Let X 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.
1. Introduction
Let X 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 given x1∈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
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
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.
If ani(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]
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].
If r=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:
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 space X 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]).
Let p>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
∥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
wp(λ)=λ(1-λ)p+λp(1-λ).
Lemma 2.2 (see [10, Lemma 1.6]).
Let X 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]).
Let X 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.
Let X 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
∥∑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
∥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
∥∑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
∥∑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.
Let X 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.
Let p∈F. For each n≥1, we note that
∥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
∥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
∥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
∥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.
Let X 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
limn→∞∥Tixn(i-1)-xn∥=0for alli=1,2,…,r,
limn→∞∥Tixn-xn∥=0for alli=1,2,…,r,
limn→∞∥xn(i)-xn∥=0for alli=1,2,…,r. Proof.
(i) Let p∈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 Lemma 2.4, there exists a continuous strictly increasing convex function g:[0,∞)→[0,∞),g(0)=0 such that
∥∑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,
∥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
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) For i∈{1,2,…,r}, we have
∥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
∥Tixn-xn∥→0asn→∞.
(iii) For i∈{1,2,…,r}, it follows from (i) that
∥xn(i)-xn∥≤∑j=1ianj(i)∥Tjxn(j-1)-xn∥→0asn→∞.Theorem 3.3.
Let X 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 that Ti0 is completely continuous where i0∈{1,2,…,r}. Then there exists a subsequence {xnk} of {xn} such that {Ti0xnk} converges.
Let limk→∞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.
Let X 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.
Let p∈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
∥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 Theorem 3.4, 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.
Let X 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 Corollary 3.5, 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.
Let X 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 Lemma 3.2(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.
For ani(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.
Let X 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 Corollary 3.8, 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].
Acknowledgments
The authors would like to thank the Commission on Higher Education, the Thailand Research Fund, the Thaksin University, and the Graduate School of Chiang Mai University, Thailand for their financial support.
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