−*Jxn* −→0*.*

3.4

2Let*u*∈*F*. Using3.1and the relative quasi-nonexpansiveness of each*Tn*, we have

ByLemma 1.4and the boundedness of{*un*}, we have

lim

*n*→ ∞*un*−*yn*0*.* 3.6

3Since

*xn*−*yn*≤ *xn*−*unun*−*ynxn*−*unTnyn*−*yn*−→0*,* 3.7

we have*ωw*{*xn*}*ωw*{*yn*}.

4 Assume that lim*n*→ ∞*xn*1 −*xn* 0. From lim*n*→ ∞*xn* −*yn* 0, we get that

lim*n*→ ∞*xn*1−*yn*0. Since*J*is uniformly norm-to-norm continuous on bounded sets, we

have

lim

*n*→ ∞*Jxn*1−*Jxnn*lim→ ∞*Jxn*1−*Jyn*0*.* 3.8

So,

*Jxn*1−*Jyn*
*Jxn*1−

*N*

*i*1

*ωn,iαn,iJxn* 1−*αn,iJSixn*

≥ *N*

*i*1

*ωn,i*1−*αn,iJxn*1−*JSixn* −*ωn,iαn,iJxn*1−*Jxn.*

3.9

From3.8, we have

*N*

*i*1

*ωn,i*1−*αn,iJxn*1−*JSixn* ≤*Jxn*1−*Jyn*
*N*

*i*1

*ωn,iαn,iJxn*1−*Jxn* −→0*.* 3.10

It follows from lim inf*n*→ ∞*ωn,i*1−*αn,i>*0 for all*i*1*,*2*, . . . , N*that

lim

*n*→ ∞*Jxn*1−*JSixn*0 3.11

for all*i*1*,*2*, . . . , N*. Since*J*−1 _{is uniformly norm-to-norm continuous on bounded sets and}
lim*n*→ ∞*xn*1−*xn*0, we have

lim

*n*→ ∞*xn*−*Sixn*0 3.12

for all*i*1*,*2*, . . . , N*, as desired.

5Assume that*xn* → *z*. From the assumption and2, we have

lim

*n*→ ∞*xn*−*unn*lim→ ∞*un*−*yn*0*.* 3.13

**Lemma 3.2**see21, Lemma 2*.*4. *LetFbe a closed convex subset of a strictly convex, smooth and*
*reflexive Banach spaceEsatisfying Kadec-Klee property. Letx*∈*Eand*{*xn*}*be a sequence inEsuch*
*thatωw*{*xn*} ⊂*Fandφxn, x*≤*φ*Π*Fx, xfor alln*∈N*. Thenxn* → *z* Π*Fx.*

Recall that a Banach space *E* satisfies Kadec–Klee property if whenever {*un*} is a

sequence in*E*with*xn x*and*xn* → *x*, it follows that*xn* → *x*.

**3.1. The CQ-Method**

**3.1. The CQ-Method**

**Theorem 3.3.** *LetCbe a nonempty closed convex subset of a uniformly smooth and uniformly convex*
*Banach spaceE. Let*{*Tn*}∞*i*1:*E* → *Ebe a family of relatively quasi-nonexpansive mappings satisfying*

*NST-condition and let*{*Si*}*Ni*1 :*C* → *Cbe a family of relatively nonexpansive mappings such that*

*F*:∞_{n}_{}_{1}*FTn*∩*Ni*1*FSi/*∅*, and*

*φu, Tnx* *φTnx, x*≤*φu, x* 3.14

*for allu*∈∞_{n}_{}_{1}*FTn,n*∈N*andx*∈*E. Let the sequence*{*xn*}*be generated by*

*x*1*x*∈*C,*

*unTnJ*−1
*N*

*i*1

*ωn,iαn,iJxn* 1−*αn,iJSixn,*

*Cn* *z*∈*C*:*φz, un*≤*φz, xn,*

*Qn*{*z*∈*C*:*xn*−*z, Jx*−*Jxn* ≥0}*,*

*xn*1 Π*Cn*∩*Qnx*

3.15

*for everyn*∈N*,*{*ωn,i*}*,*{*αn,i*} ⊂0*,*1*for alln*∈N*andi*1*,*2*, . . . , NsatisfyingNi*1*ωn,i* 1*for*

*alln*∈N*,*lim inf*n*→ ∞*ωn,i*1−*αn,i>*0*for alli*1*,*2*, . . . , N. Then*{*xn*}*converges strongly to*Π*Fx.*

*Proof.* The proof is broken into 4 steps.

