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
1and Satit Saejung
21Department of Mathematics, Mahasarakham University, Maha Sarakham 44150, Thailand
2Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand
Correspondence should be addressed to Satit Saejung,[email protected]
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
LetCbe a nonempty subset of a Banach spaceE, and letT be a mapping fromCinto itself. When{xn}is a sequence inE, we denote strong convergence of{xn}tox∈Ebyxn → xand
weak convergence byxn x. We also denote the weak∗convergence of a sequence{x∗n}to
x∗in the dualE∗byx∗
n x∗ ∗. A pointp ∈Cis an asymptotic fixed point ofT if there exists
{xn}inCsuch thatxn pandxn−Txn → 0. We denoteFTandFTby the set of fixed
points and of asymptotic fixed points ofT, respectively. A Banach spaceEis said to be strictly convex ifxy/2 < 1 forx, y ∈ SE {z ∈ E : z 1}andx /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∈SEandx−y ≥. The spaceEis said to be smooth if the limit
lim
t→0
xtx − x
exists for allx, 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.
LetEbe a smooth, strictly convex and reflexive Banach space and letCbe a nonempty closed convex subset ofE. Throughout this paper, we denote byφthe function defined by
φx, yx2−2x, Jyy2 ∀x, y∈E, 1.2
whereJis the normalized duality mapping fromEto the dual spaceE∗given by the following relation:
x, Jx x2Jx2. 1.3
We know that if Eis smooth, strictly convex, and reflexive, then the duality mappingJ is single-valued, one-to-one, and onto. The duality mappingJis said to be weakly sequentially continuous ifxn ximplies thatJxn Jx∗ see5for more details.
Following Matsushita and Takahashi6, a mappingT :C → Eis said to be relatively nonexpansive if the following conditions are satisfied:
R1FTis nonempty;
R2φu, Tx≤φu, xfor allu∈FT,x∈C;
R3FT FT.
If T satisfiesR1and R2, thenT 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 projectionΠC fromEontoCas follows:
ΠCx arg min y∈C φ
y, x ∀x∈E. 1.4
IfEis a Hilbert space, thenφy, x y−x2 andΠC becomes the metric projection of E
ontoC. 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 1.1see7, 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 1.2see17, Proposition 5. LetCbe a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach spaceE. Then
φx,ΠCyφΠCy, y≤φx, y 1.5
for allx∈Candy∈E.
Lemma 1.3see17. LetEbe a smooth and uniformly convex Banach space and letr >0. Then there exists a strictly increasing, continuous, and convex functionh:0,2r → Rsuch thath0 0
and
hx−y≤φx, y 1.6
for allx, y∈Br {z∈E:z ≤r}.
Lemma 1.4see17, 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. Iflimn→ ∞φxn, yn 0, thenlimn→ ∞xn−yn0.
Lemma 1.5see2. LetCbe a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach spaceE, letx∈E,and letz∈C. Then
z ΠCx⇐⇒y−z, Jx−Jz≤0, ∀y∈C. 1.7
Lemma 1.6see18. LetEbe a uniformly convex Banach space and letr >0. Then there exists a strictly increasing, continuous, and convex functiong :0,2r → Rsuch thatg0 0and
tx 1−ty2≤tx2 1−ty2−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}Ni1be a family of relatively quasi-nonexpansive
mappings fromCinto itself such thatFT∩Ni1FSi/∅. The mappingU:C → Eis defined by
UxTJ−1N
i1
ωiαiJx 1−αiJSix 1.9
for allx ∈ Cand {ωi},{αi} ⊂ 0,1,i 1,2, . . . , N such thatNi1ωi 1. Ifx ∈ Cand z ∈
FT∩N
i1FSi, 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.
IfEhas 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∗:0,6r → Rsuch that
g∗0 0and for each relatively quasi-nonexpansive mappingT : E → Eand each finite family of
relatively quasi-nonexpansive mappings{Si}Ni1 :C → Csuch thatFT∩ N
i1FSi/∅,
N
i1
ωiαi1−αig∗Jz−JSiz≤φu, z−φu, Uz 1.11
for allz∈C∩Br andu∈FT∩Ni1FSi∩Br, where
UxTJ−1N
i1
ωiαiJx 1−αiJSix 1.12
x∈Cand{ωi},{αi} ⊂0,1,i1,2, . . . , Nsuch thatNi1ωi 1.
