AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 965751 10.1155/2012/965751 965751 Research Article Iterative Algorithm and Δ-Convergence Theorems for Total Asymptotically Nonexpansive Mappings in CAT(0) Spaces Tang J. F. 1 Chang S. S. 2 Joseph Lee H. W. 3 Chan C. K. 3 Su Yongfu 1 Department of Mathematics Yibin University Yibin Sichuan 644007 China yibinu.cn 2 College of Statistics and Mathematics Yunnan University of Finance and Economics Kunming Yunnan 650221 China ynufe.edu.cn 3 Department of Applied Mathematics The Hong Kong Polytechnic University Hong Kong polyu.edu.hk 2012 4 9 2012 2012 09 07 2012 09 08 2012 2012 Copyright © 2012 J. F. Tang et al. 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.

The main purpose of this paper is first to introduce the concept of total asymptotically nonexpansive mappings and to prove a Δ-convergence theorem for finding a common fixed point of the total asymptotically nonexpansive mappings and the asymptotically nonexpansive mappings. The demiclosed principle for this kind of mappings in CAT(0) space is also proved in the paper. Our results extend and improve many results in the literature.

1. Introduction

A metric space X is a CAT(0) space if it is geodesically connected and if every geodesic triangle in X is at least as “thin” as its comparison triangle in the Euclidean plane. Fixed point theory in a CAT(0) space was first studied by Kirk [1, 2]. He showed that every nonexpansive mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since then the fixed point theory for various mappings in CAT(0) space has been developed rapidly and many papers have appeared . On the other hand, Browder  introduced the demiclosed principle which states that if X is a uniformly convex Banach space, C is a nonempty closed convex subset of X, and if T:CC is nonexpansive mapping, then I-T is demiclosed at each yX, that is, for any sequence {xn} in C conditions xnx weakly and (I-T)xny strongly imply that (I-T)x=y (where I is the identity mapping of X). Xu  proved the demiclosed principle for asymptotically nonexpansive mappings in the setting of a uniformly convex Banach space. Nanjaras and Panyanak  proved the demiclosed principle for asymptotically nonexpansive mappings in CAT(0) space and obtained a Δ-convergence theorem for the Krasnosel’skii-Mann iteration.

Motivated and inspired by the researches going on in this direction, especially inspired by Nanjaras and Panyanak, and so forth , the purpose of this paper is to introduce a general mapping, namely, total asymptotically nonexpansive mapping and to prove its demiclosed principle in CAT(0) space. As a consequence, we construct a hierarchical iterative algorithm to study the fixed point of the total asymptotically nonexpansive mappings and obtain a Δ-convergence theorem.

2. Preliminaries and Lemmas

Let (X,d) be a metric space and x,yX with d(x,y)=l. A geodesic path from x to y is a isometry c:[0,l]X such that c(0)=x, c(l)=y. The image of a geodesic path is called geodesic segment. A space (X,d) is a (uniquely) geodesic space if every two points of X are joined by only one geodesic segment. A geodesic triangle Δ(x1,x2,x3) in a geodesic metric space (X,d) consists of three points x1,x2,x3 in X (the vertices of Δ) and a geodesic segment between each pair of vertices (the edges of Δ). A comparison triangle for the geodesic triangle Δ(x1,x2,x3) in (X,d) is a triangle Δ¯(x1,x2,x3):=Δ(x¯1,x¯2,x¯3) in the Euclidean space 2 such that d2(x¯i,x¯j)=d(xi,xj) for i,j{1,2,3}.

A geodesic space is said to be a CAT(0) space if for each geodesic triangle Δ(x1,x2,x3) in X and its comparison triangle Δ¯:=Δ(x¯1,x¯2,x¯3) in 2, the CAT(0) inequality (2.1)d(x,y)d𝔼2(x¯,y¯). is satisfied for all x,yΔ and x¯,y¯Δ¯.

In this paper, we write (1-t)xty for the unique point z in the geodesic segment joining from x to y such that (2.2)d(x,z)=td(x,y),d(y,z)=(1-t)d(x,y). We also denote by [x,y] the geodesic segment joining from x to y, that is, [x,y]={(1-t)xty:t[0,1]}.

A subset C of a CAT(0) space X is said to be convex if [x,y]C for all x,yC.

Lemma 2.1 (see [<xref ref-type="bibr" rid="B14">14</xref>]).

A geodesic space X is a CAT(0) space, if and only if the following inequality (2.3)d((1-t)xty,z)2(1-t)d(x,z)2+td(y,z)2-t(1-t)d(x,y)2 is satisfied for all x,y,zX and t[0,1]. In particular, if x,y,z are points in a CAT(0) space and t[0,1], then (2.4)d((1-t)xty,z)(1-t)d(x,z)+td(y,z).