*Step 1*{*xn*}is well defined. First, we show that*Cn*∩*Qn*is closed and convex. Clearly,*Qn*is

closed and convex. Since

*φz, un*≤*φz, xn*⇐⇒ *un*2− *xn*2−2*z, Jun*−*Jxn* ≤0*,* 3.16

then*Cn*is closed and convex. Thus*Cn*∩*Qn*is closed and convex.

We next show that*F* ⊂*Cn*∩*Qn*. Let*u*∈*F*. Then, fromLemma 1.7,

*φu, un*≤*φu, xn.* 3.17

Next, we show by induction that*F*⊂*Cn*∩*Qn*for all*n*∈N. Since*Q*1*C*, we have

*F*⊂*C*1∩*Q*1*.* 3.18

Suppose that*F*⊂*Ck*∩*Qk*for some*k*∈N. From*xk*1 Π*Ck*∩*Qkx*∈*Ck*∩*Qk*and the definition

of the generalized projection, we have

*xk*1−*z, Jx*−*Jxk*1 ≥0 3.19

for all*z*∈*Ck*∩*Qk*. From*F* ⊂*Ck*∩*Qk*,

*xk*1−*p, Jx*−*Jxk*1

≥0 3.20

for all*p*∈*F*. Hence*F*⊂*Qk*1, and we also have*F* ⊂*Ck*1∩*Qk*1. So, we have∅*/F*⊂*Cn*∩*Qn*

for all*n*∈Nand hence the sequence{*xn*}is well defined.

*Step 2* *ωw*{*xn*} ⊂ *Ni*1*FSi*. From the definition of *Qn*, we have *xn* Π*Qnx*. Using

Lemma 1.2, we get

*φxn, x* *φ*Π*Qnx, x*

≤*φu, x*−*φu,*Π*Qnx*

≤*φu, x* 3.21

for all*u*∈*Qn*. In particular, since*xn*1∈*Qn*andΠ*Fx*∈*F*⊂*Qn*,

*φxn, x*≤*φxn*1*, x,* 3.22

*φxn, x*≤*φ*Π*Fx, x* 3.23

for all*n* ∈N. This implies that lim*n*→ ∞*φxn, x*exists and{*xn*}is bounded. Moreover, from

3.21and*xn*1∈*Qn*,

*φxn, x*≤*φxn*1*, x*−*φxn*1*, xn.* 3.24

Hence

*φxn*1*, xn*≤*φxn*1*, x*−*φxn, x*−→0*.* 3.25

It follows from*xn*1 Π*Cn*∩*Qnx*∈*Cn*that

*φxn*1*, un*≤*φxn*1*, xn*−→0*.* 3.26

From3.25,3.26, andLemma 1.4, we have

lim

So lim*n*→ ∞*xn*−*un*0. UsingLemma 3.14, we get that

lim

*n*→ ∞*xn*−*Sixn*0 3.28

for all*i*1*,*2*, . . . , N*. Since each*Si*is relatively nonexpansive,

*ωw*{*xn*} ⊂
*N*

*i*1

*FSi*
*N*

*i*1

*FSi.* 3.29

*Step 3* *ωw*{*xn*} ⊂ ∞*n*1*FTn*. Let *yn* *J*−1
*N*

*i*1*ωn,iαn,iJxn* 1 − *αn,iJSixn*. From
Lemma 3.12, we have

lim

*n*→ ∞*Tnyn*−*yn*0*,* 3.30

and*ωw*{*xn*}*ωw*{*yn*}. It follows from NST-condition that*ωw*{*xn*}*ωw*{*yn*} ⊂∞*n*1*FTn*.

*Step 4xn* → Π*Fx*. From Steps2and3, we have*ωw*{*xn*} ⊂ *F*. The conclusion follows by

Lemma 3.2and3.23.

We applyTheorem 3.3and the proof ofCorollary 2.5 and then obtain the following result.

