Volume 2007, Article ID 64363,12pages doi:10.1155/2007/64363
Research Article
Convergence Theorem for Equilibrium Problems and Fixed Point
Problems of Infinite Family of Nonexpansive Mappings
Yonghong Yao, Yeong-Cheng Liou, and Jen-Chih Yao
Received 17 March 2007; Accepted 20 August 2007
Recommended by Billy E. Rhoades
We introduce an iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of infinite nonexpan-sive mappings in a Hilbert space. We prove a strong-convergence theorem under mild assumptions on parameters.
Copyright © 2007 Yonghong Yao 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
LetH be a real Hilbert space and letCbe a nonempty closed convex subset ofH. Let
h:C×C→Rbe an equilibrium bifunction, that is,h(u,u)=0 for everyu∈C. Then one can define the equilibrium problem that is to find an elementu∈Csuch that
EP(h) :h(u,v)≥0 ∀v∈C. (1.1)
Denote the set of solutions of EP(h) by SEP(h). This problem contains fixed point problems, optimization problems, variational inequality problems, and Nash equilibrium problems as special cases, see [1]. Some methods have been proposed to solve the equi-librium problem, please consult [2–4].
improve the corresponding results announced by Combettes and Hirstoaga [2], Moudafi [5], Wittmann [6], and Tada and Takahashi [7].
In this paper, motivated and inspired by Combettes and Hirstoaga [2] and Takahashi and Takahashi [4], we introduce an iterative scheme for finding a common element of the set of solutions of EP(h) and the set of fixed points of infinite nonexpansive mappings in a Hilbert space. We obtain a strong convergence theorem which improves and extends the corresponding results of [2,4].
2. Preliminaries
LetHbe a real Hilbert space with inner product·,· and norm·. LetCbe a nonempty closed convex subset ofH. Then for anyx∈H, there exists a unique nearest point inC, denoted byPC(x), such thatx−PC(x) ≤ x−yfor all y∈C. Such aPCis called the metric projection ofHontoC. We know thatPCis nonexpansive. Further, forx∈Hand x∗∈C,
x∗=P
C(x)⇐⇒x−x∗,x∗−y≥0 ∀y∈C. (2.1)
Recall that a mappingT:C→H is called nonexpansive ifTx−T y ≤ x−yfor allx,y∈C. Denote the set of fixed points ofT byF(T). It is well known that ifCis a bounded closed convex andT:C→Cis nonexpansive, thenF(T)=∅; see, for instance, [8]. We call a mapping f :H→Hcontractive if there exists a constantα∈(0, 1) such that
f(x)−f(y) ≤αx−yfor allx,y∈H.
For an equilibrium bifunctionh:C×C→R, we callhsatisfying condition (A) ifh satisfies the following three conditions:
(i)his monotone, that is,h(x,y) +h(y,x)≤0 for allx,y∈C; (ii) for eachx,y,z∈C, limt↓0h(tz+ (1−t)x,y)≤h(x,y);
(iii) for eachx∈C,y→h(x,y) is convex and lower semicontinuous.
If an equilibrium bifunctionh:C×C→Rsatisfies condition (A), then we have the fol-lowing two important results. You can find the first lemma in [1] and the second one in [2].
Lemma2.1. LetCbe a nonempty closed convex subset ofH and lethbe an equilibrium bifunction ofC×CintoR, satisfying condition (A). Letr >0andx∈H. Then there exists
y∈Csuch that
h(y,z) +1
rz−y,y−x ≥0 ∀z∈C. (2.2)
Lemma 2.2. Assume thathsatisfies the same assumptions asLemma 2.1. Forr >0 and
x∈H, define a mappingSr:H→Cas follows:
Sr(x)=
y∈C:h(y,z) +1
rz−y,y−x ≥0,∀z∈C
(2.3)
(1)Sris single-valued andSris firmly nonexpansive, that is, for anyx,y∈H,
Srx−Sry2
≤Srx−Sry,x−y; (2.4)
(2)F(Sr)=SEP(h)andSEP(h)is closed and convex.
We also need the following lemmas for proving our main results.