Let {xn} be a bounded sequence in a CAT(0) space X. For xX, one sets (2.5)r(x,{xn})=limsupnd(x,xn). The asymptotic radius r({xn}) of {xn} is given by (2.6)r({xn})=infxX{r(x,{xn})}, the asymptotic radius rC({xn}) of {xn} with respect to CX is given by (2.7)rC({xn})=infxC{r(x,{xn})}, the asymptotic center A({xn}) of {xn} is the set (2.8)A({xn})={xX:r(x,{xn})=r({xn})}, the asymptotic center AC({xn}) of {xn} with respect to CX is the set (2.9)AC({xn})={xC:r(x,{xn})=rC({xn})}.

Recall that a bounded sequence {xn} in X is said to be regular if r({xn})=r({un}) for every subsequence {un} of {xn}.

Proposition 2.2 (see [<xref ref-type="bibr" rid="B15">15</xref>]).

If {xn} is a bounded sequence in a complete CAT(0) space X and C is a closed convex subset of X, then

there exists a unique point uC such that (2.10)r(u,{xn})=infxCr(x,{xn});

A({xn}) and AC({xn}) are both singleton.

Lemma 2.3 <xref ref-type="statement" rid="lem2.3">2.3</xref> (see [<xref ref-type="bibr" rid="B16">16</xref>]).

If C is a closed convex subset of a complete CAT(0) space X and if {xn} is a bounded sequence in C, then the asymptotic center of {xn} is in C.

Definition 2.4 (see [<xref ref-type="bibr" rid="B17">17</xref>]).

A sequence {xn} in a CAT(0) space X is said to Δ-converge to xX if x is the unique asymptotic center of {un} for every subsequence {un} of {xn}. In this case one writes Δ-limnxn=x and call x the Δ-limit of {xn}.

Lemma 2.5 (see [<xref ref-type="bibr" rid="B17">17</xref>]).

Every bounded sequence in a complete CAT(0) space always has a Δ-convergent subsequence.

Let {xn} be a bounded sequence in a CAT(0) space X and let C be a closed convex subset of X which contains {xn}. We denote the notation (2.11){xn}wiffΦ(w)=infxCΦ(x), where Φ(x):=limsupnd(xn,x).

Now one gives a connection between the “” convergence and Δ-convergence.

Proposition 2.6 <xref ref-type="statement" rid="prop2.6">2.6</xref> (see [<xref ref-type="bibr" rid="B13">13</xref>]).

Let {xn} be a bounded sequence in a CAT(0) space X and let C be a closed convex subset of X which contains {xn}. Then

Δ-limnxn=x implies that {xn}x;

{xn}x and {xn} is regular imply that Δ-limnxn=x;

Let C be a closed subset of a metric space (X,d). Recall that a mapping T:CC is said to be nonexpansive if (2.12)d(Tx,Ty)d(x,y),x,yX.T is said to be asymptotically nonexpansive if there is a sequence {kn}[1,+) with limnkn=1 such that (2.13)d(Tnx,Tny)knd(x,y),n1,x,yX.

T is said to be closed if, for any sequence {xn}C with d(xn,x)0 and d(Txn,y)0, then Tx=y.

T is called L-uniformly Lipschitzian, if there exists a constant L>0 such that (2.14)d(Tnx,Tny)Ld(x,y),x,yC,n1.

Definition 2.7.

Let (X,d) be a metric space and let C be a closed subset of X. A mapping T:CC is said to be ({vn},{μn},ζ)-total asymptotically nonexpansive if there exist nonnegative real sequences {vn},{μn} with vn0, μn0(n) and a strictly increasing continuous function ζ:[0,+)[0,+) with ζ(0)=0 such that (2.15)d(Tnx,Tny)d(x,y)+vnζ(d(x,y))+μn,n1,x,yC.

Remark 2.8.

(1) It is obvious that If T is uniformly Lipschitzian, then T is closed.

(2) From the definitions, it is to know that, each nonexpansive mapping is a asymptotically nonexpansive mapping with sequence {kn=1}, and each asymptotically nonexpansive mapping is a total asymptotically nonexpansive mapping with vn=kn-1, μn=0, for all n1, and ζ(t)=t, t0.

Lemma 2.9 (demiclosed principle for total asymptotically nonexpansive mappings).

Let C be a closed and convex subset of a complete CAT(0) space X and let T:CC be a L-uniformly Lipschitzian and ({vn},{μn},ζ)-total asymptotically nonexpansive mapping. Let {xn} be a bounded sequence in C such that limnd(xn,Txn)=0 and xnw. Then Tw=w.

Proof.

By the definition, xnw if and only if AC({xn})={w}. By Lemma 2.3, we have A({xn})={w}.

Since limnd(xn,Txn)=0, by induction we can prove that (2.16)limnd(xn,Tmxn)=0,m1.

In fact, it is obvious that, the conclusion is true for m=1. Suppose the conclusion holds for m1, now we prove that the conclusion is also true for m+1. In fact, since T is a L-uniformly Lipschitzian mapping, we have (2.17)d(xn,Tm+1xn)d(xn,Txnxn,Tm+1xn)+d(Txn,TTmxnxn,Tm+1xn)d(xn,Txnxn,Tm+1xn)+Ld(xn,Tmxnxn,Tm+1xn)0(as  n). Equation (2.16) is proved. Hence for each xC and m1 from (2.16) we have (2.18)Φ(x):=limsupnd(xn,x)=limsupnd(Tmxn,x). In (2.18) taking x=Tmw,m1, we have (2.19)Φ(Tmw)=limsupnd(Tmxn,Tmw)limsupn(d(xn,w)+vmζ(d(xn,w))+μm). Let m and taking superior limit on the both sides, it gets that (2.20)limsupmΦ(Tmw)Φ(w). Furthermore, for any n,m1 it follows from inequality (2.3) with t=1/2 that (2.21)d2(xn,wTmw2)12d2(xn,w)+12d2(xn,Tmw)-14d2(w,Tmw). Let n and taking superior limit on the both sides of the above inequality, for any m1 we get (2.22)Φ(wTmw2)212Φ(w)2+12Φ(Tmw)2-14d(w,Tmw)2. Since A({xn})={w}, we have (2.23)Φ(w)2Φ(wTmw2)212Φ(w)2+12Φ(Tmw)2-14d(w,Tmw)2,m1, which implies that (2.24)d2(w,Tmw)2Φ(Tmw)2-2Φ(w)2. By (2.20) and (2.24), we have limmd(w,Tmw)=0. This implies that limmd(w,Tm+1w)=0. Since T is uniformly Lipschitzian, T is uniformly continuous. Hence we have Tw=w. This completes the proof of Lemma 2.9.

The following proposition can be obtained from Lemma 2.9 immediately which is a generalization of Kirk and Panyanak  and Nanjaras and Panyanak .

Proposition 2.10.

Let C be a closed and convex subset of a complete CAT(0) space X and let T:CC be an asymptotically nonexpansive mapping. Let {xn} be a bounded sequence in C such that limnd(xn,Txn)=0 and Δ-limnxn=w. Then T(w)=w.

Definition 2.11 (see [<xref ref-type="bibr" rid="B18">18</xref>]).

Let X be a CAT(0) space then X is uniformly convex, that is, for any given r>0,ϵ(0,2] and λ[0,1], there exists a η(r,ϵ)=ϵ2/8 such that, for all x,y,zX, (2.25)d(x,z)rd(y,z)rd(x,y)ϵr}d((1-λ)xλy,z)(1-2λ(1-λ)ϵ28)r, where the function η:(0,)×(0,2](0,1] is called the modulus of uniform convexity of CAT(0).

Lemma 2.12 (see [<xref ref-type="bibr" rid="B14">14</xref>]).

If {xn} is a bounded sequence in a complete CAT(0) space with A({xn})={x}, {un} is a subsequence of {xn} with A({un})={u}, and the sequence {d(xn,u)} converges, then x=u.

Lemma 2.13.

Let {an},{bn}, and {δn} be sequences of nonnegative real numbers satisfying the inequality (2.26)an+1(1+δn)an+bn. If Σn=1δn< and Σn=1bn<, then {an} is bounded and limnan exists.

Lemma 2.14 (see [<xref ref-type="bibr" rid="B13">13</xref>]).

Let X be a CAT(0) space, xX be a given point and {tn} be a sequence in [b,c] with b,c(0,1) and 0<b(1-c)1/2. Let {xn} and {yn} be any sequences in X such that (2.27)limsupnd(xn,x)r,limsupnd(yn,x)r,limnd((1-tn)xntnyn,x)=r, for some r0. Then (2.28)limnd(xn,yn)=0.

3. Main Results

In this section, we will prove our main theorem.

Theorem 3.1.

Let C be a nonempty bounded closed and convex subset of a complete CAT(0) space X. Let S:CC be a asymptotically nonexpansive mapping with sequence {kn}[1,), kn1 and T:CC be a uniformly L-Lipschitzian and ({vn},{μn},ζ)-total asymptotically nonexpansive mapping such that =F(S)F(T). From arbitrary x1C, defined the sequence {xn} as follows: (3.1)yn=αnSnxn(1-αn)xn,xn+1=βnTnyn(1-βn)xn for all n1, where {βn} is a sequence in (0, 1). If the following conditions are satisfied:

Σn=1vn<;Σn=1μn<;Σn=1(kn-1)<;

there exists a constant M*>0 such that ζ(r)M*r, r0;

there exist constants b,c(0,1) with 0<b(1-c)1/2 such that {αn}[b,c];

Σn=1sup{d(z,Snz):zB}< for each bounded subset B of C.

