# Convergence Theorems for Common Fixed Points of a Finite Family of Relatively Nonexpansive Mappings in Banach Spaces

Authors: Yuanheng Wang; Weifeng Xuan
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We establish some strong convergence theorems for a common fixed point of a finite family of relatively nonexpansive mappings by using a new hybrid iterative method in mathematical programming and the generalized projection method in a Banach space. Our results improve and extend the corresponding results by many others.


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## Body

## 1. Introduction

LetE be a smooth Banach space and E* the dual of E. The function Φ:E×E→R is defined by
(1)ϕ(y,x)=∥y∥2-2〈y,Jx〉+∥x∥2,
for all x,y∈E, where J is the normalized duality mapping from E to E*. Let C be a closed convex subset of E, and let T be a mapping from C into itself. We denote by F(T) the set of fixed points of T. A point p in C is said to be an asymptotic fixed point of T (see [1]), if C contains a sequence {xn} which converges weakly to p such that the strong limn→∞(xn-Txn)=0. The set of asymptotic fixed points of T will be denoted by F^(T). A mapping T from C into itself is called nonexpansive, if
(2)∥Tx-Ty∥≤∥x-y∥,
for all x,y∈C, and relatively nonexpansive (see [2]), if F^(T)=F(T) and
(3)ϕ(p,Tx)≤ϕ(p,x),
for all x∈C and p∈F(T). The iterative methods for approximation of fixed points of nonexpansive mappings, relatively nonexpansive mappings, and other generational nonexpansive mappings have been studied by many researchers; see [3–13].Actually, Mann [14] firstly introduced Mann iteration process in 1953, which is defined as follows:
(4)x0=x∈C,xn+1=αnxn+(1-αn)Txn,n≥0.
It is very useful to approximate a fixed point of a nonexpansive mapping. However, as we all know, it has only weak convergence in a Hilbert space (see [15]). As a matter of fact, the process (3) may fail to converge for a Lipschitz pseudocontractive mapping in a Hilbert space (see [16]). For example, Reich [17] proved that if E is a uniformly convex Banach space with Fréchet differentiable norm and if {αn} is chosen such that ∑n=0∞‍αn(1-αn)=∞, then the sequence {xn} defined by (3) converges weakly to a fixed point of T.Some have made attempts to modify the Mann iteration methods, so that strong convergence is guaranteed. Nakajo and Takahashi [18] proposed the following modification of the Mann iteration method for a single nonexpansive mapping T in a Hilbert space H:
(5)x0=x∈C,yn=αnxn+(1-αn)Txn,Cn={z∈C:∥z-yn∥≤∥z-xn∥},Qn={z∈C:〈xn-z,x-xn〉≥0},xn+1=PCn⋂‍Qnx,n=0,1,2,…,
where PK denotes the metric projection from H onto a closed convex subset of H. They proved that if the sequence {αn} is bounded above from one, then {xn} defined by (5) converges strongly to PF(T)x.The ideas to generate the process (5) from Hilbert spaces to Banach spaces have been made. By using the properties available on uniformly convex and uniformly smooth Banach spaces, Matsushita and Takahashi [10] presented their idea of the following method for a single relatively nonexpansive mapping T in a Banach space E:
(6)x0=x∈C,yn=J-1(αnJxn+(1-αn)JTxn),Hn={z∈C:ϕ(z,yn)≤ϕ(z,xn)},Wn={z∈C:〈xn-z,Jx-Jxn〉≥0},xn+1=ΠHn⋂‍Wnx,n=0,1,2,…,
where J is the duality mapping on E and ΠF(T)x is the generalized projection from C onto F(T).In 2007 and 2008, Plubing and Ungchittrakool [19, 20] improved and generalized the process (6) to the new general process of two relatively nonexpansive mappings in a Banach space:
(7)x0=x∈C,yn=J-1(αnJxn+(1-αn)Jzn),zn=J-1(βn(1)Jxn+βn(2)JTxn+βn(3)JSxn),Hn={z∈C:ϕ(z,yn)≤ϕ(z,xn)},Wn={z∈C:〈xn-z,Jx-Jxn〉≥0},xn+1=ΠHn⋂‍Wnx,n=0,1,2,…,(8)x0=x∈C,yn=J-1(αnJx+(1-αn)Jzn),zn=J-1(βn(1)Jxn+βn(2)JTxn+βn(3)JSxn),Hn={z∈C:ϕ(z,yn)≤ϕ(z,xn)+αn(∥x∥2+2〈Jxn-Jx,z〉)},Wn={z∈C:〈xn-z,Jx-Jxn〉≥0},xn+1=ΠHn⋂‍Wnx,n=0,1,2,….
