Fixed point approximation of asymptotically nonexpansive mappings in hyperbolic spaces

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R E S E A R C H

Open Access

Fixed point approximation of asymptotically

nonexpansive mappings in hyperbolic spaces

Haﬁz Fukhar-ud-din

1,2

_{and Amna Kalsoom}

2*

Dedicated to Professor Wataru Takahashi on his 70th birthday.

*_{Correspondence:}

[email protected]

2_{Department of Mathematics, The}

Islamia University of Bahawalpur, Bahawalpur, 63100, Pakistan Full list of author information is available at the end of the article

Abstract

Convergence theorems are established in a hyperbolic space for the modiﬁed Noor iterations with errors of asymptotically nonexpansive mappings. The obtained results extend and improve the several known results in Banach spaces and CAT(0) spaces simultaneously.

Keywords: asymptotically nonexpansive mapping; modiﬁed Noor iterative process; (UC) hyperbolic space; convergence

1 Introduction

Nonexpansive mappings are Lipschitzian with Lipschitz constant equal to . The class of nonexpansive mappings enjoys the fixed point property and even the approximate fixed point property in the general setting of metric spaces. The importance of this class lies in its powerful applications in initial value problems of the differential equations, game-theoretic model, image recovery and minimax problems. The class of asymptotically non-expansive mappings was introduced by Goebel and Kirk [] as an important generalization of the class of nonexpansive mappings. Therefore, it is natural to extend powerful results for nonexpansive mappings to the class of asymptotically nonexpansive mappings. Itera-tive construction of fixed points of various nonlinear mappings emerged as the most pow-erful tool for solving such nonlinear problems. Approximation of fixed points of asymp-totically nonexpansive mappings has been studied extensively by many authors; see for example [–] and the references cited therein.

In , Glowinski and Le Tallec [] used a three-step iterative process to ﬁnd approxi-mate solutions of elastoviscoplasticity problem, liquid crystal theory and eigenvalue putation. They observed that the three-step iterative process gives better numerical com-putations than two-step and one-step iterative processes. In , Haubrugeet al.[] studied convergence analysis of a three-step iterative process of Glowinski and Le Tallec [] and applied this process to obtain new splitting type iterations for solving variational inequalities, separable convex programming and minimization of a sum of convex func-tions. They also proved that the three-step iterative process leads to highly parallel iter-ations under certain conditions. Thus we conclude that the three-step iterative process plays an important and signiﬁcant role in solving various numerical problems which arise in pure and applied sciences.

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In , Noor [] introduced a three-step iterative process and studied the approximate solutions of variational inclusion in Hilbert spaces. In , Xu and Noor [] presented a three-step iterative process to approximate fixed points of asymptotically nonexpansive mappings in a Banach space. Choet al.[] extended Xu and Noor’s iterative process to a three-step iterative process with errors in Banach spaces and used it to approximate fixed points of asymptotically nonexpansive mappings. In , Suantai [] proposed and analyzed the modified three-step Noor iterative process. This process was further stud-ied for different kinds of mappings by Khan and Hussain [] and Khan [] for example. Nammaneeet al.[] extended this process to the one with errors as follows.

LetCbe a nonempty convex subset of a Banach spaceXand letT:C→Cbe a given mapping. For givenx∈C, compute the sequences{xn},{yn}and{zn}by

zn=anTnxn+ ( –an–γn)xn+γnun,

yn=bnTnzn+cnTnxn+ ( –bn–cn–μn)xn+μnvn, ()

xn+=αnTnyn+βnTnzn+ ( –αn–βn–λn)xn+λnwn, n≥,

where{an},{bn},{cn},{αn},{βn},{γn},{μn}and{λn}are sequences in [, ];{un},{vn}and

{wn}are bounded sequences inC.