Proof. Let r > 0. From Lemma 1.6and E∗ is uniformly convex, then there exists a strictly increasing, continuous, and convex functiong∗:0,6r → Rsuch thatg∗0 0 and
tx∗ 1−ty∗2≤tx∗2 1−ty∗2−t
1−tg∗x∗−y∗ 1.13
for allx∗, y∗ ∈ {z∗ ∈ E∗ : z∗ ≤ 3r}and t ∈ 0,1. LetT : E → Eand {Si}Ni1 : C → C be relatively quasi-nonexpansive for all i 1,2, . . . , N such thatFT∩Ni1FSi/∅. For
z∈C∩Br andu∈FT∩Ni1FSi∩Br. It follows that
and henceSiz ≤3r. Consequently, fori1,2, . . . , N,
αiJz 1−αiJSiz2≤αiJz2 1−αiJSiz2−αi1−αig∗Jz−JSiz. 1.15
Then
φu, Uz≤φ
u, J−1N
i1
ωiαiJz 1−αiJSiz
u2−2
u, N
i1
ωiαiJz 1−αiJSiz
N
i1
ωiαiJz 1−αiJSiz
2
≤ N
i1
ωi
u2−2u, α
iJz 1−αiJSiz αiJz 1−αiJSiz2
≤ N
i1
ωi
u2−2u, α
iJz 1−αiJSiz αiJz2 1−αiJSiz2
−αi1−αig∗Jz−JSiz
N
i1
ωiαiφu, z 1−αiφu, Siz−αi1−αig∗Jz−JSiz
≤φu, z− N
i1
ωiαi1−αig∗Jz−JSiz.
1.16
Thus
N
i1
ωiαi1−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}Ni1 be a family
of relatively quasi-nonexpansive mappings fromC into itself such thatFT∩Ni1FSi/∅. The mappingU:C → Eis defined by
UxTJ−1N
i1
ωiαiJx 1−αiJSix 1.18
for allx∈Cand{ωi},{αi} ⊂0,1,i1,2, . . . , Nsuch thatNi1ωi1. Then, the following hold:
1FU FT∩Ni1FSi,
Proof. 1Clearly,FT∩Ni1FSi⊂FU. We want to show the reverse inclusion. Letz∈
FUandu∈FT∩Ni1FSi. Choose
r :max{u,z,S1z,S2z, . . . ,Smz}. 1.19
FromLemma 1.8, we have
N
i1
ωiαi1−αig∗Jz−JSiz 0. 1.20
Fromωiαi1−αi>0 for alli1,2, . . . , Nand by the properties ofg∗, we have
JzJSiz 1.21
for alli1,2, . . . , N. FromJis one to one, we have
zSiz 1.22
for alli1,2, . . . , N. Consider
zUzTJ−1N
i1
ωiαiJz 1−αiJSiz Tz. 1.23
Thusz∈FT∩Ni1FSi.
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}∞n1:E → Cbe a family of relatively quasi-nonexpansive mappings and let
{Si}Ni1:C → Cbe a family of relatively quasi-nonexpansive mappings such thatF: ∞
n1FTn∩ N
i1FSi/∅. Let the sequence{xn}be generated byx1∈C,
xn1 TnJ−1 N
i1
ωn,iαn,iJxn 1−αn,iJSixn 2.1
for anyn ∈ N,{ωn,i},{αn,i} ⊂ 0,1for alln ∈N,i 1,2, . . . , N such thatNi1ωn,i 1for all
Proof. Letu∈∞n1FTn∩Ni1FSi. Put
Un TnJ−1 N
i1
ωn,iαn,iJ 1−αn,iJSi. 2.2
FromLemma 1.7, we have
φu, xn1 φu, Unxn≤φu, xn. 2.3
Therefore limn→ ∞φu, xnexists. This implies that{φu, xn},{xn}and{Sixn}are bounded
for alli1,2, . . . , N.
Letyn≡ΠFxn. From2.3andm∈N, we have
φyn, xnm≤φyn, xn. 2.4
Consequently,
φyn, ynmφynm, xnm≤φyn, xnm≤φyn, xn. 2.5
In particular,
φyn1, xn1
≤φyn, xn. 2.6
This implies that limn→ ∞φyn, xnexists. This together with the boundedness of{xn}gives
r : supn∈Nyn < ∞. UsingLemma 1.3, there exists a strictly increasing, continuous, and
convex functionh:0,2r → Rsuch thath0 0 and
hyn−ynm≤φyn, ynm≤φyn, xn−φynm, xnm. 2.7
Since{φyn, xn}is a convergent sequence, it follows from the properties ofgthat{yn}is a
Cauchy sequence. SinceFis closed, there existsz∈Fsuch thatyn → 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}∞n1 :C → Ewith
∞
n1FTn/∅, we say that{Tn}satisfies the NST-condition
19if for each bounded sequence{zn}inC,
lim
n→ ∞zn−Tnzn0 impliesωw{zn} ⊂ ∞
n1
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}∞n1 :E → Cbe a family of relatively quasi-nonexpansive mappings
satisfying NST-condition and let{Si}Ni1 : C → Cbe a family of relatively nonexpansive mappings
such thatF:∞n1FTn∩Ni1FSi/∅and suppose that
φu, Tnx φTnx, x≤φu, x 2.9
for allu∈∞n1FTn,n∈Nandx∈E. Let the sequence{xn}be generated byx1∈C,
xn1 TnJ−1 N
i1
ωn,iαn,iJxn 1−αn,iJSixn 2.10
for anyn ∈ N,{ωn,i},{αn,i} ⊂ 0,1for alln ∈N,i 1,2, . . . , N such thatNi1ωn,i 1for all
n∈N,lim infn→ ∞ωn,iαn,i1−αn,i>0for alli1,2, . . . , N. IfJis weakly sequentially continuous, then{xn}converges weakly toz∈F, wherezlimn→ ∞ΠFxn.
