R E S E A R C H
Open Access
On the strong and
-convergence of new
multi-step and
S
-iteration processes in a
CAT()
space
Metin Ba¸sarır and Aynur ¸Sahin
**Correspondence: [email protected] Department of Mathematics, Faculty of Sciences and Arts, Sakarya University, Sakarya, 54187, Turkey
Abstract
In this paper, we introduce a new class of mappings and prove the demiclosedness principle for mappings of this type in a CAT(0) space. Also, we obtain the strong and
-convergence theorems of new multi-step andS-iteration processes in a CAT(0) space. Our results extend and improve the corresponding recent results announced by many authors in the literature.
MSC: 47H09; 47H10; 54E40; 58C30
Keywords: CAT(0) space; contractive-like mapping; strong convergence;
-convergence; new multi-step iteration;S-iteration; fixed point
1 Introduction
Contractive mappings and iteration processes are some of the main tools in the study of fixed point theory. There are many contractive mappings and iteration processes that have been introduced and developed by several authors to serve various purposes in the literature (see [–]).
Imoru and Olantiwo [] gave the following contractive definition.
Definition LetT be a self-mapping on a metric space X. The mappingT is called a contractive-like mapping if there exist a constantδ∈[, ) and a strictly increasing and continuous functionϕ: [,∞)→[,∞) withϕ() = such that, for allx,y∈X,
d(Tx,Ty)≤δd(x,y) +ϕd(x,Tx). (.)
This mapping is more general than those considered by Berinde [, ], Harder and Hicks [], Zamfirescu [], Osilike and Udomene [].
By takingδ= in (.), we define a new class of mappings as follows.
Definition The mappingTis called a generalized nonexpansive mapping if there exists a non-decreasing and continuous functionϕ: [,∞)→[,∞) withϕ() = such that, for allx,y∈X,
d(Tx,Ty)≤d(x,y) +ϕd(x,Tx). (.)
Remark Forx∈F(T) in (.), we have
d(x,Ty) =d(Tx,Ty)≤d(x,y) +ϕd(x,Tx)=d(x,y).
IfXis an interval ofR, thenF(T) is convex. The same is also true in each space with unique geodesic for each pair of points (e.g., metric trees orCAT() spaces).
In the caseϕ(t) = for allt∈[,∞), it is easy to show that every nonexpansive mapping satisfies (.), but the inverse is not necessarily true.
Example LetX= [, ],d(x,y) =|x–y|,ϕ(t) =tand defineT by
T(x) =
⎧ ⎨ ⎩
ifx= , ifx= .
By takingx= andy= ., we have
dT(),T(.)= < . =d(, .) +ϕd,T() but
dT(),T(.)= . =d(, .).
ThereforeTis a generalized nonexpansive mapping, butTis not nonexpansive mapping. Both a contractive-like mapping and a generalized nonexpansive mapping need not have a fixed point, even ifXis complete. For example, letX= [,∞),d(x,y) =|x–y|and define
T by
Tx=
⎧ ⎨ ⎩
if ≤x≤., . if . <x< +∞.
It is proved in Gürsoyet al.[] thatT is a contractive-like mapping. Similarly, one can prove thatT is a generalized nonexpansive mapping. But the mappingT has no fixed point.
By using (.), it is obvious that if a contractive-like mapping has a fixed point, then it is unique. However, if a generalized nonexpansive mapping has a fixed point, then it need not be unique. For example, letRbe the real line with the usual norm| · |, and letK= [–, ]. Define a mappingT:K→Kby
Tx=
⎧ ⎨ ⎩
x ifx∈[, ], –x ifx∈[–, ).
Now, we show thatT is a nonexpansive mapping. In fact, ifx,y∈[, ] orx,y∈[–, ), then we have
Ifx∈[, ] andy∈[–, ) orx∈[–, ) andy∈[, ], then we have
|Tx–Ty|=|x+y| ≤ |x–y|.