By diﬀerent choices of parametersan,bn,cn,αn,βn,γn,μn,λnto be zero, one can see that

one-step iterations of Mann [], two-step iterations of Ishikawa [], three-step iterations of Xu and Noor [], three-step iterations with errors of Choet al.[] and modiﬁed three-step iterations of Suantai [] all are the special cases of iteration process ().

Most of phenomena in nature are nonlinear. Therefore, mathematicians and scientists are always in pursuit of ﬁnding methods to solve nonlinear real world problems. So trans-lating a linear version of known problems into its equivalent nonlinear version has a great importance.

Keeping in mind the occurrence of such phenomena, we translate modiﬁed three-step Noor iterations with errors in a nonlinear domain, namely, hyperbolic spaces and study their convergence analysis in a new setup.

A metric space (X,d) is hyperbolic [] if there is a mappingW:X×I→Xsuch that (a) du,W(x,y,α)≤αd(u,x) + ( –α)d(u,y),

(b) dW(x,y,α),W(x,y,β)=|α–β|d(x,y), (c) W(x,y,α) =W(y,x,  –α),

(d) dW(x,z,α),W(y,w,α)≤αd(x,y) + ( –α)d(z,w)

()

for allu,w,x,y,z∈Xandα,β∈I= [, ] (see also []); the space is convex [] if only (a) is satisﬁed. A subsetCof the hyperbolic spaceXis convex ifW(x,y,α)∈Cfor allx,y∈C andα∈I. Normed spaces and their subsets are linear hyperbolic spaces while Hadamard manifolds [], the Hilbert open unit ball equipped with the hyperbolic metric [] and theCAT() spaces qualify for the criteria of nonlinear hyperbolic spaces [–].

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A mappingη: (,∞)×(, ]→(, ] such thatη(r,ε) =δfor a givenr>  andε∈(, ] (as in the deﬁnition of UC) is known as a modulus of uniform convexity. We callη mono-tone if it decreases with respect tor(for a ﬁxedε).

LetCbe a nonempty subset of a metric spaceX. A mappingT:C→Cis asymptotically nonexpansive if there exists a sequence{kn≥}withlimn→∞kn=  such that

dTnx,Tny≤knd(x,y) forx,y∈C,n≥;

it becomes nonexpansive ifkn=  for alln≥. It was shown in [] that an asymptotically

nonexpansive mapping on a nonempty, bounded, closed and convex subset of a (UC) hy-perbolic space has a ﬁxed point.

We translate () in a hyperbolic space as follows.

LetCbe a nonempty convex subset of a hyperbolic spaceXandT:C→Cbe an asymp-totically nonexpansive mapping. Then, for arbitrarily chosenx∈C, we construct the se-quences{xn},{yn}and{zn}inCas

zn=W

Tnxn,W(xn,un,θn),an

,

yn=W

Tnzn,W

Tnxn,W(xn,vn,θn),

cn

 –bn

,bn

, ()

xn+=W

Tnyn,W

Tnzn,W(xn,wn,θn), βn

 –αn

,αn

,

where{an},{bn},{cn},{αn},{βn},{γn},{μn},{λn}are sequences in [, ] and{un},{vn}and

{wn}are bounded sequences inCandθn=  – γn

–an,θn=  –

μn

–bn–cn andθn=  –

λn

–αn–βn.

Using Proposition .(a) []:W(x,y, ) =yforx,y∈X, the iteration process in () re-duces to:

(i) modiﬁed Noor iterations (withγn=μn=λn= ):

zn=W

Tnxn,xn,an

,

yn=W

Tnzn,W

Tnxn,xn,

cn

 –bn

,bn

, ()

xn+=W

Tnyn,W

Tnzn,xn, βn

 –αn

,αn

;

(ii) Noor iterations with errors (withcn=  =βn):

zn=W

Tnxn,W

xn,un,  – γn

 –an

,an

,

yn=W

Tnzn,W

xn,vn,  – μn

 –bn

,bn

, ()

xn+=W

Tnyn,W

xn,wn,  – λn

 –αn

,αn

;

