# A New Iterative Method for Common Fixed Points of a Finite Family of Nonexpansive Mappings

Authors: Suwicha Imnang; Suthep Suantai
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## Abstract

LetX be a real uniformly convex Banach space and C a closed convex nonempty subset of X. Let {Ti}i=1r be a finite family of nonexpansive self-mappings of C. For a given x1∈C, let {xn} and {xn(i)}, i=1,2,…,r, be sequences defined
xn(0)=xn,xn(1)=an1(1)T1xn(0)+(1-an1(1))xn(0),xn(2)=an2(2)T2xn(1)+an1(2)T1xn+(1-an2(2)-an1(2))xn,…,xn+1=xn(r)=anr(r)Trxn(r-1)+an(r-1)(r)Tr-1xn(r-2)+⋯+an1(r)T1xn+(1-an(r)(r)-an(r-1)(r)-⋯-an1(r))xn,n≥1, where ani(j)∈[0,1] for all j∈{1,2,…,r}, n∈ℕ and i=1,2,…,j. In this paper, weak and strong convergence theorems of the sequence {xn} to a common fixed point of a finite family of nonexpansive mappings Ti(i=1,2,…,r) are established under some certain control conditions.


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## Body

## 1. Introduction

LetX be a real Banach space, C a nonempty closed convex subset of X, and T:C→C a mapping. Recall that T is nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for all x,y∈C. Let Ti:C→C,i=1,2,…,r, be nonexpansive mappings. Let Fix(Ti) denote the fixed points set of Ti, that is, Fix(Ti):={x∈C:Tix=x}, and let F:=⋂i=1r‍Fix(Ti).For a givenx1∈C, and a fixed r∈ℕ (ℕ denote the set of all positive integers), compute the iterative sequences {xn(0)},{xn(1)},{xn(2)},…,{xn(r)} by
(1.1)xn(0)=xn,xn(1)=an1(1)T1xn(0)+(1-an1(1))xn(0),xn(2)=an2(2)T2xn(1)+an1(2)T1xn+(1-an2(2)-an1(2))xn,⋮xn+1=xn(r)=anr(r)Trxn(r-1)+an(r-1)(r)Tr-1xn(r-2)+⋯+an1(r)T1xn+(1-an(r)(r)-an(r-1)(r)-⋯-an1(r))xn,n≥1,
where ani(j)∈[0,1] for all j∈{1,2,…,r}, n∈ℕ and i=1,2,…,j. If ani(j):=0, for all n∈ℕ, j∈{1,2,…,r-1} and i=1,2,…,j, then (1.1) reduces to the iterative scheme
(1.2)xn+1=Snxn,n≥1,
where Sn:=anr(r)Tr+an(r-1)(r)Tr-1+⋯+an1(r)T1+(1-an(r)(r)-an(r-1)(r)-⋯-an1(r))I, ani(r)∈[0,1] for all i=1,2,…,r and n∈ℕ.Ifani(j):=0, for all n∈ℕ, j∈{1,2,…,r-1}, i=1,2,…,j and ani(r):=αi, for all n∈ℕ for all i=1,2,…,r, then (1.1) reduces to the iterative scheme defined by Liu et al. [1]
(1.3)xn+1=Sxn,n≥1,
where S:=αrTr+αr-1Tr-1+⋯+α1T1+(1-αr-αr-1-⋯-α1)I, αi≥0 for all i=2,3,…,r and 1-αr-αr-1-⋯-α1>0. They showed that {xn} defined by (1.3) converges strongly to a common fixed point of Ti,i=1,2,…,r, in Banach spaces, provided that Ti,i=1,2,…,r satisfy condition A. The result improves the corresponding results of Kirk [2], Maiti and Saha [3] and Sentor and Dotson [4].Ifr=2 and an1(2):=0 for all n∈ℕ, then (1.1) reduces to a generalization of Mann and Ishikawa iteration given by Das and Debata [5] and Takahashi and Tamura [6]. This scheme dealts with two mappings:
(1.4)xn(1)=an1(1)T1xn+(1-an1(1))xn,xn+1=xn(2)=an2(2)T2xn(1)+(1-an2(2))xn,n≥1,
where {an1(1)},{an2(2)} are appropriate sequences in [0,1].The purpose of this paper is to establish strong convergence theorems in a uniformly convex Banach space of the iterative sequence{xn} defined by (1.1) to a common fixed point of Ti(i=1,2,…,r) under some appropriate control conditions in the case that one of Ti(i=1,2,…,r) is completely continuous or semicompact or {Ti}i=1r satisfies condition (B). Moreover, weak convergence theorem of the iterative scheme (1.1) to a common fixed point of Ti(i=1,2,…,r) is also established in a uniformly convex Banach spaces having the Opial's condition.


