Approximating fixed points for generalized nonexpansive mapping in CAT(0) spaces

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R E S E A R C H

Open Access

Approximating fixed points for generalized

nonexpansive mapping in

CAT

()

spaces

Izhar Uddin

1

, Sumitra Dalal

2*

and Mohammad Imdad

1

*Correspondence:

mathsqueen_d@yahoo.com

2Department of Mathematics,

Faculty of Science, Jazan University, Jazan, Saudi Arabia

Full list of author information is available at the end of the article

Abstract

Takahashi and Kim (Math. Jpn. 48:1-9, 1998) used the Ishikawa iteration process to prove some convergence theorems for nonexpansive mappings in Banach spaces. The aim of this paper is to prove similar results in CAT(0) spaces for generalized nonexpansive mappings, which, in turn, generalize the corresponding results of Takahashi and Kim (Math. Jpn. 48:1-9, 1998), Laokul and Panyanak (Int. J. Math. Anal. 3(25-28):1305-1315, 2009), Razani and Salahifard (Bull. Iran. Math. Soc. 37(1):235-246, 2011) and some others.

MSC: 47H10; 54H25

Keywords: CAT(0) spaces; fixed point;

-convergence; condition (E) and Opial property

1 Introduction

A self-mappingT defined on a bounded closed and convex subsetKof a Banach spaceX

is said to be nonexpansive if

TxTyxy for allx,yK.

In an attempt to construct a convergent sequence of iterates with respect to a nonex-pansive mapping, Mann [] defined an iteration method as follows: (for anyx∈K)

xn+= ( –αn)xn+αnTxn, n∈N,

whereαn∈(, ).

In , with a view to approximate the fixed point of pseudo-contractive compact map-pings in Hilbert spaces, Ishikawa [] introduced a new iteration procedure as follows (for

x∈K):

⎧ ⎨ ⎩

yn= ( –αn)xn+αnTxn,

xn+= ( –βn)xn+βnTyn, n∈N,

whereαn,βn∈(, ).

For a comparison of the preceding two iterative schemes in one-dimensional case, we refer the reader to Rhoades [] wherein it is shown that under suitable conditions (see part (a) of Theorem ) the rate of convergence of the Ishikawa iteration is faster than that

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of the Mann iteration procedure. Iterative techniques for approximating fixed points of nonexpansive single-valued mappings have been investigated by various authors using the Mann as well as Ishikawa iteration schemes. By now, there exists an extensive literature on the iterative fixed points for various classes of mappings. For an up-to-date account of the literature on this theme, we refer the readers to Berinde [].

In , Takahashi and Kim [] consider the Ishikawa iteration procedure described as

⎧ ⎨ ⎩

yn=αnTxn+ ( –αn)xn,

xn+=βnTyn+ ( –βn)xn forn≥,

()

where the sequences{αn}and{βn}are sequences in [, ] such that one of the following holds:

(i) αn∈[a,b]andβn∈[,b]for somea,bwith <ab< ,

(ii) αn∈[a, ]andβn∈[a,b]for somea,bwith <ab< .

Utilizing the forgoing iterative scheme, Takahashi and Kim [] proved weak as well as strong convergence theorems for a nonexpansive mapping in Banach spaces.

Recently, García-Falsetet al.[] introduced two generalizations of nonexpansive map-pings which in turn include Suzuki generalized nonexpansive mapmap-pings contained in [] and also utilized the same to prove some fixed point theorems.

The following definitions are relevant to our subsequent discussions.

Definition .([]) LetCbe the nonempty subset of a Banach spaceXandT:CXbe

a single-valued mapping. ThenT is said to satisfy the condition (Eμ) (for someμ≥) if for allx,yC

xTyμxTx+xy.

We say thatTsatisfies the condition (E) wheneverTsatisfies the condition (Eμ) for some μ≥.

Definition .([]) LetCbe the nonempty subset of a Banach spaceXandT:CXbe

a single-valued mapping. ThenT is said to satisfy the condition (Cλ) (for someλ∈(, )) if for allx,yC

λxTxxyTxTyxy.

The following theorem is essentially due to García-Falsetet al.[].