(iii) Noor iterations (withcn=βn=γn=μn=λn≡):

zn=W

Tnxn,xn,an

,

yn=W

Tnzn,xn,bn

,

xn+=W

Tnyn,xn,αn

;

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(iv) Ishikawa iterations (withan=cn=βn=γn=μn=λn= ):

yn=W

Tn_x

n,xn,bn

,

xn+=W

Tnyn,xn,αn

;

()

(v) Mann iterations (withan=bn=cn=βn=γn=μn=λn= ):

xn+=W

Tnxn,xn,αn

. ()

The purpose of this paper is to establish convergence results of iteration process () for asymptotically nonexpansive mappings on a nonlinear domain ((UC) hyperbolic spaces) which includes both (UC) Banach spaces andCAT() spaces. Therefore, our results extend and improve the corresponding ones proved by Suantai [], Xu and Noor [] and others in a (UC) Banach space and are also valid inCAT() spaces, simultaneously.

In the sequel, we need the following lemmas.

Lemma .([]) Let{an},{δn}and{θn}be sequences of non-negative real numbers such

that∞_n_=θn<∞and

∞

n=δn<∞.If an+≤( +δn)an+θn,n≥,thenlimn→∞anexists. Lemma .([]) Let X be a(UC)hyperbolic space with monotone modulus of uniform convexityη. Let x∈X and{αn} be a sequence in[b,c]for some b,c∈(, ).If{xn}and

{yn} are sequences in X such thatlim supn→∞d(xn,x)≤r, lim supn→∞d(yn,x)≤r and

limn→∞d(W(xn,yn,αn),x) =r for some r≥,thenlimn→∞d(xn,yn) = .

2 Main results

The following lemma is crucial for proving the convergence results.

Lemma . Let X be a(UC)hyperbolic space with monotone modulus of uniform con-vexityη,and let C be a nonempty,bounded,closed and convex subset of X.Let T be an asymptotically nonexpansive self-mapping on C with a sequence{kn} ⊂[,∞)such that

_∞

n=(kn– ) <∞. For a given x∈C, compute{xn}, {yn} and{zn} as in ()satisfying

 <a≤αn,βn,an,bn≤b< ,

∞

n=γn<∞,

∞

n=μn<∞and

∞

n=λn<∞.

Then we have the following conclusions:

(i) Ifqis a ﬁxed point ofT,thenlimn→∞d(xn,q)exists.

(ii) If <lim inf_n_→∞αn≤lim supn→∞(αn+βn+λn) < ,then

lim

n→∞d

Tnyn,W

Tnzn,W(xn,wn,θn), βn

 –αn

= .

(iii) If <lim infn→∞αn≤lim supn→∞(αn+βn+λn) < and

 <lim infn→∞bn≤lim supn→∞(bn+cn+μn) < ,then

lim

n→∞d

Tnzn,W

Tnxn,W(xn,vn,θn),

cn

 –bn

= .

(iv) If <lim inf_n_→∞bn≤lim supn→∞(bn+cn+μn) < and

 <lim infn→∞an≤lim supn→∞(an+γn) < ,then

lim

n→∞d

Tnxn,W(xn,un,θn)

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Moreover, if  <lim infn→∞αn ≤ lim supn→∞(αn +βn +λn) < ,  <lim infn→∞bn ≤

lim sup_n_→∞(bn+cn+μn) < and <lim infn→∞an≤lim supn→∞(an+γn) < ,then

lim

n→∞d

Tnxn,xn

= lim

n→∞d

Tnzn,xn

= lim

n→∞d

Tnyn,xn

= .