Proof. Letu ∈ F. FromTheorem 2.1, limn→ ∞φu, xnexists and hence{xn}and {Sixn}are
bounded for alli1,2, . . . , N. Let
rsup
n∈N{xn,S1xn,S2xn, . . . ,SNxn}. 2.11
By Lemma 1.8, there exists a strictly increasing, continuous, and convex function g∗ :
0,2r → Rsuch thatg∗0 0 and
N
i1
ωn,iαn,i1−αn,ig∗Jxn−JSixn≤φu, xn−φu, xn1. 2.12
In particular, for alli1,2, . . . , N,
ωn,iαn,i1−αn,ig∗Jxn−JSixn≤φu, xn−φu, xn1. 2.13
Hence,
∞
n1
ωn,iαn,i1−αn,ig∗Jxn−JSixn<∞ 2.14
for all i 1,2, . . . , N. Since lim infn→ ∞ωn,iαn,i1−αn,i > 0 for alli 1,2. . . , N and the
properties ofg, we have
lim
for alli 1,2. . . , N. SinceJ−1 is uniformly norm-to-norm continuous on bounded sets, we have
lim
n→ ∞xn−Sixn0 2.16
for alli1,2. . . , N. Since{xn}is bounded, there exists a subsequence{xnk}of{xn}such that
xnk z∈C. SinceSiis relatively nonexpansive,z∈FSi FSifor alli1,2. . . , N.
We show thatz∈∞n1FTn. Let
yn J−1 N
i1
ωn,iαn,iJxn 1−αn,iJSixn. 2.17
We note from2.15that
N
i1
ωn,iαn,iJxn 1−αn,iJSixn−Jxn
≤
N
i1
ωn,i1−αn,iJSixn−Jxn −→0. 2.18
SinceJ−1is uniformly norm-to-norm continuous on bounded sets, it follows that
lim
n→ ∞yn−xnnlim→ ∞ J−1
N
i1
ωn,iαn,iJxn 1−αn,iJSixn
−J−1Jx
n
0. 2.19
Moreover, by2.9and the existence of limn→ ∞φu, xn, we have
φTnyn, yn≤φu, yn−φu, Tnyn
φ
u, J−1N
i1
ωn,iαn,iJxn 1−αn,iJSixn
−φu, xn1
≤φu, xn−φu, xn1−→0.
2.20
It follows fromLemma 1.4that limn→ ∞Tnyn−yn 0. From2.19andxnk z, we have
ynk z. Since{Tn}satisfies NST-condition, we havez∈
∞
n1FTn. Hencez∈F. Letzn ΠFxn. FromLemma 1.5andz∈F, we have
znk −z, Jxnk−Jznk ≥0. 2.21
FromTheorem 2.1, we know thatzn → z∈F. SinceJis weakly sequentially continuous, we
have
Moreover, sinceJis monotone,
z−z, Jz−Jz≤0. 2.23
Then
z−z, Jz−Jz0. 2.24
SinceE is strictly convex, z z. This implies that ωw{xn} {z} and hence xn z
limn→ ∞ΠFxn.
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. LetC be a subset of a Banach space Eand letf:C×C → Rbe a bifunction. The equilibrium problem for a bifunctionfis to findx∈C such thatfx, y≥0 for ally∈C. The set of solutions above is denoted by EPf, that is
x∈EPf ifffx, y≥0 ∀y∈C. 2.25
To solve the equilibrium problem, we usually assume that a bifunction f satisfies the following conditionsCis closed and convex:
A1fx, x 0 for allx∈C;
A2fis monotone, that is,fx, y fy, x≤0, for allx, y∈C;
A3for allx, y, z∈C, lim supt↓0ftz 1−tx, y≤fx, y;
A4for allx∈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 → Rsatisfying (A1)–(A4). Suppose thatp∈C. Thenp∈EPfif and only iffy, p≤0for ally∈C.
