AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 259470 10.1155/2013/259470 259470 Research Article Convergence Theorems for Common Fixed Points of a Finite Family of Relatively Nonexpansive Mappings in Banach Spaces 0000-0002-9846-3972 Wang Yuanheng 1 Xuan Weifeng 2 Song Yisheng 1 Department of Mathematics Zhejiang Normal University Zhejiang 321004 China zjnu.edu.cn 2 Department of Mathematics Nanjing University Nanjing 210093 China nju.edu.cn 2013 20 3 2013 2013 15 01 2013 18 02 2013 2013 Copyright © 2013 Yuanheng Wang and Weifeng Xuan. 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 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.

1. Introduction

Let E be a smooth Banach space and E* the dual of E. The function Φ:E×ER is defined by (1)ϕ(y,x)=y2-2y,Jx+x2, for all x,yE, 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 ), 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-Tyx-y, for all x,yC, and relatively nonexpansive (see ), if F^(T)=F(T) and (3)ϕ(p,Tx)ϕ(p,x), for all xC and pF(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 .

Actually, Mann  firstly introduced Mann iteration process in 1953, which is defined as follows: (4)x0=xC,xn+1=αnxn+(1-αn)Txn,n0. 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 ). As a matter of fact, the process (3) may fail to converge for a Lipschitz pseudocontractive mapping in a Hilbert space (see ). For example, Reich  proved that if E is a uniformly convex Banach space with Fréchet differentiable norm and if {αn} is chosen such that n=0αn(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  proposed the following modification of the Mann iteration method for a single nonexpansive mapping T in a Hilbert space H: (5)x0=xC,yn=αnxn+(1-αn)Txn,Cn={zC:z-ynz-xn},Qn={zC:xn-z,x-xn0},xn+1=PCnQnx,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  presented their idea of the following method for a single relatively nonexpansive mapping T in a Banach space E: (6)x0=xC,yn=J-1(αnJxn+(1-αn)JTxn),Hn={zC:ϕ(z,yn)ϕ(z,xn)},Wn={zC:xn-z,Jx-Jxn0},xn+1=ΠHnWnx,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=xC,yn=J-1(αnJxn+(1-αn)Jzn),zn=J-1(βn(1)Jxn+βn(2)JTxn+βn(3)JSxn),Hn={zC:ϕ(z,yn)ϕ(z,xn)},Wn={zC:xn-z,Jx-Jxn0},xn+1=ΠHnWnx,n=0,1,2,,(8)x0=xC,yn=J-1(αnJx+(1-αn)Jzn),zn=J-1(βn(1)Jxn+βn(2)JTxn+βn(3)JSxn),Hn={zC:ϕ(z,yn)ϕ(z,xn)+αn(x2+2Jxn-Jx,z)},Wn={zC:xn-z,Jx-Jxn0},xn+1=ΠHnWnx,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:CC be relatively nonexpansive mappings such that F:=i=1NF(Ti)ϕ. Define {xn} in the following way: (9)x0=xC,yn=J-1(αnJx+βnJxn+γnJzn),zn=J-1(λn(0)Jxn+i=1Nλn(i)JTixn),Hn={{x2}zC:ϕ(z,yn)ϕ(z,xn)+αn(x2+2Jxn-Jx,z)},Wn={zC:xn-z,Jx-Jxn0},xn+1=ΠHnWnx,n=0,1,2,, where ΠHnWn is the generalized projection from C onto the intersection set HnWn; {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 n0. 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 , Plubtieng and Ungchittrakool [19, 20], Matsushita and Takahashi , and Martinez-Yanes and Xu .

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, let E be a real Banach space. Let J denote the normalized duality mapping from E into 2E* given by (10)J(x)={fE*,x,f=xf,x=f},xE, where E* denotes the dual space of E and ·,· denotes the generalized duality pairing.