They proved that both iterations (7) and (8) converge strongly to a common fixed point of two relatively nonexpansive mappings S and T provided that the sequences satisfy some appropriate conditions.Inspired and motivated by these facts, in this paper, we aim to improve and generalize the process (7) and (8) to the new general process of a finite family of relatively nonexpansive mappings in a Banach space. Let C be a closed convex subset of a Banach space E and let  T1,T2,…,TN:C→C be relatively nonexpansive mappings such that F:=∩i=1NF(Ti)≠ϕ. Define {xn} in the following way:
(9)x0=x∈C,yn=J-1(αnJx+βnJxn+γnJzn),zn=J-1(λn(0)Jxn+∑i=1N‍λn(i)JTixn),Hn={{∥x∥2}z∈C:ϕ(z,yn)≤ϕ(z,xn)+αn(∥x∥2+2〈Jxn-Jx,z〉)},Wn={z∈C:〈xn-z,Jx-Jxn〉≥0},xn+1=ΠHn⋂‍Wnx,n=0,1,2,…,
where ΠHn⋂‍Wn is the generalized projection from C onto the intersection set  Hn⋂‍Wn; {αn},{βn},{γn},{λn(0)},{λn(1)},…,{λn(N)} are the sequences in [0,1] with αn+βn+γn=1 and ∑i=0N‍λn(i)=1 for all n≥0. We prove, under certain appropriate assumptions on the sequences, that {xn} defined by (9) converges strongly to PFx, where PF is the generalized projection from C to F.Obviously, the process (9) reduces to become (7) when N=2,αn=0 and become (8) when N=2,βn=0. So, our results extend and improve the corresponding ones announced by Nakajo and Takahashi [18], Plubtieng and Ungchittrakool [19, 20], Matsushita and Takahashi [10], and Martinez-Yanes  and Xu [21].


## 2. Preliminaries

This section collects some definitions and lemmas which will be used in the proofs for the main results in the next section.Throughout this paper, letE be a real Banach space. Let J denote the normalized duality mapping from E into 2E* given by
(10)J(x)={f∈E*,〈x,f〉=∥x∥∥f∥,∥x∥=∥f∥},∀x∈E,
where E* denotes the dual space of E and 〈·,·〉 denotes the generalized duality pairing.A Banach spaceE is said to be strictly convex if  ∥x+y∥/2<1 for ∥x∥=∥y∥=1 and x≠y. It is also said to be uniformly convex if limn→∞∥xn-yn∥=0 for any two sequences {xn}, {yn} in E such that ∥xn∥=∥yn∥=1 and limn→∞(∥xn+yn∥/2)=1. Let U={x∈E:∥x∥=1} be the unit sphere of E, then the Banach space E is said to be smooth provided that limt→0((∥x+ty∥-∥x∥)/t) exists for each x,y∈U. It is also said to be uniformly smooth if the limit is attainted uniformly for each x,y∈U. It is well known that if E is smooth, then the duality mapping J is single valued. It is also known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E. Some properties of the duality mapping have been given in [22]. A Banach space E is said to have Kadec-Klee property if a sequence {xn} of E satisfying that xn⇀x∈E and ∥xn∥→∥x∥, then xn→x. It is known that if E is uniformly convex, then E has the Kadec-Klee property; see [22] for more details.LetE be a smooth Banach space. The function Φ:E×E→R is defined by
(11)ϕ(y,x)=∥y∥2-2〈y,Jx〉+∥x∥2,
for all x,y∈E. It is obvious from the definition of the function ϕ that (1)
(∥x∥-∥y∥)2≤ϕ(y,x)≤(∥y∥2+∥x∥2),(2)
ϕ(x,y)=ϕ(x,z)+ϕ(z,y)+2〈x-z,Jz-Jx〉,(3)
ϕ(x,y)=〈x,Jx-Jy〉+〈y-x,Jy〉≤∥x∥∥Jx-Jy∥+∥y-x∥∥y∥,for all x,y∈E; see [4, 7, 23] for more details.Lemma 1 (see [4]).