Lemma2.3 (see [9]). Let{xn}and{yn}be bounded sequences in a Banach spaceXand
let{βn}be a sequence in[0, 1]with0<lim infn→∞βn≤lim supn→∞βn<1. Supposexn+1= (1−βn)yn+βnxnfor all integersn≥0andlim supn→∞(yn+1−yn − xn+1−xn)≤0. Thenlimn→∞yn−xn =0.
Lemma2.4 (see [10]). Assume{an}is a sequence of nonnegative real numbers such that
an+1≤(1−γn)an+δn, where{γn}is a sequence in(0, 1)and{δn}is a sequence such that ∞
n=1γn= ∞andlim supn→∞δn/γn≤0. Thenlimn→∞an=0.
3. Iterative scheme and strong convergence theorems
In this section, we first introduce our iterative scheme. Consequently, we will establish strong convergence theorems for this iteration scheme. To be more specific, letT1,T2,... be infinite mappings ofCintoCand letλ1,λ2,...be real numbers such that 0≤λi≤1 for everyi∈N. For anyn∈N, define a mappingWnofCintoCas follows:
Un,n+1=I,
Un,n=λnTnUn,n+1+
1−λn I,
Un,n−1=λn−1Tn−1Un,n+
1−λn−1 I,
.. .
Un,k=λkTkUn,k+1+
1−λk I,
.. .
Un,2=λ2T2Un,3+
1−λ2 I,
Wn=Un,1=λ1T1Un,2+
1−λ1 I.
(3.1)
Such a mappingWn is called theW-mapping generated byTn,Tn−1,...,T1 andλn, λn−1,...,λ1; see [11].
and{yn}∞n=1are generated iteratively by
hyn,x +r1 n
x−yn,yn−xn≥0, ∀x∈C, xn+1=αnf
xn +βnxn+γnWnyn,
(3.2)
where{αn},{βn}, and{γn}are three sequences in (0, 1) such thatαn+βn+γn=1,{rn} is a real sequence in (0,∞),his an equilibrium bifunction, andWnis theW-mapping defined by (3.1).
We have the following crucial conclusions concerningWn. You can find them in [12, 13]. Now we only need the following similar version in Hilbert spaces.
Lemma3.1. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetT1,
T2,... be nonexpansive mappings of C intoC such that∞i=1F(Ti)is nonempty, and let
λ1,λ2,...be real numbers such that0< λi≤b <1for anyi∈N. Then for everyx∈Cand k∈N, the limitlimn→∞Un,kxexists.
Remark 3.2. FromLemma 3.1, we have that ifC is bounded, then for all ε >0, there exists a common positive integer numberN0such that forn > N0,Un,kx−Uk(x)< ε for allx∈C. Indeed, by the similar argument to Lemma 3.2 in [13], letw∈∞n=1F(Tn). SinceCis bounded, there exists a constantM >0 such thatx−w ≤M for allx∈C. Fixk∈N. Then for allx∈Cand anyn∈N withn≥k, we haveUn+1,kx−Un,kx ≤ 2(ni=+1kλi)x−w ≤2M(ni=+1kλi).
Let ε >0. Then there exists n0∈N withn0≥k such that for allx∈C, bn0−k+2<
ε(1−b)/2M. So for allx∈Cand everym,nwithm > n > n0, we have Um,kx−Un,kx
≤ m−1
j=n
Uj+1,kx−Uj,kx≤m−1 j=n
2
j+1
i=k λi
x−w
≤2M m−1
j=nb
j−k+2≤2Mbn−k+2 1−b < ε.
(3.3)
Remark 3.3. Using Lemma 3.1, one can define a mapping W of C into C as Wx=
limn→∞Wnx=limn→∞Un,1x for every x∈C. Such aW is called theW-mapping gen-erated byT1,T2,...andλ1,λ2,.... We observe that if{xn}is a bounded sequence inC, then we have
lim
n→∞Wxn−Wnxn=0. (3.4)
Indeed, fromRemark 3.1, we have: for anyε >0, there isn0such thatWx−Wnx ≤ ε for allx∈ {xn} and for alln≥n0. In particular, Wxn−Wnxn ≤ε for all n≥n0. Consequently, limn→∞Wxn−Wnxn =0, as claimed.
Lemma3.4. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetT1,
T2,... be nonexpansive mappings ofC intoC such that ∞i=1F(Ti)is nonempty, and let
λ1,λ2,...be real numbers such that0< λi≤b <1for anyi∈N. ThenF(W)=∞i=1F(Ti).