Proof. Letp∈EPf, thenfp, y≥0 for ally∈C. FromA2, we get thatfy, p≤ −fp, y≤ 0 for ally∈C.
Conversely, assume thatfy, p≤0 for ally∈C. For anyy∈C, let
xtty 1−tp, fort∈0,1. 2.26
Thenfxt, p≤0 and hence
Sofxt, y≥0 for allt∈0,1. FromA3, we have
0≤lim sup
t↓0
fty 1−tp, y≤fp, y ∀y∈C. 2.28
Hencep∈EPf.
Takahashi and Zembayashi proved the following important result.
Lemma 2.4see15, Lemma 2.8. LetCbe a nonempty closed convex subset of a uniformly smooth, strictly convex and reflexive Banach spaceE. Letfbe a bifunction fromC×C → Rsatisfying (A1)– (A4). Forr >0andx∈E, define a mappingTr :E → Cas follows:
Trx
z∈C:fz, y1
r
y−z, Jz−Jx≥0∀y∈C
2.29
for allx∈E. Then, the following hold:
1Tr is single-valued;
2Tr is a firmly nonexpansive-type mapping [20], that is, for allx, y∈E
Trx−Try, JTrx−JTry≤Trx−Try, Jx−Jy; 2.30
3FTr EPf;
4EPfis closed and convex.
We now deduce Takahashi and Zembayashi’s recent result fromTheorem 2.2.
Corollary 2.5see15, Theorem 4.1. LetCbe a nonempty closed convex subset of a uniformly smooth and uniformly convex Banach spaceE. Letfbe a bifunction fromC×CtoRsatisfying (A1)– (A4) and letSbe a relatively nonexpansive mapping fromCinto itself such thatFS∩EPf/∅. Let the sequence{xn}be generated byu1 ∈E,
xn∈Csuch thatfxn, yr1 n
y−xn, Jxn−Jun≥0 ∀y∈C,
un1 J−1αnJxn 1−αnJSxn
2.31
for everyn ∈ N,{αn} ⊂ 0,1satisfyinglim infn→ ∞αn1−αn > 0and {rn} ⊂ a,∞for some
a >0. IfJ is weakly sequentially continuous, then{xn}converges weakly toz ∈ΠFS∩EPf, where
Proof. Put Tn ≡ Trn where Trn is defined by Lemma 2.4. Then
∞
n1FTn EPf. 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∈
∞
n1
FTn. 2.32
To see that {Tn} satisfies NST-condition, let {zn} be a bounded sequence in C such that
limn→ ∞zn−Tnzn 0 andp ∈ ωw{zn}. Suppose that there exists a subsequence{znk} of {zn}such thatznk p. ThenTnkznk p ∈C. SinceJ is uniformly continuous on bounded
sets andrnk ≥a, we have
lim
k→ ∞ 1
rnk
Jznk−JTnkznk0. 2.33
From the definition ofTrnk, we have
fTnkznk, y
r1
nk
y−Tnkznk, JTnkznk −Jznk
≥0 ∀y∈C. 2.34
Since
fy, Tnkznk
≤ −fTnkznk, y
≤ 1
rnk
y−Tnkznk, JTnkznk−Jznk
≤ r1
nk
y−TnkznkJTnkznk−Jznk
2.35
andfis lower semicontinuous and convex in the second variable, we have
fy, p≤lim inf
k→ ∞ f
y, Tnkznk
≤0. 2.36
Thusfy, p≤0 for ally∈C. FromLemma 2.3, we havep∈EPf. Then{Tn}satisfies the
NST-condition. FromTheorem 2.2whereN1,{xn}converges weakly toz∈FTn∩FS
EPf∩FS, wherezlimn→ ∞ΠEPf∩FSxn.
Corollary 2.6see11, Theorem 3.5. LetCbe a nonempty and closed convex subset of a uniformly convex and uniformly smooth Banach spaceE. Letf be a bifunction fromC×CtoRsatisfies (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, yr1 n
y−xn, Jxn−Jun≥0 ∀y∈C,
un1J−1
αnJxnβnJTxnγnJSxn ∀n≥1.
2.37
Assume that{αn},{βn},and{γn}are three sequences in0,1satisfying the following restrictions:
aαnβnγn1;
blim infn→ ∞αnβn>0,lim infn→ ∞αnγn>0;
c{rn} ⊂a,∞for somea >0.
IfJis weakly sequentially continuous, then{xn}converges weakly toz∈F, wherezlimn→ ∞ΠFxn.