**Corollary 3.4.** *LetC,E,f,Sbe as inCorollary 2.5. Let the sequence*{*xn*}*be generated by*

*x*1*x*∈*C,*

*ynJ*−1*αnJxn* 1−*αnJSxn,*

*un*∈*Csuch thatfun, y _{r}*1

*n*

*y*−*un, Jun*−*Jyn*≥0 ∀*y*∈*C,*

*Cn* *z*∈*C*:*φz, un*≤*φz, xn,*

*Qn*{*z*∈*C*:*xn*−*z, Jx*−*Jxn* ≥0}*,*

*xn*1 Π*Cn*∩*Qnx*

3.31

*for everyn* ∈ N*,*{*αn*} ⊂ 0*,*1*satisfying*lim sup_{n}_{→ ∞}*αn* *<* 1*and*{*rn*} ⊂*a,*∞*for somea >* 0*.*
*Then,*{*xn*}*converges strongly to*Π*FS*∩*EPfx, where*Π*FS*∩*EPfis the generalized projection ofE*

*ontoFS*∩*EPf.*

*Remark 3.5.* Corollary 3.4improves the restriction on{*αn*}of15, Theorem 3*.*1. In fact, it is

**3.2. The Monotone CQ-Method**

**3.2. The Monotone CQ-Method**

Let*C*be a closed subset of a Banach space*E*. Recall that a mapping*T* :*C* → *C*is closed if for
each{*xn*}in*C*, if*xn* → *x*and*Txn* → *y*, then*Tx* *y*. A family of mappings{*Tn*}:*C* → *E*

with∞_{n}_{}_{1}*FTn/*∅is said to satisfy the∗-condition if for each bounded sequence{*zn*}in*C*,

lim

*n*→ ∞*zn*−*Tnzn*0*, zn*−→*z*imply*z*∈
∞

*n*1

*FTn.* 3.32

*Remark 3.6.* 1If{*Tn*}satisfies NST-condition, then{*Tn*}satisfies∗-condition.

2If*Tn*≡*T* and*T*is closed, then{*Tn*}satisfies∗-condition.

**Theorem 3.7.** *Let* *C* *be a nonempty closed convex subset of a uniformly smooth and uniformly*
*convex Banach spaceE. Let*{*Tn*}∞*n*1 :*E* → *Ebe a family of relatively quasi-nonexpansive mappings*

*satisfying*∗*-condition and let*{*Si*}*Ni*1 :*C* → *Cbe a family of closed relatively quasi-nonexpansive*

*mappings such thatF*:∞_{n}_{}_{1}*FTn*∩*Ni*1*FSi/*∅*, and*

*φu, Tnx* *φTnx, x*≤*φu, x* 3.33

*for allu*∈∞_{n}_{}_{1}*FTn,n*∈N*,andx*∈*E. Let the sequence*{*xn*}*be generated by*

*x*0*x*∈*C,* *Q*0*C,*

*unTnJ*−1
*N*

*i*1

*ωn,iαn,iJxn* 1−*αn,iJSixn,*

*C*0

*z*∈*C*:*φz, u*0≤*φz, x*0

*,*

*Cnz*∈*Cn*−1∩*Qn*−1:*φz, un*≤*φz, xn,*

*Qn*{*z*∈*Cn*−1∩*Qn*−1:*xn*−*z, Jx*−*Jxn* ≥0}*,*

*xn*1 Π*Cn*∩*Qnx*

3.34

*for everyn*∈N*,*{*ωn,i*}*,*{*αn,i*} ⊂0*,*1*satisfyingiN*1*ωn,i*1*and*lim inf*n*→ ∞*ωn,i*1−*αn,i>*0*for*

*alli*1*,*2*, . . . , N. Then*{*xn*}*converges strongly to*Π*Fx.*

*Proof.*

*Step 1* {*xn*} is well defined. This step is almost the same as Step 1 of the proof of

Theorem 3.3, so it is omitted.

*Step 2*{*xn*}is a Cauchy sequence in*C*. We can follow the proof ofTheorem 3.3and conclude

that

lim

exists. Moreover, as*xnm*∈*Qn*for all*n, m*and*xn* Π*Qnx*,

*φxnm, xn* *φxnm,*Π*Qnx*

≤*φxnm, x*−*φ*Π*Qnx, x*

*φxnm, x*−*φxn, x.*

3.36

Since {*xn*} is bounded, it follows from Lemma 1.3 that there exists a strictly increasing,

continuous, and convex function*h*such that*h*0 0 and

*hxnm*−*xn*≤*φxnm, x*−*φxn, x.* 3.37

Since lim*n*→ ∞*φxn, x*exists, we have that{*xn*}is a Cauchy sequence. Therefore,*xn* → *z*for

some*z*∈*C*.