Theorem .([]) Let C be a convex subset of a Banach space X and T:CC be a

mapping satisfying conditions(E)and(Cλ)for someλ∈(, ).Further,assume that either of the following holds.

(a) Cis weakly compact andXsatisfies the Opial condition. (b) Cis compact.

(c) Cis weakly compact andXis(UCED).

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A metric space (X,d) is aCAT() space if it is geodesically connected and every geodesic triangle inXis at least as thin as its comparison triangle in the Euclidean plane. It is well known that any complete, simply connected Riemannian manifold having nonpositive sec-tional curvature is aCAT() space. Other examples are pre-Hilbert spaces, R-trees, the Hilbert ball (see []) and many others. For more details onCAT() spaces, one can consult [–]. Fixed point theory inCAT() spaces was initiated by Kirk [] who proved some theorems for nonexpansive mappings. Since then the fixed point theory for single-valued as well as multi-valued mappings has intensively been developed inCAT() spaces (e.g.

[–]). Further relevant background material onCAT() spaces is included in the next section.

The purpose of this paper is to prove some weak and strong convergence theorems of the iterative scheme () inCAT() spaces for generalized nonexpansive mappings enabling us to enlarge the class of underlying mappings as well as the class of spaces in the corre-sponding results of Takahashi and Kim [].

2 Preliminaries

In this section, to make our presentation self-contained, we collect relevant definitions and results. In a metric space (X,d), a geodesic path joiningxXandyXis a mapc

from a closed interval [,r]⊂RtoXsuch thatc() =x,c(r) =yandd(c(t),c(s)) =|st|for alls,t∈[,r]. In particular, the mappingcis an isometry andd(x,y) =r. The image ofc

is called a geodesic segment joiningxandy, which is denoted by [x,y], whenever such a segment exists uniquely. For anyx,yX, we denote the pointz∈[x,y] byz= ( –α)xαy, where ≤α≤ ifd(x,z) =αd(x,y) andd(z,y) = ( –α)d(x,y). The space (X,d) is called a geodesic space if any two points ofXare joined by a geodesic, andXis said to be uniquely geodesic if there is exactly one geodesic joiningxandyfor eachx,yX. A subsetCofX

is called convex ifCcontains every geodesic segment joining any two points inC. A geodesic triangle(x,x,x) in a geodesic metric space (X,d) consists of three points

ofX (as the vertices of) and a geodesic segment between each pair of points (as the edges of). A comparison triangle for(x,x,x) in (X,d) is a triangle(x,x,x) :=

(x,x,x) in the Euclidean plane R such that dR(xi,xj) =d(xi,xj) for i,j∈ {, , }.

A pointx∈[x,x] is said to be comparison point forx∈[x,x] ifd(x,x) =d(x,x).

Com-parison points on [x,x] and [x,x] are defined in the same way.

A geodesic metric spaceXis called aCAT() space if all geodesic triangles satisfy the following comparison axiom (CAT() inequality):

Letbe a geodesic triangle inXand letbe its comparison triangle inR. Thenis said to satisfy theCAT() inequality if for allx,y∈ and all comparison pointsx,y∈ ,

d(x,y)≤dR(x,y).

Ifx,yandyare points ofCAT() space andyis the midpoint of the segment [y,y],

then theCAT() inequality implies

d(x,y)≤

 d(x,y)

+

d(x,y)

d(y,y)

.

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(i) Any convex subset of a Euclidean spaceRn, when endowed with the induced

metric is aCAT()space.

(ii) Every pre-Hilbert space is aCAT()space.

(iii) If a normed real vector spaceXisCAT()space, then it is a pre-Hilbert space. (iv) The Hilbert ball with the hyperbolic metric is aCAT()space.

(v) IfXandXareCAT()spaces, thenX×Xis also aCAT()space.

For detailed information regarding these spaces, one can refer to [–, ].

Now, we collect some basic geometric properties which are instrumental throughout our subsequent discussions. LetXbe a completeCAT() space and{xn}be a bounded sequence inX. ForxXset

rx,{xn}=lim sup n→∞ d(x,xn).