This implies thatT is a nonexpansive mapping and soT is a generalized nonexpansive mapping withϕ(t) = for allt∈[,∞). ButF(T) ={x∈K; ≤x≤}.
Agarwalet al.[] introduced theS-iteration process which is independent of those of Mann [] and Ishikawa [] and converges faster than both of these. We apply this iteration process in aCAT() space as
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
x∈K,
xn+= ( –αn)Txn⊕αnTyn, yn= ( –βn)xn⊕βnTxn, n≥.
(.)
Gürsoyet al.[] introduced a new multi-step iteration process in a Banach space. We modify this iteration process in aCAT() space as follows.
For an arbitrary fixed orderk≥,
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈K,
xn+= ( –αn)yn⊕αnTyn, yn= ( –βn)yn⊕βnTyn,
y
n= ( –βn)yn⊕βnTyn, · · ·
yk–
n = ( –βnk–)ynk–⊕βnk–Tyk–n , yk–n = ( –βnk–)xn⊕βnk–Txn, n≥,
or, in short,
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈K,
xn+= ( –αn)yn⊕αnTyn, yi
n= ( –βni)yi+n ⊕βniTyi+n , i= , , . . . ,k– , yk–n = ( –βnk–)xn⊕βnk–Txn, n≥.
(.)
By takingk= andk= in (.), we obtain the SP-iteration process of Phuengrattana and Suantai [] and the two-step iteration process of Thianwan [], respectively.
In this paper, motivated by the above results, we prove demiclosedness principle for a new class of mappings and the -convergence theorems of the new multi-step itera-tion and theS-iteration processes for mappings of this type in aCAT() space. Also, we present the strong convergence theorems of these iteration processes for contractive-like mappings in aCAT() space.
2 Preliminaries on aCAT(0)space
point theory in aCAT() space was first studied by Kirk (see [, ]). He showed that every nonexpansive mapping defined on a bounded closed convex subset of a completeCAT() space always has a fixed point. Since then the fixed point theory in aCAT() space has been rapidly developed and many papers have appeared (see [–]). It is worth mentioning that the results in aCAT() space can be applied to anyCAT(k) space withk≤ since any
CAT(k) space is aCAT(k) space for everyk ≥k(see [, p.]).
Let (X,d) be a metric space. Ageodesic pathjoiningx∈Xtoy∈X(or more briefly, a
geodesicfromxtoy) is a mapcfrom a closed interval [,l]⊂RtoXsuch thatc() =x,
c(l) =yand d(c(t),c(t)) =|t–t| for all t,t ∈[,l]. In particular, cis an isometry and
d(x,y) =l. The image ofcis called ageodesic(ormetric)segmentjoiningxandy. When it is unique, this geodesic is denoted by [x,y]. The space (X,d) is said to be ageodesic space
if every two points ofXare joined by a geodesic andXis said to be auniquely geodesicif there is exactly one geodesic joiningxtoyfor eachx,y∈X.
Ageodesic triangle(x,x,x) in a geodesic metric space (X,d) consists of three points
inX(the vertices of) and a geodesic segment between each pair of vertices (the edges of). Acomparison trianglefor the geodesic triangle(x,x,x) in (X,d) is a triangle (x,x,x) =(x,x,x) in the Euclidean planeRsuch that
dR(xi,xj) =d(xi,xj)
fori,j∈ {, , }. Such a triangle always exists (see []).
A geodesic metric space is said to be aCAT() space [] if all geodesic triangles of appropriate size satisfy the following comparison axiom.
CAT() Letbe a geodesic triangle inX, and letbe a comparison triangle for. Then is said to satisfy theCAT()inequalityif for allx,y∈ and all comparison points
x,y∈ ,
d(x,y)≤dR(x,y).
We observe that ifx,y,yare points of aCAT() space and ifyis the midpoint of the
segment [y,y], then theCAT() inequality implies that
d(x,y)≤
d(x,y)
+
d(x,y)
–
d(y,y)
. (.)