Proof (i) Applying ()(a) withu=q∈F(T) to the sequence{zn}in (), we obtain

d(zn,q) =d

WTnxn,W(xn,un,θn),an

,q

≤and

Tnxn,q

+ ( –an)d

W(xn,un,θn),q

≤and

Tnxn,q

+ ( –an–γn)d(xn,q) +γnd(un,q)

≤anknd(xn,q) +kn( –an–γn)d(xn,q) +γnd(un,q)

≤kn( –γn)d(xn,q) +γnd(un,q). ()

Again applying ()(a) to the sequence{yn}in () and inserting (), we have

d(yn,q) =d

W

Tnzn,W

cn

 –bn

,bn

,q

≤bnd

Tnzn,q

+ ( –bn)d

W

cn

 –bn

,q

≤bnd

Tnzn,q

+cnd

Tnxn,q

+ ( –bn–cn)d

W(xn,vn,θn),q

≤bnd

Tnzn,q

+cnd

Tnxn,q

+ ( –bn–cn–μn)d(xn,q) +μnd(vn,q)

≤bnkn

kn( –γn)d(xn,q) +γnd(un,q)

+cnknd(xn,q)

≤bnkn( –γn)d(xn,q) +bnknγnd(un,q) +cnknd(xn,q)

≤bnkn( –γn)d(xn,q) +cnknd(xn,q) + ( –bn–cn–μn)d(xn,q)

+bnknγnd(un,q) +μnd(vn,q)

≤bnkn–bnknγn+cnkn

d(xn,q) +kn( –bn–cn–μn)d(xn,q)

≤bnkn–bnknγn+cnkn+kn–bnkn–cnkn–μnkn

d(xn,q)

≤k_n–bnknγn–μnkn

d(xn,q) +bnknγnd(un,q) +μnd(vn,q)

≤k_n( –bnγn–μn)d(xn,q) +bnknγnd(un,q) +μnd(vn,q).

That is,

d(yn,q)≤kn( –bnγn–μn)d(xn,q)

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Now it follows from () and () that

d(xn+,q) =d

W

Tnyn,W

 –αn

,αn

,q

≤αnd

Tnyn,q

+ ( –αn)d

W

 –αn

,q

≤αnd

Tnyn,q

+βnd

Tnzn,q

+ ( –αn–βn–λn)d(xn,q) +λnd(wn,q)

≤αnknd(yn,q) +βnknd(zn,q) + ( –αn–βn–λn)d(xn,q) +λnd(wn,q)

≤αnkn

k_n( –bnγn–μn)d(xn,q) +bnknγnd(un,q) +μnd(vn,q)

+βnkn

kn( –γn)d(xn,q) +γnd(un,q)

≤αnk_n–αnk_nbnγn–αnk_nμnd(xn,q) +αnknbnγnd(un,q)

+αnknμnd(vn,q) +

βnkn–βnknγn

d(xn,q) +βnknγnd(un,q)

≤αnkn–αnknbnγn–αnknμn

d(xn,q) +

βnkn–βnknγn

d(xn,q)

+k_n( –αn–βn–λn)d(xn,q) +αnknbnγnd(un,q)

+αnknμnd(vn,q) +βnknγnd(un,q) +λnd(wn,q)

≤αnkn–αnknbnγn–αnknμn+βnkn

–βnk_nγn+k_n–αnk_n–βnk_n–λnk_nd(xn,q)

+αnknbnγnd(un,q) +αnknμnd(vn,q) +βnknγnd(un,q) +λnd(wn,q)

≤k_n–αnknbnγn–αnknμn–βnknγn–λnkn

d(xn,q)

+αnknbnγn+βnknγn

d(un,q) +αnknμnd(vn,q) +λnd(wn,q)

≤k_nd(xn,q) +

k_n+kn

γnd(un,q) +knμnd(vn,q) +λnd(wn,q).

Therefore, we have

d(xn+,q)≤knd(xn,q) +γnA+μnB+λnC,

whereA=sup{(k

n+kn)d(un,q) :n≥},B=sup{knd(vn,q) :n≥}andC=sup{d(wn,q) :

n≥}.