*Step 3z*∈*N _{i}*

_{}

_{1}

*FSi*. Since

*xn*1 Π

*Cn*∩

*Qnx*∈

*Cn*, we have

*φxn*1*, un*≤*φxn*1*, xn*−→*φz, z* 0*.* 3.38

ByLemma 1.4and the boundedness of{*xn*}, we have

lim

*n*→ ∞*xn*1−*un*0*.* 3.39

So, we have lim*n*→ ∞*xn*−*un*0. UsingLemma 3.14, we get that

lim

*n*→ ∞*xn*−*Sixn*0 3.40

for all*i*1*,*2*, . . . , N*. Since each*Si*is closed,*z*∈*Ni*1*FSi*.

*Step 4z*∈∞_{n}_{}_{1}*FTn*. Let*yn* *J*−1*Ni*1*ωn,iαn,iJxn* 1−*αn,iJSixn*. FromLemma 3.12,
we have lim*n*→ ∞*yn*−*Tnyn*0 and*yn* → *z*. It follows from∗-condition that*z*∈*n*∞1*FTn*.

*Step 5xn* → Π*Fx*. From Steps3and4, we have*ωw*{*xn*} ⊂ *F*. The conclusion follows by

Lemma 3.2and3.23.

**Corollary 3.8**see12, Theorem 3*.*1. *LetCbe a nonempty closed convex subset of a uniformly*
*convex and uniformly smooth real Banach spaceE. Let* *T* : *C* → *Cbe a closed relatively *
*quasi-nonexpansive mapping such that* *FT/*∅*. Assume that* {*αn*} *is a sequence in* 0*,*1 *such that*

lim sup_{n}_{→ ∞}*αn* *<*1*. Define a sequence*{*xn*}*inCby the following algorithm:*

*x*0∈*Cchosen arbitrarily,*

*ynJ*−1*αnJxn* 1−*αnJTxn,*

*Cnz*∈*Cn*−1∩*Qn*−1:*φ*

*z, yn*≤*φz, xn,*

*C*0

*z*∈*C*:*φz, y*0

≤*φz, x*0

*,*

*Qn*{*z*∈*Cn*−1∩*Qn*−1:*xn*−*z, Jx*0−*Jxn* ≥0}*,*

*Q*0*C,*

*xn*1 Π*Cn*∩*Qnx*0*.*

3.41

*Then*{*xn*}*converges strongly to*Π*FTx*0*.*

Letting*Tn*identity and*N*2 yield the following result.

**Corollary 3.9**see13, Theorem 3*.*1. *LetCbe a nonempty closed convex subset of a uniformly*
*convex and uniformly smooth real Banach spaceE. LetT, Sbe two closed relatively quasi-nonexpansive*
*mappings from* *Cinto itself such thatF* : *FT*∩*FS/*∅*. Define a sequence*{*xn*}*in* *Cbe the*
*following algorithm:*

*x*0∈*Cchosen arbitrarily,*

*zn* *J*−1

*β*1

*n* *Jxnβn*2*JTxnβn*3*JSxn*

*,*

*ynJ*−1*αnJxn* 1−*αnJzn,*

*C*0

*z*∈*C*:*φz, y*0

≤*φz, x*0

*,*

*Cnz*∈*Cn*−1∩*Qn*−1:*φ*

*z, yn*≤*φz, xn,*

*Qn*{*z*∈*Cn*−1∩*Qn*−1:*xn*−*z, Jx*0−*Jxn* ≥0}*,*

*Q*0*C,*

*xn*1 Π*Cn*∩*Qnx*0

3.42

*with the conditions:βn*1*, βn*2*, βn*3∈0*,*1*withβn*1*βn*2*βn*31*and*

1lim inf*n*→ ∞*βn*1*βn*2*>*0*;*

2lim inf*n*→ ∞*βn*1*βn*3*>*0*;*

30≤*αn*≤*α <*1*for someα*∈0*,*1*.*

*Remark 3.10.* UsingTheorem 3.7, we can show that the conclusion ofCorollary 3.9remains
true under the more general restrictions on{*αn*}*,*{*βn*1}*,*{*βn*2}*,*and{*βn*3}:

1*αn, βn*1∈0*,*1are arbitrary;

2lim inf*n*→ ∞*βn*2 *>*0 and lim inf*n*→ ∞*βn*3*>*0.