The following result also follows fromTheorem 2.2.
Corollary 2.7see9, Theorem 5.3. LetEbe a uniformly smooth and uniformly convex Banach space and letCbe a nonempty closed convex subset ofE. Let{Si}Ni1 be a finite family of relatively
nonexpansive mappings fromCinto itself such thatF Ni1FSiis a nonempty and let {αn,i :
n, i ∈ N, 1 ≤ i≤ N} ⊂ 0,1and{ωn,i : n, i ∈ N, 1 ≤ i ≤ N} ⊂ 0,1be sequences such that
lim infn→ ∞αn,i1−αn,i>0andlim infn→ ∞ωn,i>0for alli∈ {1,2, . . . , N}andNi1ωn,i1for
alln∈N. LetUnbe a sequence of mappings defined by
Unx ΠCJ−1 N
i1
ωn,iαn,iJx 1−αn,iJSix 2.38
for allx∈Cand let the sequence{xn}be generated byx1x∈Cand
xn1Unxn n1,2, . . .. 2.39
Then the following hold:
1the sequence {xn} is bounded and each weak subsequential limit of {xn} belongs to N
i1FSi;
2if the duality mapping J from E into E∗ is weakly sequentially continuous, then {xn} converges weakly to the strong limit of{ΠFxn}.
Proof. Since ΠC is relatively nonexpansive, the family {ΠC} satisfies the NST-condition.
Moreover,FΠC Cand
φx,ΠCyφΠCy, 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}∞n1 : E → E and {Si}Ni1 : C → C be families of relatively
quasi-nonexpansive mappings such thatF:∞n1FTn∩Ni1FSi/∅, and
φu, Tnx φTnx, x≤φu, x 3.1
for allu∈∞n1FTn,n∈Nandx∈E. Let{xn} ⊂Cbe such that{xn}and{Sixn}are bounded for alli1,2, . . . , N, and
ynJ−1 N
i1
ωn,iαn,iJxn 1−αn,iJSixn,
unTnyn,
3.2
where{ωn,i},{αn,i} ⊂ 0,1for alln ∈Nandi 1,2, . . . , N satisfyNi1ωn,i 1for alln ∈ N, lim infn→ ∞ωn,i1−αn,i>0for alli1,2, . . . , Nandlimn→ ∞xn−un0. Then the following statements hold:
1limn→ ∞φu, xn−φu, un 0for allu∈C,
2limn→ ∞un−yn0,
3ωw{xn}ωw{yn},
4iflimn→ ∞xn1−xn0, thenlimn→ ∞xn−Sixn0for alli1,2, . . . , N,
5ifxn → z, thenun → zandyn → z.
Proof. 1 Since limn→ ∞xn − un 0 and J is uniformly norm-to-norm continuous on
bounded sets,
lim
n→ ∞Jxn−Jun0. 3.3
We note here that{un}is also bounded. For anyu∈C, we have
φu, xn−φu, unxn2− un2−2u, Jun−Jxn
≤xn2− u22|u, Jun−Jxn |
≤ xn−unxnun 2uJun−Jxn −→0.
3.4
2Letu∈F. Using3.1and the relative quasi-nonexpansiveness of eachTn, we have
ByLemma 1.4and the boundedness of{un}, we have
lim
n→ ∞un−yn0. 3.6
3Since
xn−yn≤ xn−unun−ynxn−unTnyn−yn−→0, 3.7
we haveωw{xn}ωw{yn}.
4 Assume that limn→ ∞xn1 −xn 0. From limn→ ∞xn −yn 0, we get that
limn→ ∞xn1−yn0. SinceJis uniformly norm-to-norm continuous on bounded sets, we
have
lim
n→ ∞Jxn1−Jxnnlim→ ∞Jxn1−Jyn0. 3.8
So,
Jxn1−Jyn Jxn1−
N
i1
ωn,iαn,iJxn 1−αn,iJSixn
≥ N
i1
ωn,i1−αn,iJxn1−JSixn −ωn,iαn,iJxn1−Jxn.
3.9
From3.8, we have
N
i1
ωn,i1−αn,iJxn1−JSixn ≤Jxn1−Jyn N
i1
ωn,iαn,iJxn1−Jxn −→0. 3.10
It follows from lim infn→ ∞ωn,i1−αn,i>0 for alli1,2, . . . , Nthat
lim
n→ ∞Jxn1−JSixn0 3.11
for alli1,2, . . . , N. SinceJ−1 is uniformly norm-to-norm continuous on bounded sets and limn→ ∞xn1−xn0, we have
lim
n→ ∞xn−Sixn0 3.12
for alli1,2, . . . , N, as desired.