The asymptotic radiusr({xn}) is given by

r{xn}=infr(x,xn) :xX

,

and the asymptotic centerA({xn}) of{xn}is defined as

A{xn}=xX:r(x,xn) =r

{xn}.

It is well known that in aCAT() space,A({xn}) consists of exactly one point (see Propo-sition  of []).

In , Kirk and Panyanak [] gave a concept of convergence inCAT() spaces which is the analog of the weak convergence in Banach spaces and a restriction of Lim’s concepts of convergence [] toCAT() spaces.

Definition . A sequence{xn}inX is said to-converge toxX ifxis the unique

asymptotic center of un for every subsequence {un} of {xn}. In this case we write

-limnxn=xandxis the-limit of{xn}.

Notice that given{xn} ⊂Xsuch that{xn}-converges toxand, givenyXwithy=x, by uniqueness of the asymptotic center we have

lim sup n→∞

d(xn,x) <lim sup n→∞

d(xn,y).

Thus everyCAT() space satisfies the Opial property. Now we collect some basic facts aboutCAT() spaces which will be used throughout the text frequently.

Lemma .([]) Every bounded sequence in a completeCAT()space admits a

-conver-gent subsequence.

Lemma .([]) If C is closed convex subset of a completeCAT()space and if{xn}is a bounded sequence in C,then the asymptotic center of{xn}is in C.

Lemma .([]) Let(X,d)be aCAT()space.For x,yX and t∈[, ],there exists a unique z∈[x,y]such that

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We use the notation ( –t)xtyfor the unique pointzof the above lemma.

Lemma . For x,y,zX and t∈[, ]we have

d( –t)xty,z≤( –t)d(x,z) +td(y,z).

Recently García-Falsetet al.[] introduced two generalizations of the condition (C) in Banach spaces. Now, we state their condition in the framework ofCAT() spaces.

Definition . LetT be a mapping defined on a subsetCofCAT() spaceXandμ≥,

thenTis said to satisfy the condition (Eμ), if (for allx,yC)

d(x,Ty)≤μd(x,Tx) +d(x,y).

We say thatTsatisfies the condition (E) wheneverTsatisfies the condition (Eμ) for some μ≥.

Definition . LetT be a mapping defined on a subsetCof aCAT() spaceXandλ

(, ), thenTis said to satisfy the condition (Cλ) if (for allx,yC)

λd(x,Tx)≤d(x,y) ⇒ d(Tx,Ty)≤d(x,y).

In the case  <λ<λ< , then the condition () implies the condition (). The

following example shows that the class of mappings satisfying the conditions (E) and (Cλ) (for someλ∈(, )) is larger than the class of mappings satisfying the condition (C).

Example .([]) For a givenλ∈(, ), define a mappingTon [, ] by

T(x) =

⎧ ⎨ ⎩ x

 ifx= , +λ

+λ ifx= .

Then the mappingTsatisfies the condition (Cλ) but it fails the condition (Cλ) whenever

 <λ<λ. Moreover,Tsatisfies the condition (Eμ) forμ=+λ.

The following theorem is an analog of Theorem  in [] toCAT() spaces.

Theorem . Let C be a bounded convex subset of a completeCAT()space X.If T:CC satisfies the condition(Cλ)on C for someλ∈(, ),then there exists an approximating fixed point sequence for T.

Now, we present the Ishikawa iterative scheme in the framework ofCAT() spaces. For

x∈C, the Ishikawa iteration is defined as

⎧ ⎨ ⎩

yn=αnTxn⊕( –αn)xn,

xn+=βnTyn⊕( –βn)xn forn≥,

()

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The following lemma is a consequence of Lemma . of [] which will be used to prove our main results.

Lemma . Let X be a completeCAT()space and let xX.Suppose{tn}is a sequence in

[b,c]for some b,c∈(, )and{xn},{yn}are sequences in X such thatlim supn→∞d(xn,x)≤ r,lim supn→∞d(yn,x)≤r,andlimn→∞d(( –tn)xntnyn,x) =r for some r≥.Then

lim

n→∞d(xn,yn) = .