Now we state and prove our main results.
Theorem 3.5. LetC be a nonempty closed convex subset of a real Hilbert spaceH. Let
h:C×C→Rbe an equilibrium bifunction satisfying condition (A) and let{Ti}∞i=1 be an infinite family of nonexpansive mappings of CintoC such that∞i=1F(Ti)∩SEP(h)=∅. Suppose{αn},{βn}, and{γn}are three sequences in(0, 1)such thatαn+βn+γn=1and
{rn} ⊂(0,∞). Suppose the following conditions are satisfied: (i) limn→∞αn=0and∞n=0αn= ∞;
(ii) 0<lim infn→∞βn≤lim supn→∞βn<1; (iii) lim infn→∞rn>0andlimn→∞(rn+1−rn)=0.
Let f be a contraction ofHinto itself and givenx0∈Harbitrarily. Then the sequences{xn}
and{yn}generated iteratively by (3.2) converge strongly tox∗∈i∞=1F(Ti)∩SEP(h), where
x∗=P∞
i=1F(Ti)∩SEP(h)f(x
∗).
Proof. LetQ=P∞
i=1F(Ti)∩SEP(h). Note that f is a contraction mapping with coefficientα∈ (0, 1). ThenQ f(x)−Q f(y) ≤ f(x)−f(y) ≤αx−yfor allx,y∈H. Therefore,
Q f is a contraction ofH into itself, which implies that there exists a unique element
x∗∈Hsuch thatx∗=Q f(x∗). At the same time, we note thatx∗∈C.
Letp∈∞i=1F(Ti)∩SEP(h). From the definition ofSr, we note thatyn=Srnxn. It
fol-lows thatyn−p = Srnxn−Srnp ≤ xn−p. Next, we prove that{xn}and{yn}are
bounded. From (3.1) and (3.2), we obtain xn+1−p≤αnf
xn −p+βnxn−p+γnWnyn−p
≤αnfxn −f(p)+f(p)−p +βnxn−p+γnyn−p
≤αnαxn−p+f(p)−p +1−αn xn−p
≤maxx0−p, 1
1−αf(p)−p
.
(3.5)
Therefore,{xn}is bounded. We also obtain that{yn},{Wnxn}, and{f(xn)}are all bounded. We shall useMto denote the possible different constants appearing in the fol-lowing reasoning.
Settingxn+1=βnxn+ (1−βn)znfor alln≥0, we have that
zn+1−zn=xn+21−−ββn+1xn+1
n+1 −
xn+1−βnxn 1−βn
= αn+1 1−βn+1
fxn+1 −fxn +
α
n+1 1−βn+1−
αn 1−βn
fxn
+ γn+1 1−βn+1
Wn+1yn+1−Wnyn +
γ
n+1 1−βn+1−
γn 1−βn
Wnyn.
So we have
zn+1−zn≤ ααn+1 1−βn+1
xn+1−xn
+ αn+1 1−βn+1−
αn 1−βn
fxn +Wnyn
+ γn+1 1−βn+1
Wn+1yn+1−Wnyn.
(3.7)
SinceTiandUn,iare nonexpansive from (3.1), we have
Wn+1yn−Wnyn=λ1T1Un+1,2yn−λ1T1Un,2yn
≤λ1Un+1,2yn−Un,2yn ≤λ1λ2Un+1,3yn−Un,3yn
≤ ···
≤λ1λ2···λnUn+1,n+1yn−Un,n+1yn
≤Mn i=1
λi,
(3.8)
and hence
Wn+1yn+1−Wnyn≤Wn+1yn+1−Wn+1yn+Wn+1yn−Wnyn
≤yn+1−yn+M n
i=1
λi. (3.9)
Substituting (3.9) into (3.7), we have
zn+1−zn≤ ααn+1 1−βn+1
xn+1−xn+ αn+1 1−βn+1−
αn 1−βn
fxn +Wnyn
+ γn+1 1−βn+1
yn+1−yn+ Mγn+1 1−βn+1
n
i=1
λi.