5Assume thatxn → z. From the assumption and2, we have
lim
n→ ∞xn−unnlim→ ∞un−yn0. 3.13
Lemma 3.2see21, 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≤φΠFx, xfor alln∈N. Thenxn → z ΠFx.
Recall that a Banach space E satisfies Kadec–Klee property if whenever {un} is a
sequence inEwithxn xandxn → x, it follows thatxn → x.
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}∞i1:E → Ebe a family of relatively quasi-nonexpansive mappings satisfying
NST-condition and let{Si}Ni1 :C → Cbe a family of relatively nonexpansive mappings such that
F:∞n1FTn∩Ni1FSi/∅, and
φu, Tnx φTnx, x≤φu, x 3.14
for allu∈∞n1FTn,n∈Nandx∈E. Let the sequence{xn}be generated by
x1x∈C,
unTnJ−1 N
i1
ωn,iαn,iJxn 1−αn,iJSixn,
Cn z∈C:φz, un≤φz, xn,
Qn{z∈C:xn−z, Jx−Jxn ≥0},
xn1 ΠCn∩Qnx
3.15
for everyn∈N,{ωn,i},{αn,i} ⊂0,1for alln∈Nandi1,2, . . . , NsatisfyingNi1ωn,i 1for
alln∈N,lim infn→ ∞ωn,i1−αn,i>0for alli1,2, . . . , N. Then{xn}converges strongly toΠFx.
Proof. The proof is broken into 4 steps.
Step 1{xn}is well defined. First, we show thatCn∩Qnis closed and convex. Clearly,Qnis
closed and convex. Since
φz, un≤φz, xn⇐⇒ un2− xn2−2z, Jun−Jxn ≤0, 3.16
thenCnis closed and convex. ThusCn∩Qnis closed and convex.
We next show thatF ⊂Cn∩Qn. Letu∈F. Then, fromLemma 1.7,
φu, un≤φu, xn. 3.17
Next, we show by induction thatF⊂Cn∩Qnfor alln∈N. SinceQ1C, we have
F⊂C1∩Q1. 3.18
Suppose thatF⊂Ck∩Qkfor somek∈N. Fromxk1 ΠCk∩Qkx∈Ck∩Qkand the definition
of the generalized projection, we have
xk1−z, Jx−Jxk1 ≥0 3.19
for allz∈Ck∩Qk. FromF ⊂Ck∩Qk,
xk1−p, Jx−Jxk1
≥0 3.20
for allp∈F. HenceF⊂Qk1, and we also haveF ⊂Ck1∩Qk1. So, we have∅/F⊂Cn∩Qn
for alln∈Nand hence the sequence{xn}is well defined.
Step 2 ωw{xn} ⊂ Ni1FSi. From the definition of Qn, we have xn ΠQnx. Using
Lemma 1.2, we get
φxn, x φΠQnx, x
≤φu, x−φu,ΠQnx
≤φu, x 3.21
for allu∈Qn. In particular, sincexn1∈QnandΠFx∈F⊂Qn,
φxn, x≤φxn1, x, 3.22
φxn, x≤φΠFx, x 3.23
for alln ∈N. This implies that limn→ ∞φxn, xexists and{xn}is bounded. Moreover, from
3.21andxn1∈Qn,
φxn, x≤φxn1, x−φxn1, xn. 3.24
Hence
φxn1, xn≤φxn1, x−φxn, x−→0. 3.25
It follows fromxn1 ΠCn∩Qnx∈Cnthat
φxn1, un≤φxn1, xn−→0. 3.26
From3.25,3.26, andLemma 1.4, we have
lim
So limn→ ∞xn−un0. UsingLemma 3.14, we get that
lim
n→ ∞xn−Sixn0 3.28
for alli1,2, . . . , N. Since eachSiis relatively nonexpansive,
ωw{xn} ⊂ N
i1
FSi N
i1
FSi. 3.29
Step 3 ωw{xn} ⊂ ∞n1FTn. Let yn J−1 N
i1ωn,iαn,iJxn 1 − αn,iJSixn. From Lemma 3.12, we have
lim
n→ ∞Tnyn−yn0, 3.30
andωw{xn}ωw{yn}. It follows from NST-condition thatωw{xn}ωw{yn} ⊂∞n1FTn.