Then the sequence {xn}Δ-converges to a fixed point of .

Proof.

We divide the proof of Theorem 3.1 into four steps.

(I) First we prove that for each p the following limit exists (3.2)limnd(xn,p).

In fact, for each p, we have (3.3)d(yn,p)=d(αnSnxn(1-αn)xn,p)αnd(Snxn,p)+(1-αn)d(xn,p)=αnd(Snxn,Snp)+(1-αn)d(xn,p)αnknd(xn,p)+(1-αn)d(xn,p)=(1+αn(kn-1))d(xn,p),d(xn+1,p)=d(βnTnyn(1-βn)xn,p)βnd(Tnyn,p)+(1-βn)d(xn,p)=βnd(Tnyn,Tnp)+(1-βn)d(xn,p)βn(d(yn,p)+vnζ(d(yn,p))+μn)+(1-βn)d(xn,p)βn(d(yn,p)+vnM*d(yn,p)+μn)+(1-βn)d(xn,p)d(xn,p)+βnαn(kn-1)d(xn,p)+βnvnM*(1+αn(kn-1))d(xn,p)+μn[1+(kn-1)+vnM*(1+αn(kn-1))]d(xn,p)+μn. It follows from Lemma 2.13 that {d(xn,p)} is bounded and limnd(xn,p) exists. Without loss of generality, we can assume limnd(xn,p)=c0.

(II) Next we prove that (3.4)limnd(xn,Txn)=0.

In fact, since (3.5)d(Tnyn,p)=d(Tnyn,Tnp)d(yn,p)+vnζ(d(yn,p))+μn(1+vnM*)d(yn,p)+μn(1+vnM*)(1+αn(kn-1))d(xn,p)+μn for all n and p, we have (3.6)limsupnd(Tnyn,p)c. On the other hand, since (3.7)limnd(βnTnyn(1-βn)xn,p)=limnd(xn+1,p)=c, by Lemma 2.14, we have (3.8)limnd(Tnyn,xn)=0.

From condition (iv), we have (3.9)d(xn,yn)=d(xn,(1-αn)xnαnSnxn)αnd(xn,Snxn)0(n). Hence from (3.8) and (3.9) we have that (3.10)d(xn,Tnxn)d(xn,Tnyn)+d(Tnyn,Tnxn)d(xn,Tnyn)+Ld(yn,xn)0(n). By (3.9) and (3.10) it gets that (3.11)d(xn+1,Tnxn)=d((1-βn)xnβnTnyn,Tnxn)(1-βn)d(xn,Tnxn)+βnd(Tnyn,Tnxn)(1-βn)d(xn,Tnxn)+βnLd(yn,xn)0(n). Hence from (3.10) and (3.11) we have that (3.12)d(xn,xn+1)0(n). Again since T is uniformly L-Lipschitzian, from (3.10) and (3.12) we have that (3.13)d(xn,Txn)d(xn,xn+1)+d(xn+1,Tn+1xn+1)+d(Tn+1xn+1,Tn+1xn)+d(Tn+1xn,Txn)(L+1)d(xn,xn+1)+d(xn+1,Tn+1xn+1)+Ld(Tnxn,xn)0(n). Equation (3.4) is proved.

(III) Now we prove that (3.14)wω(xn):={un}{xn}A({un})

and wω(xn) consists exactly of one point.

In fact, let uwω(xn), then there exists a subsequence {un} of {xn} such that A({un})={u}. By Lemmas 2.5 and 2.3, there exists a subsequence {νn} of {un} such that Δ-limnνn=νC. By Lemma 2.9, we have νF(T). By Lemma 2.12, u=ν. This shows that wω(xn).

Let {un} be a subsequence of {xn} with A({un})={u} and let A({xn})={x}. Since uwω(xn) and {d(xn,u)} converges, by Lemma 2.12, we have x=u. This shows that wω(xn) consists of exactly one point.

(IV) Finally we prove {xn}Δ-converges to a point of .

In fact, it follows from (3.2) that {d(xn,p)} is convergent for each p. By (3.4) limnd(xn,Txn)=0. By (3.14) wω(xn) and wω(xn) consists of exactly one point. This shows that {xn}Δ-converges to a point of .

This completes the proof of Theorem 3.1.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

All the authors contributed equally to the writing of the present article. And they also read and approved the final paper.

Acknowledgments

The authors would like to express their thanks to the referees for their helpful suggestions and comments. This study was supported by the Scientific Research Fund of Sichuan Provincial Education Department (12ZB346).

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