If we letK=max{A,B,C}, then we have

d(xn+,q)≤knd(xn,q) +K(γn+μn+λn).

Since∞_n_=(kn– ) <∞,

_∞

n=γn<∞,

_∞

n=μn<∞and

_∞

n=λn<∞, it follows from

Lemma . thatlimn→∞d(xn,q) exists.

(ii) Since C is bounded, there exists M>  such that max{d(xn,un),d(xn,vn),d(xn,

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If  <lim infn→∞αn≤lim supn→∞(αn+βn+λn) < , then there existσ,σ∈(, ) such that  <σ ≤αn≤αn+βn+λn≤σ<  for all n≥. We have shown in part (i) that limn→∞d(xn,q) exists, thereforelimn→∞d(xn+,q) =c>  (say).

That is,

lim

n→∞W

Tnyn,W

 –αn

,αn

=c. ()

From (), we have that

lim sup

n→∞

dTnyn,q

≤c. ()

Also

d

W

 –αn

,q

≤ βn

 –αn

dTnzn,q

+

 – βn

 –αn

dW(xn,wn,θn),q

≤ βn

 –αn

dTnzn,q

+ –αn–βn–λn  –αn

d(xn,q) + λn

 –αn

d(wn,q)

≤ βn

 –αn

k_nd(xn,q) +knγnM

+k_n

 – βn

 –αn

– λn  –αn

d(xn,q)

+ λn  –αn

k

n

d(xn,q) +d(xn,wn)

≤ βn

 –αn

k_nγnM+knd(xn,q) + λn

 –αn

k_nd(xn,wn)

≤ βn

 –αnk



nγnM+knd(xn,q) + λn

 –αnk



nd(xn,wn)

≤ βn

 –αn

k_nγnM+knd(xn,q) +

  –αn

k_nλnM

≤ b

 –bk 

nγnM+knd(xn,q) +

  –bk



nλnM

gives that

lim sup

n→∞ d

W

 –αn

,q

≤c. ()

The hypothesis of Lemma . is satisﬁed in (), () and (), therefore we conclude

lim

n→∞d

Tnyn,W

 –αn

= . ()

(iii) If  <lim inf_n_→∞αn≤lim supn→∞(αn+βn+λn) < , then there existσ,σ∈(, ) such that  <σ≤αn≤αn+βn+λn≤σ<  for alln≥. Similarly,  <lim infn→∞bn≤

lim sup_n_→∞(bn+cn+μn) <  gives that there existρ,ρ∈(, ) such that  <ρ≤bn≤

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Since

d(xn+,q) =d

W

Tnyn,W

 –αn

,αn

,q

≤knd(yn,q) + ( –a)d

W

Tnyn,Tnzn,W(xn,wn,θn), βn

 –αn

,

with the help of (), we have

c≤lim inf

n→∞ d(yn,q)≤lim sup_n_→∞ d(yn,q)≤c.

That is,

lim

n→∞d(yn,q) =c. ()

Obviously,

lim sup

n→∞ d

Tnzn,q

≤c ()

and

d

W

cn

 –bn

,q

≤ cn

 –bn

dTnxn,q

+

 – cn

 –bn

dW(xn,vn,θn),q

≤ cn

 –bn

knd(xn,q) +

 –bn–cn–μn

 –bn

d(xn,q) +

μn

 –bn

d(vn,q)

≤ cn

 –bn

knd(xn,q) +d(xn,q) –

cn

 –bn

knd(xn,q) – μn

 –bn

d(xn,q)

+

μn

 –bn

d(xn,q) +

μn

 –bn

d(xn,vn)

≤d(xn,q) +

μn

 –bn

d(xn,vn)≤d(xn,q) +

μn

 –b

M

gives that

lim sup

n→∞

d

W

cn

 –bn

,q

≤c. ()

Again the hypothesis of Lemma . is satisﬁed in (), () and (), therefore we get

lim

n→∞d

Tnzn,W

cn

 –bn

= . ()