Step 4xn → ΠFx. From Steps2and3, we haveωw{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
x1x∈C,
ynJ−1αnJxn 1−αnJSxn,
un∈Csuch thatfun, yr1 n
y−un, Jun−Jyn≥0 ∀y∈C,
Cn z∈C:φz, un≤φz, xn,
Qn{z∈C:xn−z, Jx−Jxn ≥0},
xn1 ΠCn∩Qnx
3.31
for everyn ∈ N,{αn} ⊂ 0,1satisfyinglim supn→ ∞αn < 1and{rn} ⊂a,∞for somea > 0. Then,{xn}converges strongly toΠFS∩EPfx, whereΠFS∩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
LetCbe a closed subset of a Banach spaceE. Recall that a mappingT :C → Cis closed if for each{xn}inC, ifxn → xandTxn → y, thenTx y. A family of mappings{Tn}:C → E
with∞n1FTn/∅is said to satisfy the∗-condition if for each bounded sequence{zn}inC,
lim
n→ ∞zn−Tnzn0, zn−→zimplyz∈ ∞
n1
FTn. 3.32
Remark 3.6. 1If{Tn}satisfies NST-condition, then{Tn}satisfies∗-condition.
2IfTn≡T andTis 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}∞n1 :E → Ebe a family of relatively quasi-nonexpansive mappings
satisfying∗-condition and let{Si}Ni1 :C → Cbe a family of closed relatively quasi-nonexpansive
mappings such thatF:∞n1FTn∩Ni1FSi/∅, and
φu, Tnx φTnx, x≤φu, x 3.33
for allu∈∞n1FTn,n∈N,andx∈E. Let the sequence{xn}be generated by
x0x∈C, Q0C,
unTnJ−1 N
i1
ωn,iαn,iJxn 1−αn,iJSixn,
C0
z∈C:φz, u0≤φz, x0
,
Cnz∈Cn−1∩Qn−1:φz, un≤φz, xn,
Qn{z∈Cn−1∩Qn−1:xn−z, Jx−Jxn ≥0},
xn1 ΠCn∩Qnx
3.34
for everyn∈N,{ωn,i},{αn,i} ⊂0,1satisfyingiN1ωn,i1andlim infn→ ∞ωn,i1−αn,i>0for
alli1,2, . . . , N. Then{xn}converges strongly toΠFx.
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 inC. We can follow the proof ofTheorem 3.3and conclude
that
lim
exists. Moreover, asxnm∈Qnfor alln, mandxn ΠQnx,
φxnm, xn φxnm,ΠQnx
≤φxnm, x−φΠQnx, 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 functionhsuch thath0 0 and
hxnm−xn≤φxnm, x−φxn, x. 3.37
Since limn→ ∞φxn, xexists, we have that{xn}is a Cauchy sequence. Therefore,xn → zfor
somez∈C.
Step 3z∈Ni1FSi. Sincexn1 ΠCn∩Qnx∈Cn, we have
φxn1, un≤φxn1, xn−→φz, z 0. 3.38
ByLemma 1.4and the boundedness of{xn}, we have
lim
n→ ∞xn1−un0. 3.39
So, we have limn→ ∞xn−un0. UsingLemma 3.14, we get that
lim
n→ ∞xn−Sixn0 3.40
for alli1,2, . . . , N. Since eachSiis closed,z∈Ni1FSi.
Step 4z∈∞n1FTn. Letyn J−1Ni1ωn,iαn,iJxn 1−αn,iJSixn. FromLemma 3.12, we have limn→ ∞yn−Tnyn0 andyn → z. It follows from∗-condition thatz∈n∞1FTn.
Step 5xn → ΠFx. From Steps3and4, we haveωw{xn} ⊂ F. The conclusion follows by
Lemma 3.2and3.23.
Corollary 3.8see12, 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 supn→ ∞αn <1. Define a sequence{xn}inCby the following algorithm:
x0∈Cchosen arbitrarily,
ynJ−1αnJxn 1−αnJTxn,
Cnz∈Cn−1∩Qn−1:φ
z, yn≤φz, xn,
C0
z∈C:φz, y0
≤φz, x0
,
Qn{z∈Cn−1∩Qn−1:xn−z, Jx0−Jxn ≥0},
Q0C,
xn1 ΠCn∩Qnx0.
3.41
Then{xn}converges strongly toΠFTx0.
LettingTnidentity andN2 yield the following result.
Corollary 3.9see13, 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:
x0∈Cchosen arbitrarily,
zn J−1
β1
n Jxnβn2JTxnβn3JSxn
,
ynJ−1αnJxn 1−αnJzn,
C0
z∈C:φz, y0
≤φz, x0
,
Cnz∈Cn−1∩Qn−1:φ
z, yn≤φz, xn,
Qn{z∈Cn−1∩Qn−1:xn−z, Jx0−Jxn ≥0},
Q0C,
xn1 ΠCn∩Qnx0
3.42
with the conditions:βn1, βn2, βn3∈0,1withβn1βn2βn31and
1lim infn→ ∞βn1βn2>0;
2lim infn→ ∞βn1βn3>0;
30≤αn≤α <1for 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},{βn1},{βn2},and{βn3}:
1αn, βn1∈0,1are arbitrary;
2lim infn→ ∞βn2 >0 and lim infn→ ∞βn3>0.