IfE is a strictly convex and smooth Banach space, then for x,y∈E, ϕ(x,y)=0 if and only if x=y.Lemma 2 (see [23]).
LetE be a uniformly convex and smooth Banach space and let {yn},{zn} be two sequences of E. If ϕ(yn,zn)→0 and either {yn} or {zn} is bounded, then yn-zn→0.LetC be a closed convex subset of E. Suppose that E is reflexive, strictly convex, and smooth. Then, for any x∈E, there exists a point x0∈C such that ϕ(x0,x)=miny∈Cϕ(y,x). The mapping ΠC:E→C defined by ΠCx=x0 is called the generalized projection (see [4, 7, 23]).Lemma 3 (see [7]).
LetC be a closed convex subset of a smooth Banach space E and x∈E. Then, x0=ΠCx if and only if
(12)〈x0-y,Jx-Jx0〉≥0,∀y∈C.Lemma 4 (see [7]).
LetE be a reflexive, strictly convex, and smooth Banach space and let C be a closed convex subset of E and x∈E. Then, ϕ(y,ΠCx)+ϕ(ΠCx,x)≤ϕ(y,x) for all y ∈ C.Lemma 5 (see [24]).
LetE be a uniformly convex Banach space and Br(0)={x∈E:∥x∥≤r} a closed ball of E. Then, there exists a continuous strictly increasing convex function g:[0,∞)→[0,∞) with g(0)=0 such that
(13)∥λx+μy+νz∥2≤λ∥x∥2+μ∥y∥2+ν∥z∥2-λμg(∥x-y∥),
for all x,y,z∈Br(0) and λ,μ,ν∈[0,1] with λ+μ+ν=1.Lemma 6 (see [19]).
LetE be a uniformly convex and uniformly smooth Banach space and let C be a closed convex subset of E. Then, for points w,x,y,z∈E and a real number a∈R, the set K:={v∈C:ϕ(v,y)≤ϕ(v,x)+〈v,Jz-Jw〉+a} is closed and convex.


## 3. Main Results

In this section, we will prove the strong convergence theorem for a common fixed point of a finite family of relatively nonexpansive mappings in a Banach space by using the hybrid method in mathematical programming. Let us prove a proposition first.Proposition 7.
LetE be a uniformly convex Banach space and Br(0)={x∈E:∥x∥≤r} a closed ball of E. Then, there exists a continuous strictly increasing convex function g:[0,∞)→[0,∞) with g(0)=0 such that
(14)∥∑i=1n‍λixi∥2≤∑i=1n‍λi∥xi∥2-1n2g(∑i=1n‍∑j=1n‍∥xi-xj∥),
for all n ≥3,xi∈Br(0) and λi∈[0,1] with ∑i=1n‍λi=1,i=1,2,…,n.Proof.
Ifλ3+λ4⋯+λn≠0, using Lemma 5 and the convexity of ∥·∥2, we have
(15)∥∑i=1n‍λixi∥2=∥λ1x1+λ2x2+(λ3+⋯+λn)×(λ3x3λ3+⋯+λn+⋯+λnxnλ3+⋯+λn)∥2≤λ1∥x1∥2+λ2∥x2∥2+(λ3+⋯+λn)×∥λ3x3λ3+⋯+λn+⋯+λnxnλ3+⋯+λn∥2-λ1λ2g(∥x1-x2∥)≤∑i=1n‍λi∥xi∥2-λ1λ2g(∥x1-x2∥).
If λ3+λ4⋯+λn=0, the last inequality above also holds obviously. By the same argument in the proof above, we obtain
(16)∥∑i=1n‍λixi∥2≤∑i=1n‍λi∥xi∥2-λiλjg(∥xi-xj∥),
for all i,j∈{1,2,…,n}. Then,
(17)n2∥∑i=1n‍λixi∥2≤n2∑i=1n‍λi∥xi∥2-∑i=1n‍∑j=1n‍λiλjg(∥xi-xj∥)≤n2∑i=1n‍λi∥xi∥2-g(∑i=1n‍∑j=1n‍λiλj∥xi-xj∥).
So,
(18)∥∑i=1n‍λixi∥2≤∑i=1n‍λi∥xi∥2-1n2g(∑i=1n‍∑j=1n‍λiλj∥xi-xj∥).Theorem 8.