(iv) If  < lim infn→∞bn ≤ lim supn→∞(bn +cn +μn) <  and  < lim infn→∞an ≤

lim sup_n_→∞(an+γn) < , then there exist ρ,ρ,τ,τ ∈(, ) such that  <ρ ≤bn≤

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Since

d(yn,q)≤d

W

Tnzn,W

cn

 –bn

,bn

,q

≤( –a)d

Tnzn,W

cn

 –bn

+knd(zn,q),

with the help of (), we have

c≤lim inf

n→∞ d(zn,q)≤lim sup_n_→∞ d(zn,q)≤c.

That is,

lim

n→∞d(zn,q) =nlim→∞d

WTnxn,W(xn,un,θn),an

,q=c. ()

Obviously,

lim sup

n→∞

dTnxn,q

≤c ()

and

dW(xn,un,θn),q

≤

 –an–γn

 –an

d(xn,q) +

γn

 –an

d(xn,un)

≤

 – γn

 –an

d(xn,q) +

γn

 –an

d(xn,un)

≤d(xn,q) – γn

 –an

d(xn,q) +

γn

 –an

d(xn,un)

≤d(xn,q) +

γn

 –b

M

provide that

lim sup

n→∞ d

W(xn,un,θn),q

≤c. ()

Finally, appealing to Lemma . (using (), (), and ()), we get that

lim

n→∞d

Tnxn,W(xn,un,θn)

= . ()

Then

dTnxn,xn

≤dTnxn,W(xn,un,θn)

+dW(xn,un,θn),xn

+

γn

 –an

d(un,xn)

+

γn

 –b

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together with () gives that

lim

n→∞d

Tnxn,xn

= . ()

Next we show thatlimn→∞d(Tnzn,xn) =  andlimn→∞d(Tnyn,xn) = .

The inequality

dTnzn,xn

≤d

Tnzn,W

cn

 –bn

+d

W

cn

 –bn

,xn

≤d

Tnzn,W

cn

 –bn

+ cn  –bn

dTnxn,xn

+

 –bn–cn

 –bn

dW(xn,vn,θn),xn

≤d

Tnzn,W

cn

 –bn

+ cn  –bn

dTnxn,xn

+ μn

 –bn

d(xn,vn)

≤d

Tnzn,W

cn

 –bn

+ cn  –bn

dTnxn,xn

+ μn

 –bn

M

together with () gives that

lim

n→∞d

Tnzn,xn

= .

Similarly,

dTnyn,xn

≤d

Tnyn,W

 –αn

+d

W

 –αn

,xn

≤d

Tnyn,W

 –αn

+ βn  –αn

dTnzn,xn

+

λn

 –αn

d(xn,wn)

≤d

Tnyn,W

 –αn

+ b  –bd

Tnzn,xn

+

λn

 –b

M

provides that

lim

n→∞d

Tnyn,xn

(11)

Hence

lim

n→∞d

Tnxn,xn

= lim

n→∞d

Tnyn,xn

= lim

n→∞d

Tnzn,xn

= .

Theorem . Let C be a nonempty bounded,closed and convex subset of a(UC)hyperbolic space with monotone modulus of uniform convexityη.Let T be a completely continuous and asymptotically nonexpansive self-mapping on C with{kn≥}satisfying

∞

n=(kn– ) <∞.

Let{an},{bn},{cn},{αn},{βn},{γn},{μn}and{λn}be control sequences in[, ]satisfying

the following conditions:

(i)  <lim inf_n_→∞αn≤lim supn→∞(αn+βn+λn) < ,

(ii)  <lim infn→∞bn≤lim supn→∞(bn+cn+μn) < ,

(iii)  <lim infn→∞an≤lim supn→∞(an+γn) < ,

(iv) ∞_n_=γn<∞,

∞

n=μn<∞and

∞

n=λn<∞.