3.3. The Shrinking Projection Method
Theorem 3.11. LetC, E, {Tn}∞n1, {Si}Ni1be as inTheorem 3.7. Let the sequence{xn}be generated
by
x0∈Echosen arbitrarily,
C1C,
x1 ΠC1x0,
unTnJ−1 N
i1
ωn,iαn,iJxn 1−αn,iJSixn,
Cn1
z∈Cn:φz, un≤φz, xn,
xn1 ΠCn1x0
3.43
for everyn∈N,{ωn,i},{αn,i} ⊂0,1for alln∈Nandi1,2, . . . , NsatisfiesNi1ωn,i1for all
n∈N,lim infn→ ∞ωn,i1−αn,i>0for alli1,2, . . . , N. Then{xn}converges strongly toΠFx.
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 byx0
x∈C,C0Cand
ynJ−1αnJxn 1−αnJSxn,
un∈Csuch thatfun, yr1 n
y−un, Jun−Jyn≥0 ∀y∈C,
Cn1
z∈Cn:φz, un≤φz, xn,
xn1 ΠCn1x0
3.44
for everyn ∈ N∪ {0}, where J is the duality mapping on E. Assume that {αn} ⊂ 0,1satisfies
lim supn→ ∞αn <1and{rn} ⊂a,∞for somea >0. Then{xn}converges strongly toΠFS∩EPfx,
whereΠFS∩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 toR 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:
x0∈Echosen arbitrarily,
C1C,
x1 ΠC1x0,
ynJ−1αnJxnβnJTxnγnJSxn,
un∈Csuch thatfun, yr1 n
y−un, Jun−Jyn≥0 ∀y∈C,
Cn1
z∈Cn:φz, un≤φz, xn,
xn1 ΠCn1x0.
3.45
Assume that{αn},{βn},and{γn}are three sequences in0,1satisfying the restrictions:
aαnβnγn1;
blim infn→ ∞αnβn>0,lim infn→ ∞αnγn>0;
c{rn} ⊂a,∞for somea >0. Then{xn}converges strongly toΠFx0.
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 infn→ ∞βn>0 and lim infn→ ∞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:
x0∈Echosen arbitrarily,
C1C,
x1 ΠC1x0,
ynJ−1δnJxn 1−δnJzn,
znJ−1αnJxnβnJTxnγnJSxn,
un∈Csuch thatfun, yr1 n
y−un, Jun−Jzn≥0 ∀y∈C,
Cn1
z∈Cn:φz, un≤φz, xn,
xn1 ΠCn1x0.
Assume that{αn},{βn},and{γn}are three sequences in0,1satisfying the following restrictions:
aαnβnγn1;
b0≤αn<1for alln∈N∪ {0}andlim supn→ ∞αn<1;
clim infn→ ∞αnβn>0,lim infn→ ∞αnγn>0;
d{rn} ⊂a,∞for somea >0.
Then{xn}and{un}converge strongly toΠFx0.
Remark 3.17. The conclusion of Corollary 3.16 remains true under the more general restrictions; that is, we replacebandcby the following one:
blim infn→ ∞βn>0 and lim infn→ ∞γn>0.
Corollary 3.18see10, Theorem 3.1. LetCbe a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E. Let {Ti}Ni1 : C → C be a family of relatively
nonexpansive mappings such thatF :Ni1FTi/∅and letx0 ∈E. ForC1 Candx1 ΠC1x0,
define a sequence{xn}ofCas follows:
ynJ−1αnJxn 1−αnJzn,
znJ−1
β1
n Jxn N
i1
βi1
n JTixn
,
Cn1
z∈Cn :φz, yn≤φz, xn,
xn1 ΠCn1x0,
3.47
where{αn},{βni} ⊂0,1satisfies the following restrictions:
i0≤αn<1for alln∈N∪ {0}andlim supn→ ∞αn<1;
ii0≤βni≤1for alli1,2, . . . , N1,Ni11β
i
n 1for alln∈N∪ {0}. If
aeitherlim infn→ ∞βn1βni1>0for alli1,2, . . . , Nor
blimn→ ∞βn10andlim infn→ ∞βnk1βnl1>0for alli /j, k, l1,2, . . . , N.
then the sequence{xn}converges strongly toΠFx0.
Remark 3.19. The conclusion of Corollary 3.18 remains true under the more general restrictions on{αn},{βni}:
1αn, βn1∈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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