A Banach space E is said to be strictly convex if x+y/2<1 for x=y=1 and xy. It is also said to be uniformly convex if limnxn-yn=0 for any two sequences {xn}, {yn} in E such that xn=yn=1 and limn(xn+yn/2)=1. Let U={xE:x=1} be the unit sphere of E, then the Banach space E is said to be smooth provided that limt0((x+ty-x)/t) exists for each x,yU. It is also said to be uniformly smooth if the limit is attainted uniformly for each x,yU. 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 . A Banach space E is said to have Kadec-Klee property if a sequence {xn} of E satisfying that xnxE and xnx, then xnx. It is known that if E is uniformly convex, then E has the Kadec-Klee property; see  for more details.

Let E be a smooth Banach space. The function Φ:E×ER is defined by (11)ϕ(y,x)=y2-2y,Jx+x2, for all x,yE. It is obvious from the definition of the function ϕ that

(x-y)2ϕ(y,x)(y2+x2),

ϕ(x,y)=ϕ(x,z)+ϕ(z,y)+2x-z,Jz-Jx,

ϕ(x,y)=x,Jx-Jy+y-x,JyxJx-Jy+y-xy,

for all x,yE; see [4, 7, 23] for more details.

Lemma 1 (see [<xref ref-type="bibr" rid="B4">4</xref>]).

If E is a strictly convex and smooth Banach space, then for x,yE, ϕ(x,y)=0 if and only if x=y.

Lemma 2 (see [<xref ref-type="bibr" rid="B23">23</xref>]).

Let E 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-zn0.

Let C be a closed convex subset of E. Suppose that E is reflexive, strictly convex, and smooth. Then, for any xE, there exists a point x0C such that ϕ(x0,x)=minyCϕ(y,x). The mapping ΠC:EC defined by ΠCx=x0 is called the generalized projection (see [4, 7, 23]).

Lemma 3 (see [<xref ref-type="bibr" rid="B7">7</xref>]).

Let C be a closed convex subset of a smooth Banach space E and xE. Then, x0=ΠCx if and only if (12)x0-y,Jx-Jx00,yC.

Lemma 4 (see [<xref ref-type="bibr" rid="B7">7</xref>]).

Let E be a reflexive, strictly convex, and smooth Banach space and let C be a closed convex subset of E and xE. Then, ϕ(y,ΠCx)+ϕ(ΠCx,x)ϕ(y,x) for all y C.

Lemma 5 (see [<xref ref-type="bibr" rid="B24">24</xref>]).

Let E be a uniformly convex Banach space and Br(0)={xE:xr} 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+νz2λx2+μy2+νz2-λμg(x-y), for all x,y,zBr(0) and λ,μ,ν[0,1] with λ+μ+ν=1.

Lemma 6 (see [<xref ref-type="bibr" rid="B19">19</xref>]).

Let E 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,zE and a real number aR, the set K:={vC: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.

Let E be a uniformly convex Banach space and Br(0)={xE:xr} 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=1nλixi2i=1nλixi2-1n2g(i=1nj=1nxi-xj), for all n 3,xiBr(0) and λi[0,1] with i=1nλi=1,i=1,2,,n.

Proof.

If λ3+λ4+λn0, using Lemma 5 and the convexity of ·2, we have (15)i=1nλixi2=λ1x1+λ2x2+(λ3++λn)×(λ3x3λ3++λn++λnxnλ3++λn)2λ1x12+λ2x22+(λ3++λn)×λ3x3λ3++λn++λnxnλ3++λn2-λ1λ2g(x1-x2)i=1nλixi2-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=1nλixi2i=1nλixi2-λiλjg(xi-xj), for all i,j{1,2,,n}. Then, (17)n2i=1nλixi2n2i=1nλixi2-i=1nj=1nλiλjg(xi-xj)n2i=1nλixi2-g(i=1nj=1nλiλjxi-xj). So, (18)i=1nλixi2i=1nλixi2-1n2g(i=1nj=1nλiλjxi-xj).

Theorem 8.