## 2. Preliminaries

In this section, we recall the well-known results and give a useful lemma that will be used in the next section.Recall that a Banach spaceX is said to satisfy Opial's condition [7] if xn→x weakly as n→∞ and x≠y imply that lim supn→∞∥xn-x∥<lim supn→∞∥xn-y∥. A finite family of mappings Ti:C→C(i=1,2,…,r) with F:=⋂i=1rFix(Ti)≠∅ is said to satisfy condition (B) [8] if there is a nondecreasing function f:[0,∞)→[0,∞) with f(0)=0 and f(t)>0 for all t∈(0,∞) such that max1≤i≤r{∥x-Tix∥}≥f(d(x,F)) for all x∈C, where d(x,F)=inf{∥x-p∥:p∈F}.Lemma 2.1 (see [9, Theorem 2]).
Letp>1, r>0 be two fixed numbers. Then a Banach space X is uniformly convex if and only if there exists a continuous, strictly increasing, and convex function g:[0,∞)→[0,∞),   g(0)=0 such that
(2.1)∥λx+(1-λ)y∥p≤λ∥x∥p+(1-λ)∥y∥p-wp(λ)g(∥x-y∥),
for all x,y in Br={x∈X:∥x∥≤r},λ∈[0,1], where
(2.2)wp(λ)=λ(1-λ)p+λp(1-λ).Lemma 2.2 (see [10, Lemma 1.6]).
LetX be a uniformly convex Banach space, C a nonempty closed convex subset of X, and T:C→C nonexpansive mapping. Then I-T is demiclosed at 0, that is, if xn→x weakly and xn-Txn→0 strongly, then x∈Fix(T).Lemma 2.3 (see [11, Lemma 2.7]).
LetX be a Banach space which satisfies Opial's condition and let {xn} be a sequence in X. Let u,v∈X be such that limn→∞∥xn-u∥ and limn→∞∥xn-v∥ exist. If {xnk} and {xmk} are subsequences of {xn} which converge weakly to u and v, respectively, then u=v.Lemma 2.4.
LetX be a uniformly convex Banach space and Br={x∈X:∥x∥≤r},r>0. Then for each n∈ℕ, there exists a continuous, strictly increasing, and convex function g:[0,∞)→[0,∞),g(0)=0 such that
(2.3)∥∑i=1n‍αixi∥2≤∑i=1n‍αi∥xi∥2-α1α2g(∥x1-x2∥),
for all xi∈Br and all αi∈[0,1](i=1,2,…,n) with ∑i=1n‍αi=1.Proof.
Clearly (2.3) holds for n=1,2, by Lemma 2.1. Next, suppose that (2.3) is true when n=k-1. Let xi∈Br and αi∈[0,1],i=1,2,…,k with ∑i=1kαi=1. Then αk-1/(1-∑i=1k-2αi)xk-1+αk/(1-∑i=1k-2αi)xk∈Br. By Lemma 2.1, we obtain that
(2.4)∥αk-11-∑i=1k-2αixk-1+αk1-∑i=1k-2αixk∥2≤αk-11-∑i=1k-2αi∥xk-1∥2+αk1-∑i=1k-2αi∥xk∥2.