LetE be a uniformly convex and uniformly smooth Banach space, and let C be a nonempty closed convex subset of E and T1,T2,…,TN:C→C relatively nonexpansive mappings such that F:=∩i=1NF(Ti)≠ϕ. The sequence {xn} is given by (9) with the following restrictions:(a)
αn+βn+γn=1,0≤αn<1,0≤βn<1,0<γn≤1 for all n≥0;(b)
limn→∞αn=0 and limsupn→∞βn<1;(c)
λni∈[0,1] with ∑i=0N‍λni=1,i=0,1,2,…,N, for all n≥0;(d)
limn→∞λn0=0 and liminfn→∞λniλnj>0,i,j=1,2,…,N; or(d′)
liminfn→∞λn0λni>0,i=1,2,…,N.Then, the sequence {xn} converges strongly to ΠFx, where ΠF is the generalized projection from C onto F.Proof.
We split the proof into seven steps.
Step
1. Show that PF is well defined for every x∈C.
It is easy to know thatF(Ti), i=1,2,…,N are closed convex sets and so is F. What is more, F is nonempty by our assumption. Therefore, PF is well defined for every x∈C.
Step
2.  Show that Hn and Wn are closed and convex for all n≥0.
From the definition ofWn, it is obvious Wn is closed and convex for each n≥0. By Lemma 6, we also know that Hn is closed and convex for each n≥0.
Step
3. Show that F⊂Hn⋂‍Wn for all n≥0.
Letu∈F and let n≥0. Then, by the convexity of ∥·∥2, we have
(19)ϕ(u,zn)=ϕ(u,J-1(λn(0)Jxn+∑i=1N‍λn(i)JTixn))=∥u∥2-2〈u,λn(0)Jxn+∑i=1N‍λn(i)JTixn〉+∥λn(0)Jxn+∑i=1N‍λn(i)JTixn∥2≤∥u∥2-2λn(0)〈u,Jxn〉-2∑i=1N‍λn(i)〈u,JTixn〉+λn(0)∥Jxn∥2+∑i=1N‍λn(i)∥JTixn∥2=λn(0)ϕ(u,xn)+∑i=1N‍λn(i)ϕ(u,Tixn)≤λn(0)ϕ(u,xn)+∑i=1N‍λn(i)ϕ(u,xn)=ϕ(u,xn),
and then,
(20)ϕ(u,yn)=ϕ(u,J-1(αnJx+βnJxn+γnJzn))=∥u∥2-2〈u,αnJx+βnJxn+γnJzn〉+∥αnJx+βnJxn+γnJzn∥2≤∥u∥2-2αn〈u,Jx〉-2βn〈u,Jxn〉-2γn〈u,Jzn〉+αn∥x∥2+βn∥xn∥2+γn∥zn∥2=αnϕ(u,x)+βnϕ(u,xn)+γnϕ(u,zn)≤αnϕ(u,x)+(1-αn)ϕ(u,xn)=ϕ(u,xn)+αn(ϕ(u,x)-ϕ(u,xn))≤ϕ(u,xn)+αn(∥x∥2+2〈Jxn-Jx,z〉).
Thus, we have u∈Hn. Therefore, we obtain F⊂Hn for all n≥0.
Next, we proveF⊂Wn for all n≥0. We prove this by induction. For n=0, we have F⊂C=W0. Assume that F⊂Wn. Since xn+1 is the projection of x onto Hn⋂‍Wn, by Lemma 3, we have
(21)〈xn+1-z,Jx-Jxn+1〉≥0,
for any z∈Hn⋂‍Wn. As F⊂Hn⋂‍Wn by the induction assumption,F⊂Wn holds, in particular, for all u∈F. This together with the definition of Wn+1 implies that F⊂Wn+1. Hence, F⊂Hn⋂‍Wn for all n≥0.
Step
4. Show that ∥xn+1-xn∥→0 as n→∞.
In view of (19) and Lemma 4, we have xn=PWnx, which means that, for any z∈Wn,
(22)ϕ(xn,x)≤ϕ(z,x).
Since xn+1∈Wn and u∈F⊂Wn, we obtain
(23)ϕ(xn,x)≤ϕ(xn+1,x),ϕ(xn,x)≤ϕ(u,x),
for all n≥0. Consequently, limn→∞ϕ(xn,x) exists and {xn} is bounded. By using Lemma 4, we have
(24)ϕ(xn+1,xn)≤ϕ(xn+1,x)-ϕ(xn,x)⟶0,
as n→∞. By using Lemma 2, we obtain ∥xn+1-xn∥→0 as n→∞.