The equality holds for the Euclidean metric. In fact (see [, p.]), a geodesic metric space is aCAT() space if and only if it satisfies the inequality (.) (which is known as the
CNinequality of Bruhat and Tits []).
Letx,y∈X, by [, Lemma .(iv)] for eacht∈[, ], there exists a unique pointz∈[x,y] such that
d(x,z) =td(x,y), d(y,z) = ( –t)d(x,y). (.)
Lemma Let X be aCAT()space.Then d( –t)x⊕ty,z≤( –t)d(x,z) +td(y,z)
for all t∈[, ]and x,y,z∈X.
Lemma Let X be aCAT()space.Then
d( –t)x⊕ty,z≤( –t)d(x,z)+td(y,z)–t( –t)d(x,y)
for all t∈[, ]and x,y,z∈X.
3 Demiclosedness principle for a new class of mappings
In Lim [] introduced the concept of convergence in a general metric space set-ting which is called -convergence. Later, Kirk and Panyanak [] used the concept of
-convergence introduced by Lim [] to prove on aCAT() space analogs of some Ba-nach space results which involve weak convergence. Also, Dhompongsa and Panyanak [] obtained the-convergence theorems for the Picard, Mann and Ishikawa iterations in a
CAT() space for nonexpansive mappings under some appropriate conditions. Now, we recall some definitions.
Let{xn}be a bounded sequence in aCAT() spaceX. Forx∈X, we set rx,{xn}
=lim sup n→∞ d(x,xn).
Theasymptotic radius r({xn}) of{xn}is given by r{xn}
=infrx,{xn}
:x∈X ,
and theasymptotic radius rK({xn}) of{xn}with respect toK⊂Xis given by rK
{xn}
=infrx,{xn}
:x∈K .
Theasymptotic center A({xn}) of{xn}is the set
A{xn}
=x∈X:rx,{xn}
=r{xn} ,
and theasymptotic center AK({xn}) of{xn}with respect toK⊂Xis the set
AK
{xn}
=x∈K:rx,{xn}
=rK
{xn} .
Proposition ([, Proposition .]) Let{xn}be a bounded sequence in a completeCAT() space X,and let K be a closed convex subset of X,then A({xn})and AK({xn})are singletons.
Lemma
(i) Every bounded sequence in a completeCAT()space always has a-convergent subsequence(see[, p.]).
(ii) LetKbe a nonempty closed convex subset of a completeCAT()space,and let{xn}
be a bounded sequence inK.Then the asymptotic center of{xn}is inK(see[,
Proposition .]).
Lemma ([, Lemma .]) If{xn}is a bounded sequence in a completeCAT()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.
Let{xn}be a bounded sequence in aCAT() spaceX, and letKbe a closed convex subset
ofXwhich contains{xn}. We denote the notation {xn}w ⇔ (w) =inf
x∈K(x), (.)
where(x) =lim supn→∞d(xn,x).
We note that{xn}wif and only ifAK({xn}) ={w}(see []).
Nanjaras and Panyanak [] gave a connection between the ‘’ convergence and
-convergence.
Proposition ([, Proposition .]) Let {xn} be a bounded sequence in a CAT() space X,and let K be a closed convex subset of X which contains{xn}.Then-limn→∞xn= p implies that{xn}p.
By using the convergence defined in (.), we obtainthe demiclosedness principle for the new class of mappings in aCAT()space.
Theorem Let K be a nonempty closed convex subset of a completeCAT()space X,and let T:K→K be a generalized nonexpansive mapping with F(T)=∅.Let{xn}be a bounded sequence in K such that{xn}w andlimn→∞d(xn,Txn) = .Then Tw=w.
Proof By the hypothesis, {xn}w. Then we haveAK({xn}) ={w}. By Lemma (ii), we
obtainA({xn}) ={w}. Sincelimn→∞d(xn,Txn) = , then we have
(x) =lim sup n→∞
d(xn,x) =lim sup n→∞
d(Txn,x) (.)
for allx∈K. By takingx=Twin (.), we have
(Tw) =lim sup n→∞
d(Txn,Tw)
≤lim sup
n→∞
d(xn,w) +ϕ
d(xn,Txn)
≤lim sup
n→∞
d(xn,w) +ϕ
lim sup n→∞
d(xn,Txn)
=lim sup n→∞ d(xn,w)
The rest of the proof closely follows the pattern of Proposition . in Nanjaras and Pa-nyanak []. HenceTw=was desired. Now, we prove the-convergence of the new multi-step iteration process for the new class of mappings in aCAT() space.
Theorem Let K be a nonempty closed convex subset of a completeCAT()space X,let T:
K→K be a generalized nonexpansive mapping with F(T)=∅,and let{xn}be a sequence defined by(.)such that{αn},{βni} ⊂[, ],i= , , . . . ,k– and{βnk–} ⊂[a,b]for some a,b∈(, ).Then the sequence{xn}-converges to the fixed point of T.
Proof Letp∈F(T). From (.), (.) and Lemma , we have
d(xn+,p) =d
( –αn)yn⊕αnTyn,p
≤( –αn)d
yn,p+αnd
Tyn,p ≤( –αn)d
yn,p+αn
dyn,p+ϕd(p,Tp) =dyn,p.
Also, we obtain
dyn,p=d –βnyn⊕βnTyn,p ≤ –βndyn,p+βndTyn,p
≤ –βndyn,p+βndyn,p+ϕd(p,Tp) =dyn,p.
Continuing the above process, we have
d(xn+,p)≤d
yn,p≤dyn,p≤ · · · ≤dyk–n ,p≤d(xn,p). (.)
This inequality guarantees that the sequence{d(xn,p)} is non-increasing and bounded
below, and so limn→∞d(xn,p) exists for all p∈F(T). Letlimn→∞d(xn,p) =r. By using
(.), we get
lim n→∞d
yk–n ,p=r.
By Lemma , we also have
dyk–n ,p=d –βnk–xn⊕βnk–Txn,p
≤ –βnk–d(xn,p)+βnk–d(Txn,p)–βnk–
–βnk–d(xn,Txn) ≤ –βnk–d(xn,p)+βnk–
d(xn,p) +ϕ
d(p,Tp) –βnk– –βnk–d(xn,Txn)
=d(xn,p)–βnk–
which implies that
d(xn,Txn)≤
a( –b)
d(xn,p)–d
yk–n ,p.
Thuslimn→∞d(xn,Txn) = . To show that the sequence{xn} -converges to a fixed point
ofT, we prove that
W(xn) =
{un}⊂{xn}
A{un}
⊆F(T)
and W(xn) consists of exactly one point. Let u∈W(xn). Then there exists a
subse-quence{un}of{xn}such thatA({un}) ={u}. By Lemma , there exists a subsequence{vn}
of{un} such that-limn→∞vn=v∈K. By Proposition and Theorem ,v∈F(T). By
Lemma , we haveu=v∈F(T). This shows thatW(xn)⊆F(T). Now, we prove that W(xn) consists of exactly one point. Let{un}be a subsequence of{xn}withA({un}) ={u},
and letA({xn}) ={x}. We have already seen thatu=vandv∈F(T). Finally, since{d(xn,v)}
converges, by Lemma ,x=v∈F(T). This shows thatW(xn) ={x}. This completes the
proof.
We give the following theorem related to the-convergence of theS-iteration process for the new class of mappings in aCAT() space.
Theorem Let K be a nonempty closed convex subset of a completeCAT()space X,let T:K→K be a generalized nonexpansive mapping with F(T)= ∅,and let{xn}be a se-quence defined by(.)such that{αn},{βn} ⊂[a,b]for some a,b∈(, ).Then the sequence {xn}-converges to the fixed point of T.
Proof Letp∈F(T). Using (.), (.) and Lemma , we have
d(xn+,p) =d
( –αn)Txn⊕αnTyn,p
≤( –αn)d(Txn,p) +αnd(Tyn,p) ≤( –αn)
d(xn,p) +ϕ
d(p,Tp) +αn
d(yn,p) +ϕ
d(p,Tp)
= ( –αn)d(xn,p) +αnd(yn,p). (.)
Also, we obtain
d(yn,p) =d
( –βn)xn⊕βnTxn,p
≤( –βn)d(xn,p) +βnd(Txn,p) ≤( –βn)d(xn,p) +βn
d(xn,p) +ϕ
d(p,Tp)
=d(xn,p). (.)
From (.) and (.), we have
This inequality guarantees that the sequence{d(xn,p)} is non-increasing and bounded
below, and solimn→∞d(xn,p) exists for allp∈F(T). Let
lim
n→∞d(xn,p) =r. (.)
Now, we prove thatlimn→∞d(yn,p) =r. By (.), we have
d(xn+,p)≤( –αn)d(xn,p) +αnd(yn,p).
This gives that
αnd(xn,p)≤d(xn,p) +αnd(yn,p) –d(xn+,p)
or
d(xn,p)≤d(yn,p) +
αn
d(xn,p) –d(xn+,p)
≤d(yn,p) +
a
d(xn,p) –d(xn+,p)
.
This gives
r≤lim inf n→∞ d(yn,p).
By (.) and (.), we obtain
lim sup n→∞
d(yn,p)≤r.
Then we get
lim
n→∞d(yn,p) =r.
By Lemma , we also have
d(yn,p) =d
( –βn)xn⊕βnTxn,p
≤( –βn)d(xn,p)+βnd(Txn,p)–βn( –βn)d(xn,Txn) ≤( –βn)d(xn,p)+βn
d(xn,p) +ϕ
d(p,Tp) –βn( –βn)d(xn,Txn)
=d(xn,p)–βn( –βn)d(xn,Txn),
which implies that
d(xn,Txn)≤
a( –b)
d(xn,p)–d(yn,p)
.
Thuslimn→∞d(xn,Txn) = . The rest of the proof follows the pattern of the above theorem
4 Strong convergence theorems for a contractive-like mapping
Now, we prove the strong convergence of the new multi-step iteration process for a contractive-like mapping in aCAT() space.
Theorem Let K be a nonempty closed convex subset of a completeCAT()space X,let T:K→K be a contractive-like mapping with F(T)= ∅,and let{xn}be a sequence defined by(.)such that{αn} ⊂[, ),
∞
n=αn=∞and{βni} ⊂[, ),i= , , . . . ,k– .Then the sequence{xn}converges strongly to the unique fixed point of T.
Proof Letpbe the unique fixed point ofT. From (.), (.) and Lemma , we have
d(xn+,p) =d
( –αn)yn⊕αnTyn,p
≤( –αn)d
yn,p+αnd
Tyn,p ≤( –αn)d
yn,p+αn
δdyn,p+ϕd(p,Tp) = ( –αn)d
yn,p+αnδd
yn,p
= –αn( –δ)
dyn,p. Also, we obtain
dyn,p=d –βnyn⊕βnTyn,p ≤ –βndyn,p+βndTyn,p
≤ –βndyn,p+βnδdyn,p+ϕd(p,Tp) = –βndyn,p+βnδdyn,p
= –βn( –δ)dyn,p. In a similar fashion, we can get
dyn,p≤ –βn( –δ)dyn,p. Continuing the above process, we have
d(xn+,p)≤
–αn( –δ)
–βn( –δ) –βn( –δ)· · ·
–βnk–( –δ)dyk–n ,p. (.) In addition, we obtain
dyk–n ,p=d –βnk–xn⊕βnk–Txn,p
≤ –βnk–d(xn,p) +βnk–d(Txn,p) ≤ –βnk–d(xn,p) +βnk–
δd(xn,p) +ϕ
d(p,Tp) = –βnk–d(xn,p) +βnk–δd(xn,p)
From (.) and (.), we have
d(xn+,p)≤
–αn( –δ)
–βn( –δ) –βn( –δ)· · ·
–βnk–( –δ) –βnk–( –δ)d(xn,p) ≤ –αn( –δ)
d(xn,p) ≤
n
j=
–αj( –δ)
d(x,p)
≤e–(–δ) n
j=αjd(x
,p). (.)
Using the fact that ≤δ< ,αj∈[, ] and
∞
n=αn=∞, we get that lim
n→∞e
–(–δ)nj=αj= .
This together with (.) implies that
lim
n→∞d(xn+,p) = .
Consequently,xn→p∈F(T) and this completes the proof.
Remark In Theorem the condition∞n=αn=∞may be replaced with
∞
n=βni=∞
for a fixedi= , , . . . ,k– .
Finally, we give the strong convergence theorem of the S-iteration process for a contractive-like mapping in aCAT() space as follows.
Theorem Let K be a nonempty closed convex subset of a completeCAT()space X,let T:K→K be a contractive-like mapping with F(T)= ∅,and let{xn}be a sequence defined by(.)such that{αn},{βn} ⊂[, ].Then the sequence{xn}converges strongly to the unique fixed point of T.
Proof Letpbe the unique fixed point ofT. From (.), (.) and Lemma , we have
d(xn+,p) =d
( –αn)Txn⊕αnTyn,p
≤( –αn)d(Txn,p) +αnd(Tyn,p) ≤( –αn)
δd(xn,p) +ϕ
d(p,Tp) +αn
δd(yn,p) +ϕ
d(p,Tp)
= ( –αn)δd(xn,p) +αnδd(yn,p). (.)
Similarly, we obtain
d(yn,p) =d
( –βn)xn⊕βnTxn,p
≤( –βn)d(xn,p) +βnd(Txn,p) ≤( –βn)d(xn,p) +βn
δd(xn,p) +ϕ
= ( –βn)d(xn,p) +βnδd(xn,p)
= –βn( –δ)
d(xn,p)
≤d(xn,p). (.)
Then, from (.) and (.), we get that
d(xn+,p)≤( –αn)δd(xn,p) +αnδd(yn,p) ≤( –αn)δd(xn,p) +αnδd(xn,p) ≤δd(xn,p)
.. .
≤δn+d(x,p).
Ifδ∈(, ), we obtain
lim
n→∞d(xn+,p) = .
Thus we havexn→p∈F(T). Ifδ= , the result is clear. This completes the proof. 5 Conclusions
The new multi-step iteration reduces to the two-step iteration and the SP-iteration pro-cesses. Also, the class of generalized nonexpansive mappings includes nonexpansive map-pings. Then these results presented in this paper extend and generalize some works for a
CAT() space in the literature.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally and significantly in writing this article. Both authors read and approved the final manuscript.
Acknowledgements
The authors are grateful to the referee for his/her careful reading and valuable comments and suggestions which led to the present form of the paper. This paper has been presented in International Conference ‘Anatolian Communications in Nonlinear Analysis (ANCNA 2013)’ in Abant Izzet Baysal University, Bolu, Turkey, July 03-06, 2013. This paper was supported by Sakarya University Scientific Research Foundation (Project number: 2013-02-00-003).
Received: 1 August 2013 Accepted: 19 September 2013 Published:07 Nov 2013
References
1. Agarwal, RP, O’Regan, D, Sahu, DR: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal.8(1), 61-79 (2007)
2. Ishikawa, S: Fixed points by a new iteration method. Proc. Am. Math. Soc.44, 147-150 (1974) 3. Mann, WR: Mean value methods in iterations. Proc. Am. Math. Soc.4, 506-510 (1953)
4. Phuengrattana, W, Suantai, S: On the rate of convergence of Mann, Ishikawa, Noor and SP iterations for continuous functions on an arbitrary interval. J. Comput. Appl. Math.235(9), 3006-3014 (2011)
5. Rhoades, BE: A comparison of various definitions of contractive mappings. Trans. Am. Math. Soc.226, 257-290 (1977) 6. Thianwan, S: Common fixed points of new iterations for two asymptotically nonexpansive nonself-mappings in a
Banach space. J. Comput. Appl. Math.224, 688-695 (2009)
7. Imoru, CO, Olantiwo, MO: On the stability of Picard and Mann iteration processes. Carpath. J. Math.19(2), 155-160 (2003)
9. Berinde, V: On the convergence of the Ishikawa iteration in the class of quasi contractive operators. Acta Math. Univ. Comen.73(1), 119-126 (2004)
10. Harder, AM, Hicks, TL: Stability results for fixed point iteration procedures. Math. Jpn.33(5), 693-706 (1988) 11. Zamfirescu, T: Fix point theorems in metric spaces. Arch. Math.23(1), 292-298 (1972)
12. Osilike, MO, Udomene, A: Short proofs of stability results for fixed point iteration procedures for a class of contractive-type mappings. Indian J. Pure Appl. Math.30(12), 1229-1234 (1999)
13. Gürsoy, F, Karakaya, V, Rhoades, BE: Data dependence results of new multi-step andS-iterative schemes for contractive-like operators. Fixed Point Theory Appl.2013, Article ID 76 (2013). doi:10.1186/1687-1812-2013-76 14. Kirk, WA: Geodesic geometry and fixed point theory. In: Seminar of Mathematical Analysis (Malaga/Seville,
2002/2003). Colecc. Abierta, vol. 64, pp. 195-225. Universidad de Sevilla Secr. Publ., Sevilla (2003)
15. Kirk, WA: Geodesic geometry and fixed point theory II. In: International Conference on Fixed Point Theory and Applications, pp. 113-142. Yokohama Publ., Yokohama (2004)
16. Dhompongsa, S, Kirk, WA, Sims, B: Fixed points of uniformly Lipschitzian mappings. Nonlinear Anal., Theory Methods Appl.65(4), 762-772 (2006)
17. Dhompongsa, S, Kirk, WA, Panyanak, B: Nonexpansive set-valued mappings in metric and Banach spaces. J. Nonlinear Convex Anal.8(1), 35-45 (2007)
18. Dhompongsa, S, Panyanak, B: On-convergence theorems in CAT(0) spaces. Comput. Math. Appl.56(10), 2572-2579 (2008)
19. ¸Sahin, A, Ba¸sarır, M: On the strong convergence of a modifiedS-iteration process for asymptotically quasi-nonexpansive mappings in a CAT(0) space. Fixed Point Theory Appl.2013, Article ID 12 (2013). doi:10.1186/1687-1812-2013-12
20. ¸Sahin, A, Ba¸sarır, M: On the strong and-convergence ofSP-iteration on a CAT(0) space. J. Inequal. Appl.2013, Article ID 311 (2013). doi:10.1186/1029-242X-2013-311
21. Bridson, M, Haefliger, A: Metric Spaces of Non-Positive Curvature. Grundlehren der Mathematischen Wissenschaften, vol. 319. Springer, Berlin (1999)
22. Bruhat, F, Tits, J: Groupes réductifs sur un corps local. Publ. Math. Inst. Hautes Études Sci.41, 5-251 (1972) 23. Lim, TC: Remarks on some fixed point theorems. Proc. Am. Math. Soc.60, 179-182 (1976)
24. Kirk, WA, Panyanak, B: A concept of convergence in geodesic spaces. Nonlinear Anal., Theory Methods Appl.68(12), 3689-3696 (2008)
25. Nanjaras, B, Panyanak, B: Demiclosed principle for asymptotically nonexpansive mappings in CAT(0) spaces. Fixed Point Theory Appl.2010, Article ID 268780 (2010). doi:10.1155/2010/268780
10.1186/1029-242X-2013-482