Let E be a uniformly convex and uniformly smooth Banach space, and let C be a nonempty closed convex subset of E and T1,T2,,TN:CC relatively nonexpansive mappings such that F:=i=1NF(Ti)ϕ. The sequence {xn} is given by (9) with the following restrictions:

αn+βn+γn=1,0αn<1,0βn<1,0<γn1 for all n0;

lim nαn=0 and limsup nβn<1;

λni[0,1] with i=0Nλni=1,i=0,1,2,,N, for all n0;

lim nλn0=0 and liminf nλniλnj>0,i,j=1,2,,N; or

liminf nλn0λni>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 xC.

It is easy to know that F(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 xC.

Step   2. Show that Hn and Wn are closed and convex for all n0.

From the definition of Wn, it is obvious Wn is closed and convex for each n0. By Lemma 6, we also know that Hn is closed and convex for each n0.

Step   3. Show that FHnWn for all n0.

Let uF and let n0. Then, by the convexity of ·2, we have (19)ϕ(u,zn)=ϕ(u,J-1(λn(0)Jxn+i=1Nλn(i)JTixn))=u2-2u,λn(0)Jxn+i=1Nλn(i)JTixn+λn(0)Jxn+i=1Nλn(i)JTixn2u2-2λn(0)u,Jxn-2i=1Nλn(i)u,JTixn+λn(0)Jxn2+i=1Nλn(i)JTixn2=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))=u2-2u,αnJx+βnJxn+γnJzn+αnJx+βnJxn+γnJzn2u2-2αnu,Jx-2βnu,Jxn-2γnu,Jzn+αnx2+βnxn2+γnzn2=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(x2+2Jxn-Jx,z). Thus, we have uHn. Therefore, we obtain FHn for all n0.

Next, we prove FWn for all n0. We prove this by induction. For n=0, we have FC=W0. Assume that FWn. Since xn+1 is the projection of x onto HnWn, by Lemma 3, we have (21)xn+1-z,Jx-Jxn+10, for any zHnWn. As FHnWn by the induction assumption,FWn holds, in particular, for all uF. This together with the definition of Wn+1 implies that FWn+1. Hence, FHnWn for all n0.

Step   4. Show that xn+1-xn0 as n.

In view of (19) and Lemma 4, we have xn=PWnx, which means that, for any zWn, (22)ϕ(xn,x)ϕ(z,x). Since xn+1Wn and uFWn, we obtain (23)ϕ(xn,x)ϕ(xn+1,x),ϕ(xn,x)ϕ(u,x), for all n0. Consequently, limnϕ(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-xn0 as n.

Step   5. Show that xn-zn0 as n.

From xn+1=PHnWnxHn, we have (25)ϕ(xn+1,yn)ϕ(xn+1,xn)+αn(x2+2Jxn-Jx,xn+1)0, as n. By Lemma 2, we also have xn+1-yn0, and then, (26)xn-ynyn-xn+1+xn-xn+10, as n. We observe that (27)ϕ(zn,xn)=ϕ(zn,yn)+ϕ(yn,xn)+2zn-yn,Jyn-Jxnϕ(zn,yn)+ϕ(yn,xn)+2zn-ynJyn-Jxn,ϕ(zn,yn)=zn2-2zn,αnJx+βnJxn+γnJzn+αnJx+βnJxn+γnJzn2αnϕ(zn,x)+βnϕ(zn,xn). So, (28)ϕ(zn,xn)αnϕ(zn,x)+βnϕ(zn,xn)+ϕ(yn,xn)+2zn-ynJyn-Jxn. Since limnαn=0, limsupnβn<1, ϕ(yn,xn)0, and zn-ynJyn-Jxn0 as n, we have (29)ϕ(zn,xn)αn1-βnϕ(zn,x)+11-βnϕ(yn,xn)+21-βnzn-ynJyn-Jxn0, as n. Using Lemma 2, we obtain xn-zn0 as n.

Step  6. Show that xn-Tixn0,i=1,2,,N.