(3.10)
On the other hand, fromyn=Srnxnandyn+1=Srn+1xn+1, we have
hyn,x + 1
rn
x−yn,yn−xn≥0 ∀x∈C, (3.11)
hyn+1,x + 1
rn+1
x−yn+1,yn+1−xn+1
Puttingx=yn+1in (3.11) andx=ynin (3.12), we have
hyn,yn+1 + 1
rn
yn+1−yn,yn−xn≥0, (3.13)
hyn+1,yn + 1
rn+1
yn−yn+1,yn+1−xn+1
≥0. (3.14)
From the monotonicity ofh, we have
hyn,yn+1 +h
yn+1,yn ≤0. (3.15)
So from (3.13), we can conclude that
yn+1−yn,ynr−xn
n −
yn+1−xn+1
rn+1
≥0, (3.16)
and hence
yn+1−yn,yn−yn+1+yn+1−xn−rrn n+1
yn+1−xn+1 ≥0. (3.17)
Since lim infn→∞rn>0, without loss of generality, we may assume that there exists a real numberτsuch thatrn> τ >0 for alln∈N. Then we have
yn+1−yn2
≤
yn+1−yn,xn+1−xn+
1−rrn
n+1
yn+1−xn+1
≤yn+1−ynxn+1−xn+1−rrn n+1
yn+1−xn+1
,
(3.18)
and hence
yn+1−yn≤xn+1−xn+M
τrn+1−rn. (3.19) Substituting (3.19) into (3.10), we have
zn+1−zn≤ ααn+1 1−βn+1
xn+1−xn+ αn+1 1−βn+1−
αn 1−βn
fxn +Wnyn
+ γn+1 1−βn+1
xn+1−xn+ γn+1 1−βn+1×
M
τrn+1−rn+ Mγn +1 1−βn+1
n
i=1
λi
≤xn+1−xn+ αn+1
1−βn+1−
αn 1−βn
fxn +Wnyn
+ γn+1 1−βn+1×
M
τrn+1−rn+1Mγ−βn+1 n+1
n
i=1
λi.
(3.20)
This together withαn→0 andrn+1−rn→0 imply that lim supn→∞(zn+1−zn − xn+1−
xn)≤0. Hence byLemma 2.3, we obtainzn−xn→0 asn→∞. Consequently,
lim
n→∞xn+1−xn=nlim→∞
From (3.19) and limn→∞(rn+1−rn)=0, we have limn→∞yn+1−yn =0. Sincexn+1=
αnf(xn) +βnxn+γnWnyn, we have
xn−Wnyn≤xn−xn+1+xn+1−Wnyn
≤xn−xn+1+αnf
xn −Wnyn+βnxn−Wnyn, (3.22)
that is,
xn−Wnyn≤ 1 1−βn
xn−xn+1+ αn 1−βn
f
xn −Wnyn. (3.23)
Hence we have limn→∞xn−Wnyn =0. For p∈∞i=1F(Ti)∩SEP(h), note that Sr is firmly nonexpansive. Then we have
yn−p2
=Srnxn−Srnp
2
≤Srnxn−Srnp,xn−p
=yn−p,xn−p
=1
2
yn−p2+xn−p2−xn−yn2
,
(3.24)
and hence
yn−p2
≤xn−p2−xn−yn2. (3.25)
Therefore, we have xn+1−p2
≤αnfxn −p2+βnxn−p2+γnWnyn−p2 ≤αnfxn −p2+βnxn−p2+γnyn−p2 ≤αnfxn −p2+βnxn−p2
+γnxn−p2−xn−yn2
≤αnfxn −p2+xn−p2−γnxn−yn2.
(3.26)
Then we have
γnxn−yn2≤αnfxn −p2+xn−p+xn+1−p
×xn−p−xn+1−p
≤αnfxn −p2+xn−xn+1xn−p+xn+1−p .
(3.27)
It is easily seen that lim infn→∞γn>0. So we have
lim
n→∞xn−yn=0. (3.28)
FromWnyn−yn ≤ Wnyn−xn+xn−yn, we also haveWnyn−yn→0. At that same time, we note that
It follows from (3.29) andRemark 3.2that limn→∞W yn−yn =0. Next, we show that
lim sup n→∞
fx∗ −x∗,x
n−x∗≤0, (3.30)
wherex∗=PF(W)∩SEP(h)f(x∗). First, we can choose a subsequence{ynj} of{yn}such
that
lim j→∞
fx∗ −x∗,y
nj−x∗
=lim sup n→∞
fx∗ −x∗,y
n−x∗. (3.31)
Since{ynj}is bounded, there exists a subsequence{ynji}of{ynj}, which converges weakly
tow. Without loss of generality, we can assume thatynj→wweakly. FromW yn−yn→0,
we obtainW ynj→wweakly. Now we showw∈SEP(h).