By the inductive hypothesis, there exists a continuous, strictly increasing and convex function g:[0,∞)→[0,∞),g(0)=0 such that
(2.5)∥∑i=1k-1‍βiyi∥2≤∑i=1k-1‍βi∥yi∥2-β1β2g(∥y1-y2∥)
for all yi∈Br and all βi∈[0,1],i=1,2,…,k-1 with ∑i=1k-1‍βi=1. It follows that
(2.6)∥∑i=1k‍αixi∥2=∥∑i=1k-2‍αixi+(1-∑i=1k-2‍αi)(αk-1xk-11-∑i=1k-2αi+αkxk1-∑i=1k-2αi)∥2≤∑i=1k-2‍αi∥xi∥2+(1-∑i=1k-2‍αi)∥αk-1xk-11-∑i=1k-2αi+αkxk1-∑i=1k-2αi∥2-α1α2g(∥x1-x2∥)≤∑i=1k-2‍αi∥xi∥2+(1-∑i=1k-2‍αi)(αk-1∥xk-1∥21-∑i=1k-2αi+αk∥xk∥21-∑i=1k-2αi)-α1α2g(∥x1-x2∥)=∑i=1k‍αi∥xi∥2-α1α2g(∥x1-x2∥).
Hence, we have the lemma.


## 3. Main Results

In this section, we prove weak and strong convergence theorems of the iterative scheme (1.1) for a finite family of nonexpansive mappings in a uniformly convex Banach space. In order to prove our main results, the following lemmas are needed.The next lemma is crucial for proving the main theorems.Lemma 3.1.
LetX be a Banach space and C a nonempty closed and convex subset of X. Let {Ti}i=1r be a finite family of nonexpansive self-mappings of C. Let ani(j)∈[0,1] for all j∈{1,2,…,r}, n∈ℕ and i=1,2,…,j. For a given x1∈C, let the sequence {xn} be defined by (1.1). If F≠∅, then ∥xn+1-p∥≤∥xn-p∥ for all n∈ℕ and limn→∞∥xn-p∥ exists for all p∈F.Proof.
Letp∈F. For each n≥1, we note that
(3.1)∥xn(1)-p∥=∥an1(1)T1xn+(1-an1(1))xn-p∥≤an1(1)∥T1xn-p∥+(1-an1(1))∥xn-p∥≤an1(1)∥xn-p∥+(1-an1(1))∥xn-p∥=∥xn-p∥.
It follows from (3.1) that
(3.2)∥xn(2)-p∥=∥an2(2)T2xn(1)+an1(2)T1xn+(1-an2(2)-an1(2))xn-p∥≤an2(2)∥T2xn(1)-p∥+an1(2)∥T1xn-p∥+(1-an2(2)-an1(2))∥xn-p∥≤an2(2)∥xn(1)-p∥+an1(2)∥xn-p∥+(1-an2(2)-an1(2))∥xn-p∥≤∥xn-p∥.
By (3.1) and (3.2), we have
(3.3)∥xn(3)-p∥=∥an3(3)T3xn(2)+an2(3)T2xn(1)+an1(3)T1xn+(1-an3(3)-an2(3)-an1(3))xn-p∥≤an3(3)∥T3xn(2)-p∥+an2(3)∥T2xn(1)-p∥+an1(3)∥T1xn-p∥+(1-an3(3)-an2(3)-an1(3))∥xn-p∥≤an3(3)∥xn(2)-p∥+an2(3)∥xn(1)-p∥+an1(3)∥xn-p∥+(1-an3(3)-an2(3)-an1(3))∥xn-p∥≤∥xn-p∥.
By continuing the above argument, we obtain that
(3.4)∥xn(i)-p∥≤∥xn-p∥∀i=1,2,…,r.
In particular, we get ∥xn+1-p∥≤∥xn-p∥ for all n∈ℕ, which implies that limn→∞∥xn-p∥ exists.Lemma 3.2.