Since {xn} is bounded and ϕ(p,Tixn)ϕ(p,xn), where pF, 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)=p2-2p,λn(0)Jxn+i=1Nλn(i)JTixn+λn(0)Jxn+i=1Nλn(i)JTixn2p2-2λn(0)p,Jxn-2i=1Nλn(i)p,JTixn+λn(0)xn2+i=1Nλn(i)Tixn2-1N2g(i=1Nj=1Nλn(i)λn(j)JTixn-JTjxn+2i=1Nλn(0)λn(i)Jxn-JTixn)=λn(0)ϕ(p,xn)+i=1Nλn(i)ϕ(p,Tixn)-1N2g(i=1Nj=1Nλn(i)λn(j)JTixn-JTjxn+2i=1Nλn(0)λn(i)Jxn-JTixn)ϕ(p,xn)-1N2g(i=1Nj=1Nλn(i)λn(j)JTixn-JTjxn+2i=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)=p2-2p,Jxn+xn2-p2+2p,Jzn-zn22pJzn-Jxn+xn2-zn20, as n. From the properties of the mapping g, we have (32)limnλn(0)λn(i)xn-Tixn=0,limnλn(i)λn(j)Tixn-Tjxn=0, for all i,j{1,2,,N}. From the condition (d), we have xn-Tixn0 immediately, as n, i=1,2,,N; from the condition (d), we can also have xn-Tixn0, as n,i=1,2,,N. In fact, since liminfnλn(i)λn(j)>0, it follows that (33)limnTixn-Tjxn=0, for all i,j{1,2,,N}. Next, we note by the convexity of ·2 and (9) that (34)ϕ(Tjxn,zn)=Tjxn2-2Tjxn,λn(0)Jxn+i=1Nλn(i)JTixn+λn(0)Jxn+i=1Nλn(i)JTixn2Tjxn2-2λn(0)Tjxn,Jxn-2i=1Nλn(i)Tjxn,JTixn+λn(0)xn2+i=1Nλn(i)Tixn2=λn(0)ϕ(Tjxn,xn)+i=1Nλn(i)ϕ(Tjxn,Tixn)0, as n. By Lemma 2, we have limnTixn-zn=0 and (35)Tixn-xnTixn-zn+xn-zn0, as n for all i{1,2,,N}.

Step  7. Show that xnΠFx, 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 n1, from xn+1=ΠHnWnx and ωFHnWn, 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^2-2x^,Jx+x2liminfn(xnk2-2xnk2,Jx+x2)=liminfnϕ(xnk,x)limsupnϕ(xnk,x)ϕ(ω,x). From the definition of ΠFx, we obtain x^=ω, and hence, limnϕ(xnk,x)=ϕ(ω,x). So, we have limkxnk=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.

Let C be a nonempty closed convex subset of a Hilbert space H and T1,T2,,TN:CC relatively nonexpansive mappings such that F:=i=1NF(Ti)ϕ. The sequence {xn} is given by (9) with the following restrictions:

αn+βn+γn=1,0αn<1,0βn<1,0γn1 for all n 0;

lim nαn=0 and limsup nβn<1;

λni[0,1] with i=0Nλni=1, i=0,1,2,,N, for all n0;

lim nλn0=0 and liminf nλniλnj>0,i,j=1,2,,N; or

liminf nλn0λni>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 projection ΠF is just the metric projection PF in Hilbert spaces.

Remark 10.

The results of Nakajo and Takahashi  and Song et al.  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 n0, then, we obtain Theorem 4.1 of Matsushita and Takahashi ; if T1=T2==TN-1 and αn=0 for all n0, then, we obtain Theorem 3.1 of Plubtieng and Ungchittrakool ; if T1=T2==TN-1 and βn=0 for all n0, then, we obtain Theorem 3.2 of Plubtieng and Ungchittrakool . So, our results improve and extend the corresponding results by many others.

Acknowledgments

This work was partially supported by the Natural Science Foundation of Zhejiang Province (Y6100696) and the National Natural Science Foundation of China (11071169, 11271330).

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