**3.3. The Shrinking Projection Method**

**3.3. The Shrinking Projection Method**

**Theorem 3.11.** *LetC, E,* {*Tn*}∞*n*1*,* {*Si*}*Ni*1*be as inTheorem 3.7. Let the sequence*{*xn*}*be generated*

*by*

*x*0∈*Echosen arbitrarily,*

*C*1*C,*

*x*1 Π*C*1*x*0*,*

*unTnJ*−1
*N*

*i*1

*ωn,iαn,iJxn* 1−*αn,iJSixn,*

*Cn*1

*z*∈*Cn*:*φz, un*≤*φz, xn,*

*xn*1 Π*Cn*1*x*0

3.43

*for everyn*∈N*,*{*ωn,i*}*,*{*αn,i*} ⊂0*,*1*for alln*∈N*andi*1*,*2*, . . . , NsatisfiesNi*1*ωn,i*1*for all*

*n*∈N*,*lim inf*n*→ ∞*ωn,i*1−*αn,i>*0*for alli*1*,*2*, . . . , N. Then*{*xn*}*converges strongly to*Π*Fx.*

*Proof.* The proof is almost the same as the proofs of Theorems3.3and3.7; so it is omitted.

In particular, applyingTheorem 3.11gives the following result.

**Corollary 3.12.** *LetC, E, f, Sbe as inCorollary 2.5. Let the sequence*{*xn*}*be generated byx*0

*x*∈*C,C*0*Cand*

*ynJ*−1*αnJxn* 1−*αnJSxn,*

*un*∈*Csuch thatfun, y _{r}*1

*n*

*y*−*un, Jun*−*Jyn*≥0 ∀*y*∈*C,*

*Cn*1

*z*∈*Cn*:*φz, un*≤*φz, xn,*

*xn*1 Π*Cn*1*x*0

3.44

*for everyn* ∈ N∪ {0}*, where* *J* *is the duality mapping on* *E. Assume that* {*αn*} ⊂ 0*,*1*satisfies*

lim sup_{n}_{→ ∞}*αn* *<*1*and*{*rn*} ⊂*a,*∞*for somea >*0*. Then*{*xn*}*converges strongly to*Π*FS*∩*EPfx,*

*where*Π*FS*∩*EPfis the generalized projection ofEontoFS*∩*EPf.*

*Remark 3.13.* Corollary 3.12improves the restriction on{*αn*}of16, Theorem 3*.*1. In fact, it is

**Corollary 3.14** see 11, Theorem 3*.*1. *Let* *C* *be a nonempty and closed convex subset of a*
*uniformly convex and uniformly smooth Banach spaceE. Let* *f* *be a bifunction from* *C*×*C* *to*R
*satisfying (A1)–(A4) and letT, S* : *C* → *Cbe two closed relatively quasi-nonexpansive mappings*
*such thatF*:*FT*∩*FS*∩*EPf/*∅*. Let the sequence*{*xn*}*be generated by the following manner:*

*x*0∈*Echosen arbitrarily,*

*C*1*C,*

*x*1 Π*C*1*x*0*,*

*ynJ*−1*αnJxnβnJTxnγnJSxn,*

*un*∈*Csuch thatfun, y _{r}*1

*n*

*y*−*un, Jun*−*Jyn*≥0 ∀*y*∈*C,*

*Cn*1

*z*∈*Cn*:*φz, un*≤*φz, xn,*

*xn*1 Π*Cn*1*x*0*.*

3.45

*Assume that*{*αn*}*,*{*βn*}*,and*{*γn*}*are three sequences in*0*,*1*satisfying the restrictions:*

a*αnβnγn*1*;*

blim inf*n*→ ∞*αnβn>*0*,*lim inf*n*→ ∞*αnγn>*0*;*

c{*rn*} ⊂*a,*∞*for somea >*0*.*
*Then*{*xn*}*converges strongly to*Π*Fx*0*.*

*Remark 3.15.* The conclusion of Corollary 3.14 remains true under the more general
assumption; that is, we can replacebby the following one:

blim inf*n*→ ∞*βn>*0 and lim inf*n*→ ∞*γn>*0.