LetX be a uniformly convex Banach space and C a nonempty closed and convex subset of X. Let {Ti}i=1r be a finite family of nonexpansive self-mappings of C with F≠∅ and ani(j)∈[0,1] for all j∈{1,2,…,r}, n∈ℕ and i=1,2,…,j such that ∑i=1j‍ani(j) are in [0,1] for all j∈{1,2,…,r} and n∈ℕ. For a given x1∈C, let {xn} be defined by (1.1). If  0<lim infn→∞ani(r)≤lim supn→∞(an(r)(r)+an(r-1)(r)+⋯+an1(r))<1, then(i)
limn→∞∥Tixn(i-1)-xn∥=0for alli=1,2,…,r,(ii)
limn→∞∥Tixn-xn∥=0for alli=1,2,…,r,(iii)
limn→∞∥xn(i)-xn∥=0for alli=1,2,…,r. Proof.
(i) Letp∈F, by Lemma 3.1,    supn∥xn-p∥<∞. Choose a number s>0such that supn∥xn-p∥<s, it follows by (3.4) that {xn(i)-p},{Tixn(i-1)-p}⊆Bs, for all i∈{1,2,…,r}.By Lemma2.4, there exists a continuous strictly increasing convex function g:[0,∞)→[0,∞),g(0)=0 such that
(3.5)∥∑i=1n‍αixi∥2≤∑i=1n‍αi∥xi∥2-α1α2g(∥x1-x2∥),
for all xi∈Bs,αi∈[0,1](i=1,2,…,n) with ∑i=1n‍αi=1. By (3.4) and (3.5), we have for i=1,2,…,r,
(3.6)∥xn+1-p∥2=∥anr(r)Trxn(r-1)+an(r-1)(r)Tr-1xn(r-2)+⋯+an1(r)T1xn+(1-an(r)(r)-an(r-1)(r)-⋯-an1(r))xn-p∥2≤anr(r)∥Trxn(r-1)-p∥2+an(r-1)(r)∥Tr-1xn(r-2)-p∥2+⋯+an1(r)∥T1xn-p∥2+(1-an(r)(r)-an(r-1)(r)-⋯-an1(r))∥xn-p∥2-ani(r)(1-an(r)(r)-an(r-1)(r)-⋯-an1(r))g(∥Tixn(i-1)-xn∥)≤anr(r)∥xn(r-1)-p∥2+an(r-1)(r)∥xn(r-2)-p∥2+⋯+an1(r)∥xn-p∥2+(1-an(r)(r)-an(r-1)(r)-⋯-an1(r))∥xn-p∥2-ani(r)(1-an(r)(r)-an(r-1)(r)-⋯-an1(r))g(∥Tixn(i-1)-xn∥)≤anr(r)∥xn-p∥2+an(r-1)(r)∥xn-p∥2+⋯+an1(r)∥xn-p∥2+(1-an(r)(r)-an(r-1)(r)-⋯-an1(r))∥xn-p∥2-ani(r)(1-an(r)(r)-an(r-1)(r)-⋯-an1(r))g(∥Tixn(i-1)-xn∥)=∥xn-p∥2-ani(r)(1-an(r)(r)-an(r-1)(r)-⋯-an1(r))g(∥Tixn(i-1)-xn∥).
Therefore
(3.7)ani(r)(1-an(r)(r)-an(r-1)(r)-⋯-an1(r))g(∥Tixn(i-1)-xn∥)≤∥xn-p∥2-∥xn+1-p∥2
for all i=1,2,…,r. Since 0<lim infn→∞ani(r)≤lim supn→∞(an(r)(r)+an(r-1)(r)+⋯+an1(r))<1, it implies by Lemma 3.1 that limn→∞g(∥Tixn(i-1)-xn∥)=0. Since g is strictly increasing and continuous at 0 with g(0)=0, it follows that limn→∞∥Tixn(i-1)-xn∥=0 for all  i=1,2,…,r.(ii) Fori∈{1,2,…,r}, we have
(3.8)∥Tixn-xn∥≤∥Tixn-Tixn(i-1)∥+∥Tixn(i-1)-xn∥≤∥xn-xn(i-1)∥+∥Tixn(i-1)-xn∥≤∑j=1i-1‍anj(i-1)∥Tjxn(j-1)-xn∥+∥Tixn(i-1)-xn∥.