Then{xn},{yn}and{zn}in()converge to the same ﬁxed point of T.

Proof By Lemma ., we have

lim

n→∞d

Tn_x n,xn

= lim

n→∞d

Tn_y n,xn

= lim

n→∞d

Tn_z n,xn

= .

Since

d(xn+,xn) =d

W

Tnyn,W

 –αn

,αn

,xn

≤αnd

Tnyn,xn

+ ( –αn)d

W

Tn_z

n,W(xn,wn,θn), βn

 –αn

,xn

≤αnd

Tnyn,xn

+βnd

Tnzn,xn

+λnd(xn,wn),

we have

dxn+,Tnxn+

≤d(xn+,xn) +d

Tnxn+,Tnxn,

+dTnxn,xn

≤d(xn+,xn) +knd(xn+,xn) +d

Tnxn,xn

≤( +kn)d(xn+,xn) +d

Tnxn,xn

≤( +kn)αnd

Tnyn,xn

+ ( +kn)βnd

Tnzn,xn

+ ( +kn)λnd(xn,wn) +d

Tnxn,xn

.

This together with Lemma . implies that

lim

n→∞d

xn+,Tnxn+

= .

Moreover, the estimate

d(xn+,Txn+)≤d

xn+,Tn+xn+

+dTxn+,Tn+xn+

≤dxn+,Tn+xn+

+kd

xn+,Tnxn+

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implies that

lim

n→∞d(xn,Txn) = . ()

SinceT is completely continuous and{xn} ⊂C is bounded, there exists a subsequence

{xnk} of {xn} such that {Txnk} converges. Therefore from (), {xnk} converges. Let

limk→∞xnk =q. By the continuity of T and (), we have that Tq=q, so q is a ﬁxed

point of T. By Lemma .(i), limn→∞d(xn,q) exists. But limk→∞d(xnk,q) = . Thus

limn→∞d(xn,q) = . Further the inequalities d(yn,xn)≤bnd(Tnzn,xn) +cnd(Tnxn,xn) + μnd(vn,xn) andd(zn,xn)≤and(Tnxn,xn) +γnd(un,xn) give thatlimn→∞d(yn,xn) =  and

lim_n_→∞d(zn,xn) = , respectively.

That is,

lim

n→∞yn=q and nlim→∞zn=q.

Forγn=μn=λn= , Theorem . reduces to the following.

Corollary . Let C be a nonempty bounded, closed and convex subset of a(UC)

hy-perbolic space with monotone modulus of uniform convexity η. Let T be a completely continuous and asymptotically nonexpansive self-mapping on C with{kn≥}satisfying

∞

n=(kn– ) <∞.Let{an},{bn},{cn},{αn}and{βn}be in[, ]with bn+cn,αn+βn∈[, ]

for all n≥and

(i)  <lim inf_n_→∞bn≤lim supn→∞(bn+cn) < ,

(ii)  <lim infn→∞αn≤lim supn→∞αn< .

Then{xn},{yn}and{zn}in()converge to the same ﬁxed point of T.

Forcn=βn=γn=μn=λn≡ in Theorem ., we obtain the following result.

Corollary . Let C be a nonempty bounded,closed and convex subset of a(UC) hy-perbolic space with monotone modulus of uniform convexity η. Let T be a completely continuous and asymptotically nonexpansive self-mapping on C with{kn≥}satisfying

_∞

n=(kn– ) <∞.Let{an},{bn}and{αn}be in[, ]satisfying

(i)  <lim inf_n_→∞bn≤lim supn→∞bn< ,and

(ii)  <lim infn→∞αn≤lim supn→∞αn< .

Then{xn},{yn}and{zn}in()converge to the same ﬁxed point of T.

Foran=cn=βn=γn=μn=λn≡ in Theorem ., we can obtain the Ishikawa-type

convergence result.