We also deduce the following result.

**Corollary 3.16** see 14, Theorem 3*.*1. *Let* *C, E, f, T, S* *be as in* *Corollary 3.14. Let the*
*sequences*{*xn*}*,*{*yn*}*,*{*zn*}*,and*{*un*}*be generated by the following:*

*x*0∈*Echosen arbitrarily,*

*C*1*C,*

*x*1 Π*C*1*x*0*,*

*ynJ*−1*δnJxn* 1−*δnJzn,*

*znJ*−1*αnJxnβnJTxnγnJSxn,*

*un*∈*Csuch thatfun, y _{r}*1

*n*

*y*−*un, Jun*−*Jzn*≥0 ∀*y*∈*C,*

*Cn*1

*z*∈*Cn*:*φz, un*≤*φz, xn,*

*xn*1 Π*Cn*1*x*0*.*

*Assume that*{*αn*}*,*{*βn*}*,and*{*γn*}*are three sequences in*0*,*1*satisfying the following restrictions:*

a*αnβnγn*1*;*

b0≤*αn<*1*for alln*∈N∪ {0}*and*lim sup*n*→ ∞*αn<*1*;*

clim inf*n*→ ∞*αnβn>*0*,*lim inf*n*→ ∞*αnγn>*0*;*

d{*rn*} ⊂*a,*∞*for somea >*0*.*

*Then*{*xn*}*and*{*un*}*converge strongly to*Π*Fx*0*.*

*Remark 3.17.* The conclusion of Corollary 3.16 remains true under the more general
restrictions; that is, we replacebandcby the following one:

blim inf*n*→ ∞*βn>*0 and lim inf*n*→ ∞*γn>*0.

**Corollary 3.18**see10, Theorem 3*.*1. *LetCbe a nonempty closed convex subset of a uniformly*
*convex and uniformly smooth Banach space* *E. Let* {*Ti*}*Ni*1 : *C* → *C* *be a family of relatively*

*nonexpansive mappings such thatF* :*N _{i}*

_{}

_{1}

*FTi/*∅

*and letx*0 ∈

*E. ForC*1

*Candx*1 Π

*C*1

*x*0

*,*

*define a sequence*{*xn*}*ofCas follows:*

*ynJ*−1*αnJxn* 1−*αnJzn,*

*znJ*−1

*β*1

*n* *Jxn*
*N*

*i*1

*βi*1

*n* *JTixn*

*,*

*Cn*1

*z*∈*Cn* :*φz, yn*≤*φz, xn,*

*xn*1 Π*Cn*1*x*0*,*

3.47

*where*{*αn*}*,*{*βni*} ⊂0*,*1*satisfies the following restrictions:*

i0≤*αn<*1*for alln*∈N∪ {0}*and*lim sup_{n}_{→ ∞}*αn<*1*;*

ii0≤*βni*≤1*for alli*1*,*2*, . . . , N*1*,Ni*11*β*

*i*

*n* 1*for alln*∈N∪ {0}*. If*

a*either*lim inf*n*→ ∞*βn*1*βni*1*>*0*for alli*1*,*2*, . . . , Nor*

blim*n*→ ∞*βn*10*and*lim inf*n*→ ∞*βnk*1*βnl*1*>*0*for alli /j, k, l*1*,*2*, . . . , N.*

*then the sequence*{*xn*}*converges strongly to*Π*Fx*0*.*

*Remark 3.19.* The conclusion of Corollary 3.18 remains true under the more general
restrictions on{*αn*}*,*{*βni*}:

1*αn, βn*1∈0*,*1are arbitrary.

**Acknowledgments**

The authors would like to thank the referee for their comments on the manuscript. The first author is supported by grant fund under the program Strategic Scholarships for Frontier Research Network for the Ph.D. Program Thai Doctoral degree from the Oﬃce of the Higher Education Commission, Thailand, and the second author is supported by the Thailand Research Fund under Grant MRG5180146.

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