It follows from (i) that
(3.9)∥Tixn-xn∥→0asn→∞.(iii) Fori∈{1,2,…,r}, it follows from (i) that
(3.10)∥xn(i)-xn∥≤∑j=1i‍anj(i)∥Tjxn(j-1)-xn∥→0asn→∞.Theorem 3.3.
LetX be a uniformly convex Banach space and C a nonempty closed and convex subset of X. Let {Ti}i=1r be a finite family of nonexpansive self-mappings of C with F≠∅. Let the sequence {ani(j)}n=1∞ be as in Lemma 3.2. For a given x1∈C, let sequences {xn} and {xn(i)}(i=0,1,…,r) be defined by (1.1). If one of {Ti}i=1r is completely continuous then {xn} and {xn(j)} converge strongly to a common fixed point of {Ti}i=1r for all j=1,2,…,r.Proof.
Suppose thatTi0 is completely continuous where i0∈{1,2,…,r}. Then there exists a subsequence {xnk} of {xn} such that {Ti0xnk} converges.Letlimk→∞Ti0xnk=q for some q∈C. By Lemma 3.2 (ii), limn→∞∥Ti0xn-xn∥=0. It follows that limk→∞xnk=q. Again by Lemma 3.2(ii), we have limn→∞∥Tixn-xn∥=0 for all  i=1,2,…,r. It implies that limk→∞Tixnk=q. By continuity of Ti, we get Tiq=q, i=1,2,…,r. So q∈F. By Lemma 3.1, limn→∞∥xn-q∥ exists, it follows that limn→∞∥xn-q∥=0. By Lemma 3.2(iii), we have limn→∞∥xn(j)-xn∥=0 for each j∈{1,2,…,r}. It follows that limn→∞xn(j)=q for all j=1,2,…,r.Theorem 3.4.
LetX be a uniformly convex Banach space and C a nonempty closed and convex subset of X. Let {Ti}i=1r be a finite family of nonexpansive self-mappings of C with F≠∅. Let the sequence {ani(j)}n=1∞ be as in Lemma 3.2. For a given x1∈C, let sequences {xn} and {xn(i)}(i=0,1,…,r) be defined by (1.1). If the family {Ti}i=1r satisfies condition (B) then {xn} and {xn(j)} converge strongly to a common fixed point of {Ti}i=1r for all j=1,2,…,r.Proof.
Letp∈F. Then by Lemma 3.1, limn→∞∥xn-p∥ exists and ∥xn+1-p∥≤∥xn-p∥ for all n≥1. This implies that d(xn+1,F)≤d(xn,F)foralln≥1, therefore, we get limn→∞d(xn,F) exists. By Lemma 3.2(ii), we have limn→∞∥Tixn-xn∥=0 for each i=1,2,…,r. It follows, by the condition (B) that limn→∞f(d(xn,F))=0. Since f is nondecreasing and f(0)=0, therefore, we get limn→∞d(xn,F)=0. Next we show that {xn} is a Cauchy sequence. Since limn→∞d(xn,F)=0, given any ϵ>0, there exists a natural number n0 such that d(xn,F)<ϵ/2 for all n≥n0. In particular, d(xn0,F)<ϵ/2. Then there exists q∈F such that ∥xn0-q∥<ϵ/2. For all n≥n0 and m≥1, it follows by Lemma 3.1 that
(3.11)∥xn+m-xn∥≤∥xn+m-q∥+∥xn-q∥≤∥xn0-q∥+∥xn0-q∥<ϵ.