Step
5. Show that ∥xn-zn∥→0 as n→∞.
Fromxn+1=PHn⋂‍Wnx∈Hn, we have
(25)ϕ(xn+1,yn)≤ϕ(xn+1,xn)+αn(∥x∥2+2〈Jxn-Jx,xn+1〉)⟶0,
as n→∞. By Lemma 2, we also have ∥xn+1-yn∥→0, and then,
(26)∥xn-yn∥≤∥yn-xn+1∥+∥xn-xn+1∥⟶0,
as n→∞. We observe that
(27)ϕ(zn,xn)=ϕ(zn,yn)+ϕ(yn,xn)+2〈zn-yn,Jyn-Jxn〉≤ϕ(zn,yn)+ϕ(yn,xn)+2∥zn-yn∥∥Jyn-Jxn∥,ϕ(zn,yn)=∥zn∥2-2〈zn,αnJx+βnJxn+γnJzn〉+∥αnJx+βnJxn+γnJzn∥2≤αnϕ(zn,x)+βnϕ(zn,xn).
So,
(28)ϕ(zn,xn)≤αnϕ(zn,x)+βnϕ(zn,xn)+ϕ(yn,xn)+2∥zn-yn∥∥Jyn-Jxn∥.
Since limn→∞αn=0, limsupn→∞βn<1, ϕ(yn,xn)→0, and ∥zn-yn∥∥Jyn-Jxn∥→0 as n→∞, we have
(29)ϕ(zn,xn)≤αn1-βnϕ(zn,x)+11-βnϕ(yn,xn)+21-βn∥zn-yn∥∥Jyn-Jxn∥⟶0,
as n→∞. Using Lemma 2, we obtain ∥xn-zn∥→0 as n→∞.
Step6. Show that ∥xn-Tixn∥→0,i=1,2,…,N.
Since{xn} is bounded and ϕ(p,Tixn)≤ϕ(p,xn), where p∈F, i=1,2,…,N, we also obtain that {Jxn},{JT1xn},…,{JTNxn} are bounded, and hence, there exists r>0 such that {Jxn},{JT1xn},…,{JTNxn}⊂Br(0). Therefore, Proposition 7 can be applied and we observe that
(30)ϕ(p,zn)=∥p∥2-2〈p,λn(0)Jxn+∑i=1N‍λn(i)JTixn〉+∥λn(0)Jxn+∑i=1N‍λn(i)JTixn∥2≤∥p∥2-2λn(0)〈p,Jxn〉-2∑i=1N‍λn(i)〈p,JTixn〉+λn(0)∥xn∥2+∑i=1N‍λn(i)∥Tixn∥2-1N2g(∑i=1N‍∑j=1N‍λn(i)λn(j)∥JTixn-JTjxn∥+2∑i=1N‍λn(0)λn(i)∥Jxn-JTixn∥)=λn(0)ϕ(p,xn)+∑i=1N‍λn(i)ϕ(p,Tixn)-1N2g(∑i=1N‍∑j=1N‍λn(i)λn(j)∥JTixn-JTjxn∥+2∑i=1N‍λn(0)λn(i)∥Jxn-JTixn∥)≤ϕ(p,xn)-1N2g(∑i=1N‍∑j=1N‍λn(i)λn(j)∥JTixn-JTjxn∥+2∑i=1N‍λn(0)λn(i)∥Jxn-JTixn∥),
where g:[0,∞)→[0,∞) is a continuous strictly increasing convex function with g(0)=0. And
(31)ϕ(p,xn)-ϕ(p,zn)=∥p∥2-2〈p,Jxn〉+∥xn∥2-∥p∥2+2〈p,Jzn〉-∥zn∥2≤2∥p∥∥Jzn-Jxn∥+∥xn∥2-∥zn∥2⟶0,
as n→∞. From the properties of the mapping g, we have
(32)limn→∞λn(0)λn(i)∥xn-Tixn∥=0,limn→∞λn(i)λn(j)∥Tixn-Tjxn∥=0,
for all i,j∈{1,2,…,N}. From the condition (d′), we have ∥xn-Tixn∥→0 immediately, as n→∞, i=1,2,…,N; from the condition (d), we can also have ∥xn-Tixn∥→0, as n→∞,i=1,2,…,N. In fact, since liminfn→∞λn(i)λn(j)>0, it follows that
(33)limn→∞∥Tixn-Tjxn∥=0,
for all i,j∈{1,2,…,N}. Next, we note by the convexity of ∥·∥2 and (9) that
(34)ϕ(Tjxn,zn)=∥Tjxn∥2-2〈Tjxn,λn(0)Jxn+∑i=1N‍λn(i)JTixn〉+∥λn(0)Jxn+∑i=1N‍λn(i)JTixn∥2≤∥Tjxn∥2-2λn(0)〈Tjxn,Jxn〉-2∑i=1N‍λn(i)〈Tjxn,JTixn〉+λn(0)∥xn∥2+∑i=1N‍λn(i)∥Tixn∥2=λn(0)ϕ(Tjxn,xn)+∑i=1N‍λn(i)ϕ(Tjxn,Tixn)⟶0,
as n→∞. By Lemma 2, we have limn→∞∥Tixn-zn∥=0 and
(35)∥Tixn-xn∥≤∥Tixn-zn∥+∥xn-zn∥⟶0,
as n→∞ for all i∈{1,2,…,N}.