In this paper, we prove that the sequence{xn}described by ()-converges to a fixed point ofTif one of the following conditions holds:

⎧ ⎨ ⎩

(i) αn∈[a,b] and βn∈[,b] for somea,bwith  <ab<  or

(ii) αn∈[a, ] and βn∈[a,b] for somea,bwith  <ab< .

()

This result is an analog of a result of the weak convergence theorem of Takahashi and Kim [] for a generalized nonexpansive mapping in a Banach space. A strong convergence theorem is also proved. In the process, the corresponding results of Takahashi and Kim [], Laokul and Panyanak [], Razani and Salahifard [], and others are generalized and improved.

3 Main results

Before proving our main results, firstly we re-write Theorem . in the setting ofCAT() spaces which is essentially Theorem . of [].

Theorem . Let C be a bounded,closed,and convex subset of a completeCAT()space X.

If T:CC satisfies the conditions(E)and(Cλ)for someλ∈(, ),then T has a fixed point.

Now, to accomplish our main results, we prove the following lemma.

Lemma . Let C be a nonempty closed convex subset of a completeCAT()space X and

T :CC be a mapping which satisfies the condition(Cλ)for someλ∈(, ).If{xn}is a sequence defined by()and the sequences{αn}and{βn}satisfy the condition described in

(),thenlimn→∞d(xn,x)exists for all xF(T).

Proof SinceTsatisfies the condition (Cλ) (for someλ∈(, )) andxF(T), we have

λd(x,Tx) = ≤d(x,yn)

and

λd(x,Tx) = ≤d(x,xn) for alln≥,

so that

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Now consider

d(xn+,x) =d

βnTyn⊕( –βn)xn,x

βnd(Tyn,x) + ( –βn)d(xn,x) ≤βnd(yn,x) + ( –βn)d(xn,x)

=βnd

αnTxn⊕( –αn)xn,x

+ ( –βn)d(xn,x)

βnαnd(Txn,x) +βn( –αn)d(xn,x) + ( –βn)d(xn,x) ≤d(xn,x),

which shows that the sequence d(xn,x) is decreasing and bounded below so that

limn→∞d(xn,x) exists.

Lemma . Let C be a nonempty closed convex subset of a completeCAT()space X and let T:CC satisfy the conditions(Cλ)and(E)on C.If{xn}is a sequence defined by()and the sequences{αn}and{βn}satisfy the condition described in(),then F(T)is nonempty if and only if{xn}is bounded andlimn→∞d(Txn,xn) = .

Proof Suppose that the fixed point setF(T) is nonempty andxF(T). Then by Lemma .,

limn→∞d(xn,x) exists and let us take it to becbesides that{xn}is bounded. We have

λd(x,Tx) = ≤d(x,yn)

and

λd(x,Tx) = ≤d(x,xn) for alln≥.

Owing to the condition (Cλ), we haved(Tx,Tyn)≤d(x,yn) andd(Tx,Txn)≤d(x,xn).

Therefore, we have

d(Tyn,x)≤d(yn,x)

=dαnTxn⊕( –αn)xn,x

αnd(Txn,x) + ( –αn)d(xn,x) ≤αnd(xn,x) + ( –αn)d(xn,x)

=d(xn,x),

so that

lim sup n→∞

d(Tyn,x)≤lim sup n→∞

d(yn,x)≤c.

Also, we observe that

lim n→∞d

βnTyn⊕( –βn)xn,x

= lim

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Case I: If  <aβnb<  and ≤αn≤, then by the foregoing discussion and by

Lemma ., we havelimn→∞d(Tyn,xn) = .

For eachαn∈[,b],

d(Txn,xn)≤d(Txn,yn) +d(yn,xn)

=dTxn,αnTxn⊕( –αn)xn

+d(yn,xn)

≤( –αn)d(Txn,xn) +d(yn,xn),

so that

αnd(Txn,xn)≤d(yn,xn).

Asαn∈[,b], in view of the condition (Cλ) we haved(Txn,Tyn)≤d(xn,yn).

Now, consider

d(Txn,xn)≤d(Txn,Tyn) +d(Tyn,xn)

d(xn,yn) +d(Tyn,xn) ≤αnd(xn,Txn) +d(Tyn,xn).