By yn=Srnxn, we haveh(yn,x) + (1/rn)x−yn,yn−xn ≥0 for allx∈C. From the
monotonicity ofh, we have (1/rn)x−yn,yn−xn ≥ −h(yn,x)≥h(x,yn), and hence
x−ynj,
ynj−xnj
rnj
≥hx,ynj . (3.32)
Since (ynj−xnj)/rnj→0 andynj→wweakly, from the lower semicontinuity ofh(x,y) on
the second variabley, we haveh(x,w)≤0 for allx∈C. Fortwith 0< t≤1 andx∈C, letxt=tx+ (1−t)w. Sincex∈Candw∈C, we havext∈C, and henceh(xt,w)≤0. So from the convexity of equilibrium bifunctionh(x,y) on the second variabley, we have
0=hxt,xt ≤thxt,x + (1−t)hxt,w ≤thxt,x . (3.33)
Henceh(xt,x)≥0. Therefore, we haveh(w,x)≥0 for allx∈C, and hencew∈SEP(h). We will showw∈F(W). Assumew∈F(W). Sinceynj→wweakly andw=Ww, from
Opial’s condition, we have
lim inf j→∞
yn
j−w<lim infj→∞ ynj−Ww ≤lim inf
j→∞ ynj−W ynj+W ynj−Ww
≤lim inf
j→∞ ynj−w.
(3.34)
This is a contradiction. So we getw∈F(W)=i∞=1F(Ti). Therefore,w∈ ∞
i=1F(Ti)∩ SEP(h). Sincex∗=P∞
i=1F(Ti)∩SEP(h)f(x∗), we have
lim sup n→∞
fx∗ −x∗,x
n−x∗=limj→∞fx∗ −x∗,xnj−x∗
=lim j→∞
fx∗ −x∗,y
nj−x∗
=fx∗ −x∗,w−x∗≤0.
First, we prove that{xn}converges strongly tox∗∈∞i=1F(Ti)∩SEP(h). From (3.2), we have
xn+1−x∗2
≤βnxn−x∗ +γnWnyn−x∗ 2+ 2αnfxn −x∗,xn+1−x∗
≤βnxn−x∗+γnWnyn−x∗ 2+ 2αnfx∗ −x∗,xn+1−x∗
+ 2αnfxn −fx∗ ,xn+1−x∗
≤βnxn−x∗+γnyn−x∗2+ 2ααnxn−x∗xn+1−x∗
+ 2αnfx∗ −x∗,xn+1−x∗
≤1−αn 2xn−x∗2+ααnxn+1−x∗2+xn−x∗2
+ 2αnfx∗ −x∗,xn+1−x∗
,
(3.36)
which implies that
xn+1−x∗2
≤
1−αn 2+ααn 1−ααn
xn−x∗2 + 2αn
1−ααn
fx∗ −x∗,xn
+1−x∗
=1−2αn+ααn
1−ααn
xn−x∗2 + α
2 n 1−ααn
xn−x∗2
+ 2αn 1−ααn
fx∗ −x∗,x
n+1−x∗
≤
1−2(1−α)αn
1−ααn
xn−x∗2+2(11−−ααα)αn n
× Mαn
2(1−α)+ 1 1−α
fx∗ −x∗,xn
+1−x∗
=1−ϕn xn−x∗2+φnϕn,
(3.37)
whereϕn=2(1−α)αn/(1−ααn) andφn=Mαn/2(1−α) + 1/(1−α)f(x∗)−x∗,xn+1−
x∗ . It is easily seen that∞
n=0ϕn= ∞and lim supn→∞φn≤0. Now applyingLemma 2.4 and (3.35) to (3.37), we conclude thatxn→x∗ (n→∞). Consequently, from (3.28), we haveyn→x∗(n→∞). This completes the proof. Corollary3.6. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Let
h:C×C→Rbe an equilibrium bifunction satisfying condition (A) such that SEP(h)=∅. Let{αn},{βn}, and{γn}be three sequences in(0, 1)such thatαn+βn+γn=1and{rn} ⊂ (0,∞)is a real sequence. Suppose the following conditions are satisfied:
Let f be a contraction ofH into itself and givenx0∈H arbitrarily. Let{xn}and{yn}be
sequences generated iteratively by
hyn,x +r1 n
x−yn,yn−xn≥0, ∀x∈C, xn+1=αnfxn +βnxn+γnyn.
