FOR A FINITE FAMILY OF NONEXPANSIVE
MAPPINGS IN BANACH SPACES
FENG GU AND JING LU
Received 18 January 2006; Revised 22 August 2006; Accepted 23 August 2006
The purpose of this paper is to study the weak and strong convergence of implicit iter-ation process with errors to a common fixed point for a finite family of nonexpansive mappings in Banach spaces. The results presented in this paper extend and improve the corresponding results of Chang and Cho (2003), Xu and Ori (2001), and Zhou and Chang (2002).
Copyright © 2006 F. Gu and J. Lu. 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 and preliminaries
Throughout this paper we assume thatEis a real Banach space andT:E→Eis a map-ping. We denote byF(T) andD(T) the set of fixed points and the domain ofT, respec-tively.
Recall thatEis said to satisfyOpial condition[11], if for each sequence{xn}inE, the
condition that the sequencexn→xweakly implies that
lim inf
n→∞ xn−x<lim infn→∞ xn−y (1.1) for all y∈Ewith y=x. It is well known that (see, e.g., Dozo [9]) inequality (1.1) is equivalent to
lim sup
n→∞
x
n−x<lim sup n→∞
x
n−y. (1.2)
Definition 1.1. LetDbe a closed subset ofEand letT:D→Dbe a mapping.
(1)Tis said to bedemiclosedat the origin, if for each sequence{xn}inD, the
condi-tionsxn→x0weakly andTxn→0 strongly implyTx0=0.
Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2006, Article ID 82738, Pages1–11
(2)T is said to besemicompact, if for any bounded sequence {xn} inD such that
xn−Txn →0 (n→ ∞), then there exists a subsequence{xni} ⊂ {xn}such that xni→ x∗∈D.
(3)Tis said to benonexpansive, ifTx−T y ≤ x−y, for alln≥1 for allx,y∈D. Let E be a Hilbet space, let K be a nonempty closed convex subset of E, and let {T1,T2,...,TN}:K→KbeN nonexpansive mappings. In 2001, Xu and Ori [19]
intro-duced the following implicit iteration process{xn}defined by
xn=αnxn−1+ (1−αn)Tn(modN)xn, ∀n≥1, (1.3)
wherex0∈Kis an initial point,{αn}n≥1is a real sequence in (0, 1) and proved the weakly
convergence of the sequence{xn} defined by (1.3) to a common fixed point p∈F= N
i=1F(Ti).
Recently concerning the convergence problems of an implicit (or nonimplicit) itera-tive process to a common fixed point for a finite family of asymptotically nonexpansive mappings (or nonexpansive mappings) in Hilbert spaces or uniformly convex Banach spaces have been considered by several authors (see, e.g., Bauschke [1], Chang and Cho [3], Chang et al. [4], Chidume et al. [5], Goebel and Kirk [6], G ´ornicki [7], Halpern [8], Lions [10], Reich [12], Rhoades [13], Schu [14], Shioji and Takahashi [15], Tan and Xu [16,17], Wittmann [18], Xu and Ori [19], and Zhou and Chang [20]).
In this paper, we introduce the following new implicit iterative sequence{xn}with
errors:
x1=α1x0+β1T1
α1x0+β1T1x1+γ1v1
+γ1u1,
x2=α2x1+β2T2
α2x1+β2T2x2+γ2v2
+γ2u2,
.. .
xN=αNxN−1+βNTNαNxN−1+βNTNxN+γNvN+γNuN,
xN+1=αN+1xN+βN+1T1
αN+1xN+βN+1T1xN+1+γN+1vN+1
+γN+1uN+1,
.. . x2N=α2Nx2N−1+β2NTN
α2Nx2N−1+β2NTNx2N+γ2Nv2N
+γ2Nu2N,
x2N+1=α2N+1x2N+β2N+1T1
α2N+1x2N+β2N+1T1x2N+1+γ2N+1v2N+1
+γ2N+1u2N+1,
.. .
(1.4)
for a finite family of nonexpansive mappings{Ti}Ni=1:K→K, where {αn},{βn},{γn},
{αn},{βn}, and{γn}are six sequences in [0, 1] satisfyingαn+βn+γn=αn+βn+γn=1
inK, which can be written in the following compact form:
xn=αnxn−1+βnTn(modN)yn+γnun,
yn=αnxn−1+βnTn(modN)xn+γnvn, ∀n≥1.
(1.5)
Especially, if{Ti}Ni=1:K→KareNnonexpansive mappings,{αn},{βn},{γn}are three
sequences in [0, 1], andx0is a given point inK, then the sequence{xn}defined by
xn=αnxn−1+βnTn(modN)xn−1+γnun, ∀n≥1 (1.6)
is called the explicit iterative sequence for a finite family of nonexpansive mappings {Ti}Ni=1.
The purpose of this paper is to study the weak and strong convergence of iterative sequence {xn}defined by (1.5) and (1.6) to a common fixed point for a finite family
of nonexpansive mappings in Banach spaces. The results presented in this paper not only generalized and extend the corresponding results of Chang and Cho [3], Xu and Ori [19], and Zhou and Chang [20], but also in the case ofγn=γn=0 orβn=γn=0 are also new.
In order to prove the main results of this paper, we need the following lemmas. Lemma1.2 [2]. Let Ebe a uniformly convex Banach space, letK be a nonempty closed convex subset ofE, and letT:K→K be a nonexpansive mapping withF(T)= ∅. Then I−Tis semiclosed at zero, that is, for each sequence{xn}inK,if{xn}converges weakly to
q∈Kand{(I−T)xn}converges strongly to0, then(I−T)q=0.
Lemma1.3 [17]. Let{an}and{bn}be two nonnegative real sequences satisfying the
fol-lowing condition:an+1≤an+bn for alln≥n0, where n0 is some nonnegative integer. If ∞
n=0bn<∞, thenlimn→∞anexists. If in addition{an}has a subsequence which converges
strongly to zero, thenlimn→∞an=0.
Lemma1.4 [14]. LetEbe a uniformly convex Banach space, letbandcbe two constants with 0< b < c <1. Suppose that{tn}is a sequence in[b,c]and{xn}and{yn}are two sequence in
Esuch thatlimn→∞tnxn+ (1−tn)yn =d,lim supn→∞xn ≤d, andlim supn→∞yn ≤
dhold for somed≥0, thenlimn→∞xn−yn =0.
Lemma1.5. LetEbe a real Banach space, letK be a nonempty closed convex subset ofE, and let{T1,T2,...,TN}:K→K beN nonexpansive mappings withF=Ni=1F(Ti)= ∅.
Let{un}and{vn}be two bounded sequences inK, and let{αn},{βn},{γn},{αn},{βn},
and{γn}be six sequences in[0, 1]satisfying the following conditions:
(i)αn+βn+γn=αn+βn+γn=1, for alln≥1;
(ii)τ=sup{βn:n≥1}<1;
(iii)∞n=1γn<∞, ∞
n=1γn<∞.
If{xn}is the implicit iterative sequence defined by (1.5), then for eachp∈F=Ni=1F(Ti)
Proof. SinceF=N
n=1F(Ti)= ∅, for any givenp∈F, it follows from (1.5) that xn−p=1−βn−γn
xn−1+βnTn(modN)yn+γnun−p
≤1−βn−γnxn−1−p+βnTn(modN)yn−p+γnun−p
=1−βn−γnxn−1−p+βnTn(modN)yn−Tn(modN)p+γnun−p
≤1−βnxn−1−p+βnyn−p+γnun−p.
(1.7) Again it follows from (1.5) that
yn−p=1−βn−γnxn−1+βnTn(modN)xn+γnvn−p
≤1−βn−γnxn−1−p+βnTn(modN)xn−p+γnvn−p
=1−βn−γnxn−1−p+βnTn(modN)xn−Tn(modN)p+γnvn−p
≤1−βnxn−1−p+βnxn−p+γnvn−p.
(1.8) Substituting (1.8) into (1.7), we obtain that
xn−p≤
1−βnβnxn−1−p+βnβnxn−p
+βnγnvn−p+γnun−p.
(1.9)
Simplifying we have
1−βnβnxn−p≤1−βnβnxn−1−p+σn, (1.10)
whereσn=βnγnvn−p+γnun−p.By condition (iii) and the boundedness of the
sequences{βn},{un−p}, and{vn−p}, we have ∞
n=1σn<∞.From condition (ii)
we know that
βnβn≤βn≤τ <1 and so 1−βnβn≥1−τ >0; (1.11)
hence, from (1.10) we have
x
n−p≤xn−1−p+ σn
1−τ=xn−1−p+bn, (1.12) wherebn=σn/(1−τ) with
∞
i=1bn<∞.
Takingan= xn−1−pin inequality (1.12), we havean+1≤an+bn, for alln≥1, and
satisfied all conditions inLemma 1.3. Therefore the limit limn→∞xn−pexists. This
completes the proof ofLemma 1.5.
2. Main results
Theorem2.1. LetEbe a real Banach space, letK be a nonempty closed convex subset of E, and let{T1,T2,...,TN}:K→KbeNnonexpansive mappings withF=
N
i=1F(Ti)= ∅
(the set of common fixed points of {T1,T2,...,TN}). Let {un} and{vn}be two bounded
sequences inK, and let{αn},{βn},{γn},{αn},{βn}, and {γn}be six sequences in [0, 1]
satisfying the following conditions:
(i)αn+βn+γn=αn+βn+γn=1, for alln≥1;
(ii)τ=sup{βn:n≥1}<1;
(iii)∞n=1γn<∞, ∞
n=1γn<∞.
Then the implicit iterative sequence{xn}defined by (1.5) converges strongly to a common
fixed pointp∈F=N
i=1F(Ti)if and only if
lim inf
n→∞ d(xn,F)=0. (2.1)
Proof. The necessity of condition (2.1) is obvious.
Next we prove the sufficiency ofTheorem 2.1. For any given p∈F, it follows from (1.12) inLemma 1.5that
xn−p≤xn−1−p+bn ∀n≥1, (2.2)
wherebn=σn/(1−τ) with ∞
n=1bn<∞. Hence, we have
dxn,F
≤dxn−1,F
+bn ∀n≥1. (2.3)
It follows from (2.3) andLemma 1.3that the limit limn→∞d(xn,F) exists. By condition
(2.1), we have limn→∞d(xn,F)=0.
Next we prove that the sequence{xn}is a Cauchy sequence inK. In fact, for any
posi-tive integersmandn, from (2.2), it follows that
x
n+m−p≤xn+m−1−p+bn+m≤xn+m−2−p+bn+m−1+bn+m
≤ ··· ≤xn−p+ n+m
i=n+1
bi≤xn−p+
∞
i=n+1
bi.
(2.4)
Since limn→∞d(xn,F)=0 and ∞
n=1bn<∞, for any given>0, there exists a positive
integern0such thatd(xn,F)</8, ∞
i=n+1bi</2, for alln≥n0. Therefore there exists
p1∈Fsuch thatxn−p1</4, for alln≥n0. Consequently, for anyn≥n0and for all
m≥1, from (2.4), we have
x
n+m−xn≤xn+m−p1+xn−p1 ≤2xn−p1+
∞
i=n+1
bi<2·4+2=.
(2.5)
This implies that {xn} is a Cauchy sequence in K. By the completeness of K, we can
assume that limn→∞xn=x∗∈K. Moreover, since the set of fixed points of a nonexpansive
mapping is closed, so isF; thusx∗∈Ffrom limn→∞d(xn,F)=0, that is,x∗is a common
Theorem2.2. LetEbe a real Banach space, letK be a nonempty closed convex subset of E, and let{T1,T2,...,TN}:K→KbeNnonexpansive mappings withF=
N
i=1F(Ti)= ∅
(the set of common fixed points of{T1,T2,...,TN}). Let{un}be a bounded sequence inK,
and let{αn},{βn}, and{γn}be three sequences in[0, 1]satisfying the following conditions:
(i)αn+βn+γn=1, for alln≥1;
(ii)τ=sup{βn:n≥1}<1;
(iii)∞n=1γn<∞.
Then the explicit iterative sequence{xn}defined by (1.6) converges strongly to a common
fixed pointp∈F=N
i=1F(Ti)if and only iflim infn→∞d(xn,F)=0.
Proof. Taking βn =γn=0, for all n≥1 in Theorem 2.1, then the conclusion of Theorem 2.2can be obtained fromTheorem 2.1immediately. This completes the proof
ofTheorem 2.2.
Theorem2.3. LetEbe a real uniformly convex Banach space satisfying Opial condition, let K be a nonempty closed convex subset ofE, and let{T1,T2,...,TN}:K→K beN
nonex-pansive mappings withF=N
i=1F(Ti)= ∅. Let{un}and{vn}be two bounded sequences
inK, and let{αn},{βn},{γn},{αn},{βn}, and{γn}be six sequences in[0, 1]satisfying the
following conditions:
(i)αn+βn+γn=αn+βn+γn=1, for alln≥1;
(ii) 0< τ1=inf{βn:n≥1} ≤sup{βn:n≥1} =τ2<1;
(iii)βn→0 (n→ ∞);
(iv)∞n=1γn<∞, ∞
n=1γn<∞.
Then the implicit iterative sequence{xn}defined by (1.5) converges weakly to a common
fixed point of{T1,T2,...,TN}.
Proof. First, we prove that
lim
n→∞xn−Tn(modN)+jxn=0, ∀j=1, 2,...,N. (2.6)
Letp∈F. Putd=limn→∞xn−p. It follows from (1.5) that
xn−p=1−βn xn−1−p+γn
un−xn−1
+βn Tn(modN)yn−p+γn
un−xn−1−→d, n−→ ∞.
(2.7)
Again since limn→∞xn−pexists, so{xn} is a bounded sequence inK. By virtue of
condition (iv) and the boundedness of sequences{xn}and{un}we have
lim sup
n→∞
x
n−1−p+γn
un−xn−1
≤lim sup
n→∞
x
n−1−p+ lim sup n→∞ γn
u
n−xn−1=d, p∈F.
It follows from (1.8) and condition (iii) that lim sup
n→∞
Tn(modN)yn−p+γn
un−xn−1
≤lim sup
n→∞
yn−p+ lim sup n→∞ γn
un−xn−1
=lim sup
n→∞
yn−p
≤lim sup
n→∞
1−βnxn−1−p+βnxn−p+γnvn−p
≤lim sup
n→∞
1−βnxn−1−p+ lim sup n→∞
βnxn−p+ lim sup n→∞ γn
vn−p
=d, p∈F.
(2.9)
Therefore, from condition (ii), (2.7)–(2.9), andLemma 1.4we know that
lim
n→∞Tn(modN)yn−xn−1=0. (2.10) From (1.5) and (2.10) we have
xn−xn−1=βn Tn(modN)yn−xn−1
+γnun−xn−1
≤βnTn(modN)yn−xn−1+γnun−xn−1−→0, n−→ ∞,
(2.11)
which implies that
lim
n−→∞xn−xn−1=0 (2.12)
and so
lim
n→∞xn−xn+j=0 ∀j=1, 2,...,N. (2.13) On the other hand, we have
xn−Tn(modN)xn≤xn−xn−1+xn−1−Tn(modN)yn
+Tn(modN)yn−Tn(modN)xn.
(2.14)
Now, we consider the third term on the right-hand side of (2.14). From (1.5) we have
Tn(modN)yn−Tn(modN)xn
≤yn−xn=αn
xn−1−xn
+βn
Tn(modN)xn−xn
+γn
vn−xn
≤αnxn−1−xn+βnTn(modN)xn−xn+γnvn−xn.
(2.15)
Substituting (2.15) into (2.14), we obtain that
xn−Tn(modN)xn≤
1 +αnxn−xn−1+xn−1−Tn(modN)yn
+βnTn(modN)xn−xn+γnvn−xn.
Hence, by virtue of conditions (iii), (iv), (2.10), (2.12) and the boundedness of sequences {Tn(modN)xn−xn}and{vn−xn}we have
lim
n→∞xn−Tn(modN)xn=0. (2.17) Therefore, from (2.13) and (2.17), for anyj=1, 2,...,N, we have
xn−Tn(modN)+jxn≤xn−xn+j+xn+j−Tn(modN)+jxn+j
+Tn(modN)+jxn+j−Tn(modN)+jxn
≤2xn−xn+j+xn+j−Tn(modN)+jxn+j−→0, n−→ ∞.
(2.18)
That is, (2.6) holds.
SinceEis uniformly convex, every bounded subset ofEis weakly compact. Again since {xn}is a bounded sequence inK, there exists a subsequence{xnk} ⊂ {xn}such that{xnk} converges weakly toq∈K.
Without loss of generality, we can assume thatnk=i(modN), whereiis some positive
integer in{1, 2,...,N}. Otherwise, we can take a subsequence{xnk j} ⊂ {xnk} such that nkj=i(modN). For anyl∈ {1, 2,...,N}, there exists an integerj∈ {1, 2,...,N}such that nk+j=l(modN). Hence, from (2.18) we have
lim
k→∞
x
nk−Tlxnk=0, l=1, 2,...,N. (2.19)
ByLemma 1.2, we know thatq∈F(Tl). By the arbitrariness ofl∈ {1, 2,...,N}, we know
thatq∈F=N j=1F(Tj).
Finally, we prove that{xn}converges weakly toq. In fact, suppose the contrary, then
there exists some subsequence{xnj} ⊂ {xn}such that{xnj}converges weakly toq1∈K andq1=q. Then by the same method as given above, we can also prove thatq1∈F= N
j=1F(Tj).
Taking p=q and p=q1 and by using the same method given in the proof of Lemma 1.5, we can prove that the following two limits exist and limn→∞xn−q =d1
and limn→∞xn−q1 =d2, whered1andd2are two nonnegative numbers. By virtue of
the Opial condition ofE, we have
d1=lim sup nk−→∞
xn
k−q<lim sup
nk→∞
xn
k−q1=d2
=lim sup
nj→∞
xn
j−q1<lim sup
nj→∞
xn
j−q=d1.
(2.20)
This is a contradiction. Henceq1=q. This implies that{xn}converges weakly toq. This
completes the proof ofTheorem 2.3.
Theorem2.4. LetEbe a real uniformly convex Banach space satisfying Opial condition, let K be a nonempty closed convex subset ofE, and let{T1,T2,...,TN}:K→K beN
nonex-pansive mappings withF=N
{αn},{βn}, and{γn}be three sequences in[0, 1]satisfying the following conditions:
(i)αn+βn+γn=1,∀n≥1;
(ii) 0< τ1=inf{βn:n≥1} ≤sup{βn:n≥1} =τ2<1;
(iii)∞n=1γn<∞.
Then the explicit iterative sequence{xn}defined by (1.6) converges weakly to a common
fixed point of{T1,T2,...,TN}.
Proof. Taking βn =γn=0, for all n≥1 in Theorem 2.3, then the conclusion of Theorem 2.4can be obtained fromTheorem 2.3immediately. This completes the proof
ofTheorem 2.4.
Theorem2.5. LetEbe a real uniformly convex Banach space, letKbe a nonempty closed convex subset ofE, and let{T1,T2,...,TN}:K→KbeNnonexpansive mappings withF= N
i=1F(Ti)= ∅and there exists anTj, 1≤j≤N, which is semicompact (without loss of
generality, assume thatT1is semicompact). Let{un}and{vn}be two bounded sequences in
K, and let{αn},{βn},{γn},{αn},{βn}, and{γn}be six sequences in[0, 1]satisfying the
following conditions:
(i)αn+βn+γn=αn+βn+γn=1, for alln≥1;
(ii) 0< τ1=inf{βn:n≥1} ≤sup{βn:n≥1} =τ2<1;
(iii)βn→0 (n→ ∞);
(iv)∞n=1γn<∞, ∞
n=1γn<∞.
Then the implicit iterative sequence{xn}defined by (1.5) converges strongly to a common
fixed point of{T1,T2,...,TN}inK.
Proof. For any givenp∈F=N
i=1F(Ti), by the same method as given in provingLemma 1.5and (2.19), we can prove that
lim
n→∞xn−p=d, (2.21)
whered≥0 is some nonnegative number, and
lim
k→∞
x
nk−Tlxnk=0, l=1, 2,...,N. (2.22)
Especially, we have
lim
k→∞
x
nk−T1xnk=0. (2.23)
By the assumption,T1is semicompact; therefore it follows from (2.23) that there exists a
subsequence{xnki} ⊂ {xnk}such thatxnki→x∗∈K. Hence from (2.22) we have that
x∗−T
lx∗=lim ki→∞
xn
ki−Tlxnki=0 ∀l=1, 2,...,N, (2.24)
which implies thatx∗∈F=N
i=1F(Ti).Take p=x∗ in (2.21), similarly we can prove
Theorem2.6. LetEbe a real uniformly convex Banach space, letKbe a nonempty closed convex subset of E, and let {T1,T2,...,TN}:K→K be N nonexpansive mappings with
F=N
i=1F(Ti)= ∅and there exists anTj, 1≤j≤N, which is semicompact (without loss
of generality, assume thatT1is semicompact). Let{un}be a bounded sequence inK, and let
{αn},{βn}, and{γn}be three sequences in[0, 1]satisfying the following conditions:
(i)αn+βn+γn=1, for alln≥1;
(ii) 0< τ1=inf{βn:n≥1} ≤sup{βn:n≥1} =τ2<1;
(iii)∞n=1γn<∞.
Then the explicit iterative sequence{xn}defined by (1.6) converges strongly to a common
fixed point of{T1,T2,...,TN}inK.
Proof. Taking βn =γn=0, for all n≥1 in Theorem 2.5, then the conclusion of Theorem 2.6can be obtained fromTheorem 2.5immediately. This completes the proof
ofTheorem 2.6.
Remark 2.7. Theorems2.3–2.6improve and extend the corresponding results in Chang and Cho [3, Theorem 3.1] and Zhou and Chang [20, Theorem 3], and the implicit it-erative process{xn}defined by (1.3) is replaced by the more general implicit or explicit
iterative process{xn}defined by (1.5) or (1.6).
Remark 2.8. Theorems2.3–2.6generalize and improve the main results of Xu and Ori [19] in the following aspects.
(1) The class of Hilbert spaces is extended to that of Banach spaces satisfying Opial’s or semicompactness condition.
(2) The implicit iterative process{xn}defined by (1.3) is replaced by the more general
implicit or explicit iterative process{xn}defined by (1.5) or (1.6).
Remark 2.9. The iterative algorithm used in this paper is different from those in [1,8,10,
14,18].
Acknowledgments
The present studies were supported by the Natural Science Foundation of Zhejiang Province (Y605191), the Natural Science Foundation of Heilongjiang Province (A0211), and the Scientific Research Foundation from Zhejiang Province Education Committee (20051897).
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Feng Gu: Department of Mathematics, Institute of Applied Mathematics, Hangzhou Teacher’s College, Hangzhou, Zhejiang 310036, China E-mail address:[email protected]