Corollary . Let C be a nonempty bounded,closed and convex subset of a(UC)

hyper-bolic space with monotone modulus of uniform convexityη.Let T be a completely continu-ous asymptotically nonexpansive self-mapping of C with{kn≥}satisfying

_∞

n=(kn– ) <

∞.Let{αn}and{bn}be real sequences in[, ]satisfying

(i)  <lim infn→∞αn≤lim supn→∞αn< ,and

(ii)  <lim inf_n_→∞bn≤lim supn→∞bn< .

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Foran=bn=cn=βn=γn=μn=λn≡, Theorem . reduces to the Mann-type

con-vergence result.

Corollary . Let C be a nonempty bounded,closed and convex subset of a uniformly convex hyperbolic space with monotone modulus of uniform convexityη.Let T be a com-pletely continuous asymptotically nonexpansive self-map of C with{kn}satisfying kn≥

and∞_n_=(kn– ) <∞.Let{αn}be real sequences in[, ]satisfying <lim infn→∞αn≤

lim sup_n_→∞αn< .Then{xn}in()converge to a ﬁxed point of T.

As a direct consequence of Theorem ., we formulate the following result inCAT() spaces.

Corollary . Let C be a nonempty bounded,closed and convex subset of aCAT()space. Let T be a completely continuous and asymptotically nonexpansive self-mapping on C with

{kn≥}satisfying∞n=(kn– ) <∞.Let{an},{bn},{cn},{αn},{βn},{γn},{μn}and{λn}be

control sequences in[, ]satisfying the following conditions: (i)  <lim infn→∞αn≤lim supn→∞(αn+βn+λn) < ,

(ii)  <lim infn→∞bn≤lim supn→∞(bn+cn+μn) < ,

(iii)  <lim infn→∞an≤lim supn→∞(an+γn) < ,

(iv) ∞_n_=γn<∞,

_∞

n=μn<∞and

_∞

n=λn<∞.

For a given x∈C,compute{xn},{yn}and{zn}as

zn=anTnxn⊕( –αn)  – γn

 –an

xn⊕

γn

 –an

un

,

yn=bnTnzn⊕( –bn)

cn

 –bn

Tnxn⊕

 – cn

 –bn

×  – μn  –bn–cn

xn⊕

μn

 –bn–cn

vn

,

xn+=αnTnyn⊕( –αn) βn

 –αn

Tnzn⊕

 – βn

 –αn

θnxn⊕( –θn)wn

,

whereλx⊕( –λ)y is the geodesic path between x and y in X.Then{xn},{yn}and{zn}

converge to the same ﬁxed point of T.

Proof AnyCAT() space is a (UC) hyperbolic space (takeW(x,y,λ) =λx⊕( –λ)y), there-fore conclusion follows from Theorem ..

Remark . () Our Theorem . and its corollaries extend and generalize corresponding theorems in a uniformly convex Banach space to a hyperbolic space. Some of these are given below:

(i) Theorem . itself is a nonlinear version of Theorem . in []. (ii) Corollary . extends and generalizes Theorem . in []. (iii) Corollary . extends and generalizes Theorem . in []. (iv) Corollary . extends and generalizes Theorem  in [].

(v) Corollary . is a generalization and reﬁnement of Theorem  in [], Theorem . in [] and Theorem . in [].

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Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi}

Arabia. 2_{Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur, 63100, Pakistan.}

Acknowledgements

The first author owes a lot to Professor Wataru Takahashi from whom he received his doctorate at Tokyo Institute of Technology, Tokyo, Japan. He is extremely indebted to Professor Takahashi and wishes him a long healthy active life. The first author is also grateful to King Fahd University of Petroleum and Minerals for supporting research project IN121055. The second author Amna Kalsoom gratefully acknowledges Higher Education Commission (HEC) of Pakistan for financial support during this research.

Received: 31 October 2013 Accepted: 27 February 2014 Published:17 Mar 2014 References

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