(3.38)
Then the sequences{xn}and{yn}generated by (3.38) converge strongly tox∗∈SEP(h),
wherex∗=PSEP(h)f(x∗).
Proof. TakeTix=xfor alli=1, 2,...and for allx∈Cin (3.1). ThenWnx=xfor all x∈C. The conclusion follows from Theorem 3.1. This completes the proof. Corollary3.7. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Let
{Ti}∞i=1be an infinite family of nonexpansive mappings ofCintoCsuch that ∞
i=1F(Ti)=∅. Let{αn},{βn}, and{γn}be three sequences in(0, 1)such thatαn+βn+γn=1. Suppose the
following conditions are satisfied:
(i) limn→∞αn=0and∞n=0αn= ∞; (ii) 0<lim infn→∞βn≤lim supn→∞βn<1.
Let f be a contraction ofH into itself and givenx0∈H arbitrarily. Let{xn}be a sequence
generated iteratively by
xn+1=αnfxn +βnxn+γnWnPCxn. (3.39)
Then the sequence{xn}converges strongly tox∗=P∞
i=1F(Ti)f(x∗).
Proof. Seth(x,y)=0 for allx,y∈Candrn=1 for alln∈N. Then we haveyn=PCxn. From (3.2), we have
xn+1=αnf
xn +βnxn+γnWnPCxn. (3.40)
Then the conclusion follows fromTheorem 3.5. This completes the proof.
Acknowledgments
The authors thank the referee for his/her helpful comments and suggestions, which im-proved the presentation of this manuscript. The research was partially supported by the Grant NSC 96-2221-E-230-003.
References
[1] E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium prob-lems,”The Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994.
[2] P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,”Journal of Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 117–136, 2005.
[3] S. D. Fl˚am and A. S. Antipin, “Equilibrium programming using proximal-like algorithms,” Mathematical Programming, vol. 78, no. 1, pp. 29–41, 1997.
[5] A. Moudafi, “Viscosity approximation methods for fixed-points problems,”Journal of Mathe-matical Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000.
[6] R. Wittmann, “Approximation of fixed points of nonexpansive mappings,”Archiv der Mathe-matik, vol. 58, no. 5, pp. 486–491, 1992.
[7] A. Tada and W. Takahashi, “Strong convergence theorem for an equilibrium problem and a nonexpansive mapping,” inNonlinear Analysis and Convex Analysis, W. Takahashi and T. Tanaka, Eds., pp. 609–617, Yokohama, Yokohama, Japan, 2007.
[8] W. Takahashi,Nonlinear Functional Analysis, Yokohama, Yokohama, Japan, 2000.
[9] T. Suzuki, “Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals,”Journal of Mathematical Analysis and Ap-plications, vol. 305, no. 1, pp. 227–239, 2005.
[10] H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,”Journal of Mathe-matical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004.
[11] W. Takahashi, “Weak and strong convergence theorems for families of nonexpansive mappings and their applications,”Annales Universitatis Mariae Curie-Skłodowska. Sectio A, vol. 51, no. 2, pp. 277–292, 1997.
[12] W. Takahashi and K. Shimoji, “Convergence theorems for nonexpansive mappings and feasibil-ity problems,”Mathematical and Computer Modelling, vol. 32, no. 11–13, pp. 1463–1471, 2000. [13] K. Shimoji and W. Takahashi, “Strong convergence to common fixed points of infinite
nonex-pansive mappings and applications,”Taiwanese Journal of Mathematics, vol. 5, no. 2, pp. 387– 404, 2001.
Yonghong Yao: Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China Email address:[email protected]
Yeong-Cheng Liou: Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan
Email address:simplex [email protected]
Jen-Chih Yao: Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan
