R E S E A R C H
Open Access
On the convergence of fixed points for
Lipschitz type mappings in hyperbolic spaces
Shin Min Kang
1*, Samir Dashputre
2, Bhuwan Lal Malagar
2and Arif Rafiq
3*Correspondence:
1Department of Mathematics and
RINS, Gyeongsang National University, Jinju, 660-701, Korea Full list of author information is available at the end of the article
Abstract
In this paper, we prove strong and
-convergence theorems for a class of mappings which is essentially wider than that of asymptotically nonexpansive mappings on hyperbolic space through theS-iteration process introduced by Agarwalet al. (J. Nonlinear Convex Anal. 8:61-79, 2007) which is faster and independent of the Mann (Proc. Am. Math. Soc. 4:506-510, 1953) and Ishikawa (Proc. Am. Math. Soc. 44:147-150, 1974) iteration processes. Our results generalize, extend, and unify the corresponding results of Abbaset al.(Math. Comput. Model. 55:1418-1427, 2012), Agarwalet al. (J. Nonlinear Convex Anal. 8:61-79, 2007), Dhompongsa and Panyanak (Comput. Math. Appl. 56:2572-2579, 2008), and Khan and Abbas (Comput. Math. Appl. 61:109-116, 2011).
MSC: 47H10
Keywords: S-iteration process; uniformly convex hyperbolic space; nearly asymptotically nonexpansive mapping
1 Introduction
The class of asymptotically nonexpansive mappings, introduced by Goebel and Kirk [] in , is an important generalization of the class of nonexpansive mapping and they proved thatif C is a nonempty closed and bounded subset of a uniformly convex Banach space, then every asymptotically nonexpansive self-mapping of C has a fixed point.
There are numerous papers dealing with the approximation of fixed points of nonex-pansive and asymptotically nonexnonex-pansive mappings in uniformly convex Banach spaces through modified Mann and Ishikawa iteration processes (see,e.g., [–] and references therein). The class of Lipschitz mappings is larger than the classes of nonexpansive and asymptotically nonexpansive mappings. However, the theory of the computation of fixed points of non-Lipschitz mappings is equally important and interesting. There are few a results in this direction (see,e.g., [–]).
In , Lim [] introduced a concept of convergence in a general metric space set-ting which he called ‘-convergence’. In , Kirk and Panyanak [] specialized Lim’s concept toCAT() spaces and showed that many Banach space results involving weak con-vergence have precise analogs in this setting. Since then, the existence problem and the
-convergence problem of iterative sequences to a fixed point for nonexpansive mapping, asymptotically nonexpansive mapping, nearly asymptotically nonexpansive, asymptoti-cally nonexpansive mapping in intermediate sense, asymptotiasymptoti-cally nonexpansive nonself-mapping via Picard, Mann [], Ishikawa [], Agarwalet al. [] in the framework of
CAT() space have been rapidly developed and many papers have appeared in this direc-tion (see,e.g., [–]).
The purpose of the paper is to establish-convergence as well as strong convergence through theS-iteration process for a class of mappings which is essentially wider than that of asymptotically nonexpansive mappings on a nonlinear domain, uniformly convex hyperbolic space which includes both uniformly convex Banach spaces andCAT() spaces. Therefore, our results extend and improve the corresponding ones proved by Abbaset al. [], Dhompongsa and Panyanak [], Khan and Abbas [] and many other results in this direction.
2 Preliminaries
LetF(T) ={Tx=x:x∈C} denotes the set of fixed point. We begin with the following definitions.
Definition . LetCbe a nonempty subset of metric spaceXandT:C→Ca mapping. A sequence{xn}inCis said to be anapproximating fixed point sequenceofTif
lim
n→∞d(xn,Txn) = .
Definition . LetCbe a nonempty subset of a metric spaceX. The mappingT:C→C is said to be
() uniformlyL-Lipschitzianif for eachn∈N, there exists a positive numberL> such that
dTnx,Tny≤Ld(x,y) for allx,y∈C;
() asymptotically nonexpansiveif there exists a sequence{kn}in[,∞)with limn→∞kn= such that
dTnx,Tny≤( +kn)d(x,y) for allx,y∈Candn∈N;
() asymptotically quasi-nonexpansiveifF(T)=∅and there exists a sequence{kn}in [,∞)withlimn→∞kn= such that
dTnxn,p≤( +kn)d(xn,p) for allx∈C,p∈F(T)andn∈N.
The class of nearly Lipschitzian mappings is an important generalization of the class of Lipschitzian mappings and was introduced by Sahu [].
LetCbe a nonempty subset of a metric spaceXand fix a sequence{an}in [,∞) with an→. A mappingT:C→Cis said to benearly Lipschitzianwith respect to{an}if for eachn∈N, there exists a constantkn≥ such that
dTnx,Tny≤kn
d(x,y) +an
for allx,y∈C. (.)
The infimum of the constantsknfor which (.) holds is denoted byη(Tn) and is called thenearly Lipschitz constantofTn.
() nearly nonexpansiveifη(Tn) = for alln∈N;
() nearly asymptotically nonexpansiveifη(Tn)≥for alln∈Nandlim
n→∞η(Tn) = ;
() nearly uniformlyk-Lipschitzianifη(Tn)≤kfor alln∈N.
Definition . LetCbe a nonempty subset of a metric spaceXand fix a sequence{an} in [,∞) withan→. A mappingT :C→Cis said to benearly asymptotically quasi-nonexpansivewith respect to{an}ifF(T)= ∅and there exists a sequence{un}in [,∞) withlimn→∞un= such that
dTnx,p≤( +un)d(x,p) +an
for allx∈C,p∈F(T) andn∈N.
In fact, ifTis a nearly asymptotically nonexpansive mapping andF(T) is nonempty, then Tis a nearly asymptotically quasi-nonexpansive mapping. The following is an example of a nearly asymptotically quasi-nonexpansive mapping withF(T)=φ.
Example .[] LetX=R,C= (–∞, ] andT:C→Cbe a mapping defined by
Tx= ⎧ ⎨ ⎩
x ifx∈(–∞, ],
x– ifx∈(, ].
Here,F(T) ={}and also,Tis nearly asymptotically quasi-nonexpansive mapping with {un}={,,,, . . .}and{an}={,,,, . . .}.
A nearly asymptotically quasi-nonexpansive mapping is called a nearly quasi-nonexpan-sive (asymptotically quasi-nonexpanquasi-nonexpan-sive mapping) if un = for all n∈N(an= for all n∈N). Notice that every nearly asymptotically quasi-nonexpansive mapping with bounded domain is nearly quasi-nonexpansive. Indeed, ifCis a bounded subset of a metric space andT:C→Ca nearly asymptotically quasi-nonexpansive mapping with sequence {(an,un)}, then
dTnx,p≤( +un)d(x,p) +an≤d(x,p) +
unsup
x,y∈Cd(x,y) +an
for allx∈C,p∈F(T) andn∈N.
The following example shows thatT is a nearly quasi-nonexpansive mapping but not Lipschitzian and quasi-nonexpansive.
Example .[] LetX=R,C= [–π,π] andk∈(, ). LetT:C→Cbe a mapping de-fined by
Tx= ⎧ ⎨ ⎩
ifx= ,
kxsinx ifx= .
define
fn(x) = Tnx –|x| for allx∈C.
Fix a sequence{an}inRdefined by
an=max
sup
x∈Cfn(x),
for alln∈N.
It is clear thatan≥ for alln∈Nandan→, sinceTnx→ uniformly. By the definition of{an}, we have
Tnx ≤ |x|+an for allx∈Candn∈N.
Clearly, T is a nearly quasi-nonexpansive mapping with respect to {an} and it is not Lipschitz and not quasi-nonexpansive.
Lemma .[, Lemma .] Let C be a nonempty subset of a metric space (X,d)and T:C→C a quasi-L-Lipschitzian,i.e.,F(T)=∅and there exists a constant L> such that
d(Tx,Ty)≤Ld(x,y) for all x∈C and y∈F(T).
If{xn}is a sequence in C such thatlimn→∞d(xn,F(T)) = andlimn→∞xn=x∈C,where d(x,F(T)) =inf{d(x,p) :p∈F(T)},then x is a fixed point of T.
Throughout this paper we consider the following definition of a hyperbolic space in-troduced by Kohlenbach []. It is worth noting that they are different from the Gromov hyperbolic space [] or from other notions of hyperbolic space that can be found in the literature (see,e.g., [–]).
Definition . A metric space (X,d) is a hyperbolic space if there exists a mapW:X×
[, ]→Xsatisfying
(i) d(u,W(x,y,α))≤αd(u,x) + ( –α)d(u,y), (ii) d(W(x,y,α),W(x,y,β)) =|α–β|d(x,y), (iii) W(x,y,α) =W(y,x, ( –α)),
(iv) d(W(x,z,α),W(y,w,α))≤αd(x,y) + ( –α)d(z,w) for allx,y,z,w∈Xandα,β∈[, ].
An important example of a hyperbolic space is aCAT() space. It is nonlinear in nature and its brief introduction is as follows.
A metric space (X,d) is alength spaceif any two points ofXare joined by a rectifiable path (that is, a path of finite length) and the distance between any two points ofXis taken to be the infimum of the lengths of all rectifiable paths joining them. In this case, dis known as alength metric(otherwise aninner metricorintrinsic metric). In the case that no rectifiable path joins two points of the space, the distance between them is taken to be∞.
isometry andd(x,y) =l. The imageαofcis called a geodesic (or metric)segmentjoining xandy. The space (X,d) is said to be ageodesic spaceif any two points ofXare joined by a geodesic path andXis said to beuniquely geodesicif there is exactly one geodesic path denoted byαx⊕( –α)yjoiningxandyfor eachx,y∈X. The set{αx⊕( –α)y:α∈[, ]} will be denoted by [x,y], called the segment joiningxtoy. A subsetCof a geodesic space Xis convex if for anyx,y∈C, we have [x,y]⊂C.
A geodesic triangle(x,x,x) in a geodesic metric space (X,d) is defined to be a
col-lection of three points inX(theverticesof) and three geodesic segments between each pair of vertices (theedgesof). Acomparison trianglefor geodesic triangle(x,x,x)
in (X,d) is a triangle(x,x,x) :=(x¯,x¯,x¯) inR such thatdR(xi,¯ xj) =¯ d(xi,xj) for i,j∈ {, , }and such a triangle always exists (see []).
A geodesic metric space is aCAT() space if all geodesic trianglesinXwith a com-parison triangle⊂Rsatisfy theCAT() inequality
d(x,y)≤dR(x,¯ y)¯
for all x,y∈and for all comparison pointsx,¯ ¯y∈. LetX be aCAT() space. Define W:X×[, ]→XbyW(x,y,α) =αx⊕( –α)y. ThenW satisfies the four properties of
a hyperbolic space. Also ifXis a Banach space andW(x,y,α) =αx+ ( –α)y, thenXis a hyperbolic space. Therefore, our hyperbolic space represents a unified approach for both linear and nonlinear structures simultaneously.
To elaborate that there are hyperbolic spaces which are not imbedded in any Banach space, we give the following example.
Example . LetBbe the open unit ball in complex Hilbert space with respect to the Poincaré metric (also called ‘Poincaré distance’)
dB(x,y) =arg tanh
–x–xy¯y =arg tanh –σ(x,y),
where
σ(x,y) =( –|x|
)( –|y|)
| –xy¯| for allx,y∈B.
ThenBis a hyperbolic space which is not imbedded in any Banach space.
A metric space (X,d) is called a convex metric space introduced by Takahashi [] if it satisfies only (i). A subsetC of a hyperbolic spaceXisconvexifW(x,y,α)∈C for all x,y∈Candα∈[, ].
A hyperbolic space (X,d,W) isuniformly convex[] if for anyu,x,y∈X,r> and ∈(, ], there exists aδ∈(, ] such thatd(W(x,y,),u)≤( –δ)rwheneverd(x,u)≤r, d(y,u)≤randd(x,y)≥ r.
A mappingη: (,∞)×(, ]→(, ] which provides such aδ=η(r, ) for givenr> and ∈(, ], is known as modulus of uniform convexity. We callηmonotone if it decreases withr(for a fixed ).
of hyperbolic spaces also contains Hadamard manifolds, Hilbert balls equipped with the hyperbolic metric [],R-trees and Cartesian products of Hilbert balls as special cases.
LetCbe a nonempty subset of hyperbolic spaceX. Let{xn}be a bounded sequence in a hyperbolic spaceX. Forx∈X, define a continuous functionalra(·,{xn}) :X→[,∞) by
rax,{xn}=lim sup
n→∞ d(xn,x).
The asymptotic radiusr({xn}) of{xn}is given by
r{xn}
=infra
x,{xn}
:x∈X.
The asymptotic centerAC({xn}) of a bounded sequence of{xn}with respect to a subset of CofXis the set
AC{xn}=x∈X:rax,{xn}≤ray,{xn}for anyy∈C.
This is the set of minimizers of the functionalra(·,{xn}). If the asymptotic center is taken with respect toX, then it is simply denoted byA({xn}).
It is well known that uniformly convex Banach spaces and evenCAT() spaces enjoy the property that bounded sequences have unique asymptotic centers with respect to closed convex subsets. The following lemma is due to Leustean [] and ensures that this property also holds in a complete uniformly convex hyperbolic space.
Lemma . [] Let (X,d,W) be a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity.Then every bounded sequence {xn}in X has a unique asymptotic center with respect to any nonempty closed convex subset C of X.
Recall that a sequence{xn}inXis said to-converge tox∈X, ifxis the unique asymp-totic center of{un}for every subsequence{un}of{xn}. In this case, we write-limnxn=x and callxthe-limitof{xn}.
Lemma .[] Let C be a nonempty closed convex subset of a uniformly convex hyper-bolic space and{xn}a bounded sequence in C such that AC({xn}) ={y}and r({xn}) =ρ.If {ym}is another sequence in C such thatlimm→∞ra(ym,{xn}) =ρ,thenlimm→∞ym=y.
Lemma .[] Let(X,d,W)be a uniformly convex hyperbolic space with monotone modulus of uniform convexityη.Let x∈X and{tn}be a sequence in[a,b]for some a,b∈ (, ).If{xn}and{yn}are sequences in X such that
lim sup
n→∞ d(xn,x)≤c, lim supn→∞ d(yn,x)≤c, n→∞lim d
W(xn,yn,tn),x=c
for some c≥,thenlimn→∞d(xn,yn) = .
Lemma .[] Let{δn},{βn},and{γn}be three sequences of nonnegative numbers such that
δn+≤βnδn+γn for all n∈N.
Ifβn≥for all n∈N,
∞
3 Strong and
-convergence theorems in hyperbolic space
In this section, we approximate fixed point for nearly asymptotically nonexpansive map-pings in a hyperbolic space. More briefly, we established-convergence and strong con-vergence theorems for iteration scheme (.).
First, we define theS-iteration process in hyperbolic space as follows.
LetCbe a nonempty closed convex subset of a hyperbolic spaceXandT:C→Cbe a nearly asymptotically nonexpansive mapping. Then, for arbitrarily chosenx∈C, we
construct the sequence{xn}inCsuch that ⎧
⎨ ⎩
xn+=W(Tnxn,Tnyn,αn),
yn=W(xn,Tnxn,βn), n∈N,
(.)
where{αn}and{βn}are sequences in (, ) is called anS-iteration process.
Lemma . Let C be a nonempty convex subset of a hyperbolic space X and T:C→C a nearly asymptotically quasi-nonexpansive mapping with sequence{(an,un)} such that
∞
n=an<∞and
∞
n=un<∞.Let{xn}be a sequence in C defined by(.),where{αn} and{βn}are sequences in(, ).Thenlimn→∞d(xn,p)exists for each p∈F(T).
Proof First, we show thatlimn→∞d(xn,p) exists for eachp∈F(T), we have
d(xn+,p) =d
WTnxn,Tnyn,αn
,p
≤( –αn)d
Tnxn,p
+αnd
Tnyn,p
≤( –αn)( +un)d(xn,p) +an
+αn
( +un)d(yn,p) +an
≤( +un)( –αn)d(xn,p) +αnd(yn,p)+an (.)
and
d(yn,p) =dWxn,Tnxn,βn
,p
≤( –βn)d(xn,p) +βndTnxn,p ≤( –βn)d(xn,p) +βn
( +un)d(xn,p) +an
≤( +βnun)d(xn,p) +anβn, (.)
from (.) and (.), we have
d(xn+,p)
≤( +un)( –αn)d(xn,p) +αn
( +βnun)d(xn,p) +anβn
+an
≤ +un +αnβn+αnβnun
d(xn,p)
+an + ( +un)αnβn
, n∈N.
It follows that
for someM,M≥.{un}is bounded. By Lemma ., we find thatlimn→∞d(xn,p)
ex-ists.
Lemma . Let C be a nonempty and closed convex subset of a uniformly convex hyperbolic space X with monotone modulus of uniform convexityηand let T:C→C be a nearly asymptotically quasi-nonexpansive mapping with sequences{(an,un)}such that∞n=an< ∞and∞n=un<∞.Let F(T)=∅,then for the sequence{xn}in C defined by(.),we have
limn→∞d(xn,Tnxn) = .
Proof From Lemma ., we find thatlimn→∞d(xn,p) exists for eachp∈F(T). We suppose thatlimn→∞d(xn,p) =c≥. Since
dTnxn,p≤( +un)d(xn,p) +an for alln∈N,
we have
lim sup
n→∞ d
Tnxn,p≤c.
Also
d(yn,p)≤dWxn,Tnxn,βn
,p
≤( –βn)d(xn,p) +βndTnxn,p ≤( +βnun)d(xn,p) +anβn,
which yields
lim sup
n→∞ d(yn,p)≤c. (.)
Hence
lim sup
n→∞ d
Tnyn,p
≤lim sup
n→∞
( +un)d(yn,p) +an
≤c. (.)
Since
c= lim
n→∞d(xn+,p) =n→∞limd
WTnxn,Tnyn,αn
,p,
it follow from Lemma . that
lim
n→∞d
Tnxn,Tnyn= . (.)
From (.) and (.), we have
dxn+,Tnxn
=dWTnxn,Tnyn,αn
,Tnxn
≤αndTnxn,Tnyn
Hence, from (.) and (.), we have
dxn+,Tnyn
≤dxn+,Tnxn
+dTnxn,Tnyn→ asn→ ∞. (.)
Now using (.), we have
d(xn+,p)≤d
xn+,Tnyn
+dTnyn,p
≤dxn+,Tnyn
+( +un)d(yn,p) +an
, (.)
which gives from (.)
c≤lim inf
n→∞ d(yn,p). (.)
From (.) and (.), we obtain
c= lim
n→∞d(yn,p) =d
Wxn,Tnxn,βn
,p. (.)
Apply Lemma . in (.), and we obtain
lim
n→∞d
xn,Tnxn= .
Theorem . Let C be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexityηand let T:C→C be a uniformly continuous nearly asymptotically nonexpansive mapping with F(T)=∅and sequence{(an,η(Tn))}such that∞
n=(η(Tn) – ) <∞and
∞
n=an<∞.For arbitrary x∈
C,let{xn}be a sequence in C defined by(.),where{αn}and{βn}are sequences in(, ). Then{xn}is-convergent to an element of F(T).
Proof By Lemma .,limn→∞d(xn,Tnxn) = . By uniform continuity ofT,d(xn,Tnxn)→ implies thatd(Txn,Tn+xn)→, observe that
d(xn+,xn)≤d
WTnxn,Tnyn,αn
,xn
≤( –αn)dxn,Tnxn+αndTnyn,xn ≤dxn,Tnxn+αndTnyn,Tnxn ≤dxn,Tnxn+ηTnd(xn,yn) +an ≤dxn,Tnxn+ηTndxn,Tnxn+an → asn→ ∞.
Also
d(xn,Txn)≤d(xn,xn+) +d
xn+,Tn+xn+
+dTn+xn+,Tn+xn
+dTn+xn,Txn
≤ +ηTnd(xn,xn+)
+dxn+,Tnxn+
and hence
lim
n→∞d(xn,Txn) = . (.)
Next, we have to show that{xn}is-convergent to an element ofF(T).
Since {xn} is bounded (by Lemma .) therefore, Lemma . asserts that{xn} has a unique asymptotic center. That is, A({xn}) ={x} (say). LetA({yn}) ={v}. Then by (.),
limn→∞d(yn,Tyn) = .T is a nearly asymptotically nonexpansive mapping with sequence {(an,η(Tn))}. By uniform continuity ofT
lim
n→∞d
Tiyn,Ti+yn= fori= , , , . . . . (.)
Now we claim thatvis a fixed point ofT. For this, we define a sequence{zn}inCby zm=Tmv,m∈N. For integersm,n∈N, we have
d(zm,yn)≤dTmv,Tmyn+dTmyn,Tm–yn+· · ·+d(Tyn,yn)
≤ηTmd(v,yn) +am+ m–
i=
dTiyn,Ti+yn. (.)
Then, by (.) and (.), we have
razm,{yn}=lim sup
m→∞ d(zm,yn)
≤ηTmrav,{yn}+am.
Hence
lim sup
m→∞ ra
zm,{yn}
≤ra
v,{yn}
. (.)
SinceAC({yn}) ={v}, by definition of asymptotic centerAC({yn}) of a bounded sequence {yn}with respect toC⊂X, we have
ra
v,{yn}
≤ra
y,{yn}
, ∀y∈C.
This implies that
lim inf
m→∞ra
zm,{yn}≥rav,{yn}, (.)
therefore, from (.) and (.), we have
lim
m→∞ra
zm,{yn}=rav,{yn}.
It follows from Lemma . thatTmv→v. By uniform continuity ofT, we have
Tv=T
lim
m→∞T
mv=Tm+v=v,
Next, we claim thatvis the unique asymptotic center for each subsequence{yn}of{xn}. Assume contrarily, that is,x=v. Sincelimn→∞d(xn,v) exists by Lemma ., therefore, by the uniqueness of asymptotic centers, we have
lim sup
n→∞ d(yn,v) <lim supn→∞ d(yn,x)
≤lim sup
n→∞ d(xn,x)
<lim sup
n→∞ d(xn,v)
=lim supd(yn,v),
a contradiction and hencex=v. Since{yn}is an arbitrary subsequence of{xn}, therefore, AC({yn}) ={v}for all subsequence of{yn}of{xn}. This proves that{xn}-converges to a
fixed point ofT.
We now discuss the strong convergence for theS-iteration process defined by (.) for Lipschitzian type mappings in a uniformly convex hyperbolic space setting.
Theorem . Let C be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexityηand let T:C→C be a nearly asymptotically quasi-nonexpansive mapping with sequence{(an,un)} such that
∞
n=an<∞and
∞
n=un<∞.Assume that F(T)is a closed set.Let{xn}be a sequence in
C defined by(.),where{αn}and{βn}are sequences in(, ).Then{xn}converges strongly to a fixed point of T if and only iflim infn→∞d(xn,F(T)) = .
Proof Necessity is obvious.
Conversely, suppose thatlim infn→∞d(xn,F(T)) = . From (.), we have
dxn+,F(T)
≤( +Mun)d
xn,F(T)
+Man, n∈N,
so limn→∞d(xn,F(T)) exists. It follows thatlimn→∞d(xn,F(T)) = . Next, we show that {xn} is a Cauchy sequence. The following arguments are similar to those given in [, Lemma ] and [, Theorem .], and we obtain the following inequality:
d(xn+m,p)≤L
d(xn,p) + ∞
j=n bj
for every p∈F(T) and for all m,n≥, where L=eM( n+m–
j=n uj)> and bj=M
aj. As,
∞
n=un<∞ soL∗ =eM( ∞
n=un)≥L=eM( n+m–
j=n uj) > . Let > be arbitrarily chosen.
Sincelimn→∞d(xn,F) = and∞n=an<∞, there exists a positive integernsuch that
d(xn,F) <
L∗ and ∞
j=n bj<
L∗, ∀n≥n.
In particular,inf{d(xn,p) :p∈F}<L∗. Thus there must existp∗∈Fsuch that
Hence forn≥n, we have
d(xn+m,xn)≤d
xn+m,p∗
+dp∗,xn
≤L∗
dxn,p∗+ ∞
j=n bj
< L∗
L∗ +L∗
= .
Hence {xn} is a Cauchy sequence in closed subset C of a complete hyperbolic space and so it must converge strongly to a point qin C. Now, limn→∞d(xn,F(T)) = gives
d(q,F(T)) = . SinceF(T) is closed, we haveq∈F(T).
In the next result, the closedness assumption onF(T) is not required.
Theorem . Let C be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity ηand T :C→C an asymptotically quasi-nonexpansive mapping with sequence{un}such that∞n=un<∞. Let{xn}be a sequence in C defined by(.),where{αn}and{βn}are sequences in(, ). Then{xn}converges strongly to a fixed point of T iflim infn→∞d(xn,F(T)) = .
Proof Following an argument similar to those of Theorem ., we see that{xn}is a Cauchy sequence inC. Letlimn→∞xn=x. Since an asymptotically quasi-nonexpansive mapping is quasi-L-Lipschitzian, it follows from Lemma . thatxis a fixed point ofT.
Theorem . Let C be a nonempty closed convex subset of a complete uniformly con-vex hyperbolic space X with monotone modulus of uniform concon-vexityηand T:C→C a uniformly continuous nearly asymptotically nonexpansive mapping with F(T)=∅and sequence{an,η(Tn)}such that∞
n=(η(Tn) – ) <∞and an<∞.For arbitrary x∈C,let
{xn}be a sequence in C defined by(.),where{αn}and{βn}are sequences in(, ).If T is uniformly continuous and Tmis demicompact for some m∈N,it follows that{x
n}converges strongly to a fixed point of T.
Proof By (.), we havelimn→∞d(xn,Txn) = . By the uniformly continuous ofT, we have
d(xn,Txn)→ ⇒ dTx,Txn→ ⇒ · · · ⇒ dTixn,Ti+xn→
for alli∈N. It follows that
dxn,Tmxn≤ m–
i=
dTixn,Ti+xn→ asn→ ∞.
Sinced(xn,Tmxn)→, andTm is demicompact, there exists a subsequence{xnj}of{xn} such thatlimj→∞Tmxn
j=x∈C. Note that
d(xnj,x)≤d
xnj,T mxn
j
+dTmxnj,x
Sincelimn→∞d(xn,Txn) = , we getx∈F(T). Sincelimn→∞d(xn,x) exists by Lemma ., andlimj→∞d(xnj,x) = , we conclude thatxn→x.
Recall that a mappingTfrom a subset of a metric space (X,d) into itself withF(T)=∅is said tosatisfy condition(A) (see []) if there exists a nondecreasing functionf: [,∞)→ [,∞) withf() = ,f(t) > fort∈(,∞) such that
d(x,Tx)≥fdx,F(T) for allx∈C.
Theorem . Let C be a nonempty closed convex subset of a complete uniformly con-vex hyperbolic space X with monotone modulus of uniform concon-vexityηand T:C→C a uniformly continuous nearly asymptotically nonexpansive mapping with F(T)=∅and sequence{(an,η(Tn))}such that∞
n=η(Tn– ) <∞and
∞
n=an<∞.For arbitrary x∈C,
let{xn}be a sequence in C defined by(.),where{αn}and{βn}are sequences in(, ). Sup-pose that T satisfies the condition(A).Then{xn}converges strongly to a fixed point of T.
Proof By (.), we havelimn→∞d(xn,Txn) = Further, by condition (A),
lim
n→∞d(xn,Txn)≥n→∞lim f
dxn,F(T).
It follows thatlimn→∞d(xn,F(T)) = . Therefore, the result follows from Theorem ..
4 Conclusion
. We prove strong and-convergence of theS-iteration process, which is faster than the iteration processes used by Abbaset al.[], Dhompongsa and Panyanak [], and Khan and Abbas [].
. Theorem . extends Agarwalet al.[, Theorem .] from a uniformly convex Banach space to a uniformly convex hyperbolic space.
. Theorem . extends Dhompongsa and Panyanak [, Theorem .] from the class of nonexpansive mappings to the class of mappings which are not necessarily
Lipschitzian.
. Theorem ., extends corresponding results of Beg [], Chang [], Khan and Takahashi [] and Osilike and Aniagbosor [] for a more general class of non-Lipschitzian mappings in the framework of a uniformly convex hyperbolic space. It also extends the corresponding results of Dhomponsga and Panyanak [] from the class of nonexpansive mappings to a more general class of non-Lipschitzian mappings in the same space setting.
. Theorem . extends Sahu and Beg [, Theorem .] from a Banach to a uniformly convex hyperbolic space.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Author details
1Department of Mathematics and RINS, Gyeongsang National University, Jinju, 660-701, Korea.2Department of Applied
Mathematics, Shri Shankaracharya Group of Institutions, Junwani, Bhilai, 490020, India.3Department of Mathematics,
Acknowledgements
The authors would like to thank the editor and all referees for their valuable comments and suggestions for improving the paper.
Received: 17 April 2014 Accepted: 31 October 2014 Published:12 Nov 2014
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Cite this article as:Kang et al.:On the convergence of fixed points for Lipschitz type mappings in hyperbolic spaces.