Making use of the observation ( –b)d(Txn,xn)≤( –αn)d(xn,Txn)≤d(Tyn,xn), we have limn→∞d(Txn,xn)≤–blimn→∞d(Tyn,xn) = .

Case :  <aβn≤ and  <aαnb< .

Since we haved(Txn,x)≤d(xn,x), for alln≥, we get

lim sup n→∞

d(Txn,x)≤c.

Consider

d(xn+,x)≤βnd(Tyn,x) + ( –βn)d(xn,x) ≤βnd(yn,x) + ( –βn)d(xn,x),

which amounts to saying that

d(xn+,x) –d(xn,x)

αn

d(yn,x) –d(xn,x).

Takinglim infn→∞of both sides of the above inequality, we have

lim inf n→∞

d(xn+,x) –d(xn,x)

αn

≤lim inf n→∞

d(yn,x) –d(xn,x)

.

Aslimn→∞d(xn+,x) =limn→∞d(xn,x) =c, we have

≤lim inf n→∞

d(yn,x) –d(xn,x)

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while, owing tod(yn,x) –d(xn,x)≤, we have

lim inf n→∞

d(yn,x) –d(xn,x)

≤.

Therefore,lim infn→∞(d(yn,x) –d(xn,x)) = . Thus, we get lim inf

n→∞ d(xn,x)≤lim infn→∞ d(yn,x).

That is,c≤lim infn→∞d(yn,x). By combining the foregoing observations, we have c≤lim inf

n→∞ d(yn,x)≤lim supn→∞ d(yn,x)≤c,

so that

c= lim

n→∞d(yn,x) =nlim→∞d

αnTxn⊕( –αn)xn,x

.

Again in view of Lemma ., we havelimn→∞d(Txn,xn) = .

Conversely, suppose that{xn}is bounded andlimn→∞d(xn,Txn) = . LetA({xn}) ={x}.

Then, by Lemma .,xC. AsTsatisfies the condition (Eμ) onC, there existsμ>  such that

d(xn,Tx)≤μd(xn,Txn) +d(xn,x),

which implies that

lim sup

n→∞ d(xn,Tx)≤lim supn→∞

μd(xn,Txn) +d(xn,x)

=lim sup

n→∞

d(xn,x).

Owing to the uniqueness of asymptotic center,Tx=x, so thatxis fixed point ofT.

Theorem . Let C be a nonempty closed convex subset of a completeCAT()space X and

T :CC be a mapping which satisfies conditions(Cλ)for someλ∈(, )and(E)on C with F(T)=∅.If the sequences{xn},{αn}and{βn}are described as in()and(),then the sequence{xn}-converges to a fixed point of T.

Proof By Lemma ., we observe that{xn}is a bounded sequence and

lim

n→∞d(xn,Txn) = .

LetWω({xn}) =: A({un}), where the union is taken over all subsequence{un}over{xn}. To show the-convergence of{xn}to a fixed point ofT, we show thatWω({xn})⊂F(T) andWω({xn}) is a singleton set. To show thatWω({xn})⊂F(T) letyWω({xn}). Then there exists a subsequence{yn}of{xn}such thatA({yn}) =y. By Lemmas . and ., there exists a subsequence{zn}of{yn}such that-limnzn=zandzC. Sincelimn→∞d(zn,Tzn) = 

andTsatisfies the condition (E), there exists aμ≥ such that

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By taking thelim supof both sides, we have

lim sup n→∞

d(zn,Tz)≤lim sup n→∞

μd(zn,Tzn) +d(zn,z)

≤lim sup

n→∞ d(zn,z).

As-limnzn=z, by the Opial property

lim sup

n→∞ d(zn,z)≤lim supn→∞ d(zn,Tz).

HenceTz=z,i.e. zF(T). Now, we claim thatz=y. If not, by Lemma .,limnd(xn,z)

exists and, owing to the uniqueness of the asymptotic centers,

lim sup n→∞

d(zn,z) <lim sup n→∞

d(zn,y)

≤lim sup

n→∞

d(yn,y)

< lim sup n→∞ d(yn,z)

=lim sup

n→∞

d(xn,z)

=lim sup

n→∞ d(zn,z),

which is a contradiction. Hencey=z. To assert thatWω({xn}) is a singleton let{yn}be a subsequence of{xn}. In view of Lemmas . and ., there exists a subsequence{zn}of

{yn}such that-limnzn=z. LetA({yn}) =yandA({xn}) =x. Earlier, we have shown that y=z. Therefore it is enough to showz=x. Ifz=x, then in view of Lemma .{d(xn,z)}is

convergent. By uniqueness of the asymptotic centers

lim sup n→∞

d(zn,z) <lim sup n→∞

d(zn,x)

≤lim sup

n→∞ d(xn,x)

< lim sup n→∞

d(xn,z)

=lim sup

n→∞

d(zn,z),

which is a contradiction so that the conclusion follows. In view of Theorem ., we have the following corollary of the preceding theorem.

Corollary . Let C be a nonempty bounded, closed and convex subset of a complete

CAT()space X and T:CC be a mapping which satisfies conditions(Cλ)for some

λ∈(, )and(E)on C.If sequences{xn},{αn},and{βn}are described by()and(),then the sequence{xn}-converges to a fixed point of T.

Theorem . Let C be a nonempty closed convex subset of a completeCAT()space X and

T :CC be a mapping which satisfies conditions(Cλ)for someλ∈(, )and(E)on C.

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Proof To show that the fixed point setF(T) is closed, let{xn}be a sequence inF(T) which converges to some pointzC. As

λd(xn,Txn) = ≤d(xn,z),

in view of the condition (Cλ), we have

d(xn,Tz) =d(Txn,Tz)≤d(xn,z).

By taking the limit of both sides we obtain

lim

n→∞d(xn,Tz)≤nlim→∞d(xn,z) = .

In view of the uniqueness of the limit, we havez=Tz, so thatF(T) is closed. Observe that by Lemma ., we havelimn→∞d(xn,Txn) = . It follows from the condition (I) that

lim n→∞f

dxn,F(T)

≤ lim

n→∞d(xn,Txn) = ,

so thatlimn→∞f(d(xn,F(T))) = . Sincef : [,∞]→[,∞) is a nondecreasing mapping

satisfyingf() =  for allr∈(,∞), we havelimn→∞d(xn,F(T)) = . This implies that

there exists a subsequence{xnk}of{xn}such that

d(xnk,pk)≤

k for allk≥

wherein{pk}is inF(T). By Lemma ., we have

d(xnk+,pk)≤d(xnk,pk)≤

 k,

so that

d(pk+,pk)≤d(pk+,xnk+) +d(xnk+,pk)

≤ 

k+ +

 k<

 k–,

which implies that{pk}is a Cauchy sequence. SinceF(T) is closed,{pk}is a convergent sequence. Write limk→∞pk=p. Now, in order to show that{xn} converges top let us

proceed as follows:

d(xnk,p)≤d(xnk,pk) +d(pk,p)→ ask→ ∞,

so thatlimk→∞d(xnk,p) = . Sincelimn→∞d(xn,p) exists, we havexnp.

Corollary . Let C be a nonempty bounded, closed,and convex subset of a complete

CAT()space X and let T :CC be a mapping which satisfies the conditions(Cλ)for someλ∈(, )and(E)on C.Moreover,T satisfies the condition(I).If the sequences{xn},

(12)

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors have contributed in this work on an equal basis. All authors read and approved the final manuscript.

Author details

1Department of Mathematics, Aligarh Muslim University, Aligarh, Uttar Pradesh 202002, India.2Department of

Mathematics, Faculty of Science, Jazan University, Jazan, Saudi Arabia.

Acknowledgements

The authors thank the anonymous referees and handling editor of the manuscript for their valuable suggestions and fruitful comments. The first author is grateful to University Grants commission, India for the financial assistance in the form of Maulana Azad National Fellowship. The research of second author is supported by Jazan University, Saudi Arabia.

Received: 23 January 2014 Accepted: 16 April 2014 Published:02 May 2014 References

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10.1186/1029-242X-2014-155

Cite this article as:Uddin et al.:Approximating fixed points for generalized nonexpansive mapping inCAT(0)spaces.

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