This shows that {xn} is a Cauchy sequence in C, hence it must converge to a point of C. Let limn→∞xn=p*. Since limn→∞d(xn,F)=0 and F is closed, we obtain p*∈F. By Lemma 3.2(iii), limn→∞∥xn(j)-xn∥=0 for each j∈{1,2,…,r}. It follows that limn→∞xn(j)=p* for all j=1,2,…,r.In Theorem3.4, if ani(j):=0, for all n∈ℕ, j∈{1,2,…,r-1} and i=1,2,…,j, we obtain the following result. Corollary 3.5.
LetX be a uniformly convex Banach space and C a nonempty closed and convex subset of X. Let {Ti}i=1r be a finite family of nonexpansive self-mappings of C with F≠∅ and ani(r)∈[0,1] for all i=1,2,…,r and n∈ℕ such that ∑i=1r‍ani(r) are in [0,1] for all n∈ℕ. For a given x1∈C, let the sequence {xn} be defined by (1.2). If the family {Ti}i=1r satisfies condition (B) and 0<lim infn→∞ani(r)≤lim supn→∞(an(r)(r)+an(r-1)(r)+⋯+an1(r))<1, then the sequence {xn} converges strongly to a common fixed point of {Ti}i=1r.Remark 3.6.
In Corollary3.5, if ani(r)=αi, for all n∈ℕ and for all i=1,2,…,r, the iterative scheme (1.2) reduces to the iterative scheme (1.3) defined by Liu et al. [1] and we obtain strong convergence of the sequence {xn} defined by Liu et al. when {Ti}i=1r satisfies condition (B) which is different from the condition (A) defined by Liu et al. and we note that the result of Senter and Dotson [4] is a special case of Theorem 3.4 when r=1.In the next result, we prove weak convergence for the iterative scheme (1.1) for a finite family of nonexpansive mappings in a uniformly convex Banach space satisfying Opial's condition.Theorem 3.7.
LetX be a uniformly convex Banach space which satisfies Opial's condition and C a nonempty closed and convex subset of X. Let {Ti}i=1r be a finite family of nonexpansive self-mappings of C with F≠∅. For a given x1∈C, let {xn} be the sequence defined by (1.1). If the sequence {ani(j)}n=1∞ is as in Lemma 3.2, then the sequence {xn} converges weakly to a common fixed point of {Ti}i=1r.Proof.
By Lemma3.2(ii), limn→∞∥Tixn-xn∥=0 for all  i=1,2,…,r. Since X is uniformly convex and {xn} is bounded, without loss of generality we may assume that xn→u weakly as n→∞ for some u∈C. By Lemma 2.2, we have u∈F. Suppose that there are subsequences {xnk} and {xmk} of {xn} that converge weakly to u and v, respectively. From Lemma 2.2, we have u,v∈F. By Lemma 3.1, limn→∞∥xn-u∥ and limn→∞∥xn-v∥ exist. It follows from Lemma 2.3 that u=v. Therefore {xn} converges weakly to a common fixed point of {Ti}i=1r.Forani(j):=0, for all n∈ℕ, j∈{1,2,…,r-1} and i=1,2,…,j in Theorem 3.7, we obtain the following result.Corollary 3.8.
LetX be a uniformly convex Banach space which satisfies Opial's condition and C a nonempty closed and convex subset of X. Let {Ti}i=1r be a finite family of nonexpansive self-mappings of C with F≠∅ and ani(r)∈[0,1] for all i=1,2,…,r and n∈ℕ such that ∑i=1r‍ani(r) are in [0,1] for all n∈ℕ. For a given x1∈C, let {xn} be the sequence defined by (1.2). If 0<lim infn→∞ani(r)≤lim supn→∞(an(r)(r)+an(r-1)(r)+⋯+an1(r))<1, then the sequence {xn} converges weakly to a common fixed point of {Ti}i=1r.Remark 3.9.
In Corollary3.8, if ani(r)=αi, for all n∈ℕ and for all i=1,2,…,r, then we obtain weak convergence of the sequence {xn} defined by Liu et al. [1].



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