Step  7. Show that xn→ΠFx, as n→∞.
From the result of Step 6, we know that if{xnk} is a subsequence of {xn} such that {xnk}⇀x^∈C, then x^∈∩i=1NF^(Ti)=∩i=1NF(Ti). Because E is a uniformly convex and uniformly smooth Banach space and {xn} is bounded, so we can assume {xnk} is a subsequence of {xn} such that {xnk}⇀x^∈F and ω=ΠFx. For any n≥1, from xn+1=ΠHn⋂‍Wnx and ω∈F⊂Hn⋂‍Wn, we have
(36)ϕ(xn+1,x)≤ϕ(ω,x).
On the other hand, from weakly lower semicontinuity of the norm, we have
(37)ϕ(x^,x)=∥x^∥2-2〈x^,Jx〉+∥x∥2≤liminfn→∞(∥xnk∥2-2〈∥xnk∥2,Jx〉+∥x∥2)=liminfn→∞ϕ(xnk,x)≤limsupn→∞ϕ(xnk,x)≤ϕ(ω,x).
From the definition of ΠFx, we obtain x^=ω, and hence, limn→∞ϕ(xnk,x)=ϕ(ω,x). So, we have limk→∞∥xnk∥=∥ω∥. Using the Kadec-klee property of E, we obtain that {xnk} converges strongly to ΠFx. Since {xnk} is an arbitrary weakly convergent sequence of {xn}, we can conclude that {xn} converges strongly to ΠFx.Corollary 9.
LetC be a nonempty closed convex subset of a Hilbert space H and T1,T2,…,TN:C→C relatively nonexpansive mappings such that F:=∩i=1NF(Ti)≠ϕ. The sequence {xn} is given by (9) with the following restrictions: (a)
αn+βn+γn=1,0≤αn<1,0≤βn<1,0≤γn≤1 for all n ≥ 0;(b)
limn→∞αn=0 and limsupn→∞βn<1;(c)
λni∈[0,1] with ∑i=0N‍λni=1, i=0,1,2,…,N, for all n≥0;(d)
limn→∞λn0=0 and liminfn→∞λniλnj>0,i,j=1,2,…,N; or(d′)
liminfn→∞λn0λni>0,i=1,2,…,N.Then, the sequence {xn} converges strongly to PFx, where PF is the metric projection from C onto F.Proof.
It is true because the generalized projectionΠF is just the metric projection PF in Hilbert spaces.Remark 10.
The results of Nakajo and Takahashi [18] and Song et al. [11] are the special cases of our results in Corollary 9. And in our results of Theorem 8, if T1=T2=⋯=TN,λn(0) and αn=0 for all n≥0, then, we obtain Theorem 4.1 of Matsushita and Takahashi [10]; if T1=T2=⋯=TN-1 and αn=0 for all n≥0, then, we obtain Theorem 3.1 of Plubtieng and Ungchittrakool [19]; if T1=T2=⋯=TN-1 and βn=0 for all n≥0, then, we obtain Theorem 3.2 of Plubtieng and Ungchittrakool [19]. So, our results improve and extend the corresponding results by many others.



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*Source: 259470-2013-03-20.xml*