Strong convergence of an Ishikawa-type algorithm in CAT(0) spaces

R E S E A R C H

Open Access

Strong convergence of an Ishikawa-type

algorithm in

CAT

()

spaces

Hafiz Fukhar-ud-din

*_{Correspondence:} [email protected]; [email protected]

Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur, 63100, Pakistan

Abstract

We study strong convergence of an Ishikawa-type algorithm of two asymptotically nonexpansive type maps to their common fixed point on a CAT(0) space. Our work provides an affirmative answer to the question of Tan and Xu (Proc. Am. Math. Soc. 122:733-739, 1994); in particular, strong convergence of an Ishikawa-type algorithm of two asymptotically nonexpansive maps without the rate of convergence condition is obtained on a nonlinear domain.

MSC: Primary 47H09; 47H10; secondary 49M05

Keywords: asymptotically nonexpansive type map; common fixed point; Ishikawa-type algorithm; uniform equicontinuity; strong convergence

1 Introduction

ACAT() space is simply a geodesic metric space whose each geodesic triangle is at least as thin as its comparison triangle in the Euclidean plane. In , Kirk [] proved a fixed point theorem for a nonexpansive map defined on a subset of aCAT() space. Since then, approximation of fixed points of nonlinear maps on aCAT() space has rapidly developed (see,e.g., [–]).

We describe briefly the needed details for aCAT() space. A metric space (X,d) is said to be 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 said to be alength metric (otherwise known as aninner metricorintrinsic metric). In case no rectifiable path joins two points of the space, the distance between them is taken to be∞.

Ageodesic pathjoiningx∈Xtoy∈X(or, more briefly, ageodesicfromxtoy) is a map

cfrom a closed interval [,l]⊂RtoXsuch thatc() =x,c(l) =y, andd(c(t),c(t)) =|t–t| for allt,t∈[,l]. In particular,cis an isometry andd(x,y) =l. The imageαofcis called a geodesic (or metric)segmentjoiningxandy. We say thatXis: (i) ageodesic spaceif any two points ofXare joined by a geodesic, and (ii)uniquely geodesicif there is exactly one geodesic joiningxandyfor eachx,y∈X, which we will denote by [x,y], called the segment joiningxtoy.

Ageodesic triangle(x,x,x) in a geodesic metric space (X,d) consists of three points

inX(theverticesof) and a geodesic segment between each pair of vertices (theedges

of ). A comparison triangle for geodesic triangle (x,x,x) in (X,d) is a triangle

(x,x,x) :=(x¯,x¯,x¯) inR such thatdR(x¯_i,x¯_j) =d(x_i,x_j) fori,j∈ {, , }. Such a triangle always exists (see []).

A geodesic metric space is said to be aCAT() space if all geodesic triangles of appro-priate size satisfy the followingCAT() comparison axiom.

Letbe a geodesic triangle inXand let⊂R_{be a comparison triangle for. Then}

is said to satisfy theCAT()inequalityif for allx,y∈and all comparison pointsx¯,y¯∈, d(x,y)≤d(x¯,y¯).

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

we will denote byy⊕y

 , then theCAT() inequality implies

d

x,y⊕y 



≤ 

d(x,y)

₊

d(x,y)



–

d(y,y)

_.

The above inequality is the (CN) inequality of Bruhat and Titz [] and it was extended in [] as follows:

dz,αx⊕( –α)y≤αd(z,x)+ ( –α)d(z,y) –α( –α)d(x,y)

for anyα∈[, ] andx,y,z∈X.

Let us recall thata geodesic metric space is aCAT()space if and only if it satisfies the

(CN)inequality(see[],p.). Moreover, ifXis aCAT() metric space andx,y∈X, then

for anyα∈[, ], there exists a unique pointαx⊕( –α)y∈[x,y] such that dz,αx⊕( –α)y≤αd(z,x) + ( –α)d(z,y)

for anyz∈Xand [x,y] ={αx⊕( –α)y:α∈[, ]}.

A subsetCof aCAT() spaceXis convex if for anyx,y∈C, we have [x,y]⊂C. CompleteCAT() spaces are known asHadamard spaces(see[]). The reader inter-ested in a more general nonlinear domain, namely -uniformly convex hyperbolic space containing a CAT() space as a special case, is referred to Dehaish [] and Dehaishet al.[].

LetCbe a nonempty subset of a metric space (X,d). Then a selfmapTonCis:

(i) uniformlyL-Lipschitzian if for someL> ,d(Tn_x,Tn_{y)≤Ld(x,y)forx,y∈C,}

n≥;

(ii) uniformly Hölder continuous if for some positive constantsLandα,

d(Tn_x,Tn_y)≤Ld(x,y)α_{forx,y∈C,n≥;}

(iii) uniformly equicontinuous if for anyε> , there existsδ> such that

d(Tn_x,Tn_{y)≤εwheneverd(x,y)≤δforx,y∈C,n≥or, equivalently,Tis} uniformly equicontinuous if and only ifd(Tn_x

n,Tnyn)→wheneverd(xn,yn)→ asn→ ∞;

(iv) asymptotically nonexpansive if there is a sequence{kn} ⊂[,∞)with

limn→∞kn= such thatd(Tnx,Tny)≤knd(x,y)forx,y∈C,n≥;

(vi) of asymptotically nonexpansive type in the sense of Xu [] if

lim sup_n→∞sup_x∈C{d(Tn_x,Tn_{y) –d(x,y)} ≤for eachy∈C,n≥;} (vii) of asymptotically nonexpansive type in the sense of Changet al.[] if

lim sup_n→∞sup_x∈C{d(Tn_x,Tn_y)_–d(x,y)_{} ≤for eachy∈C,n≥.}

The mapT is semi-compact if for any bounded sequence{xn}inCwithd(xn,Txn)→

asn→ ∞, there is a subsequence{xni}of{xn}such thatxni→x∗∈Casni→ ∞.

It is not difficult to see that nonexpansive map, asymptotically nonexpansive map, asymptotically nonexpansive map in the intermediate sense and asymptotically pansive type map in the sense of Xu [] all are special cases of asymptotically nonex-pansive type map in the sense of Changet al.[]. Moreover, a uniformlyL-Lipschitzian map is uniformly Hölder continuous, and a uniformly Hölder continuous map is uniformly equicontinuous. However, the converse statements are not true as indicated below.

Example . TakeX=RandC= [, ]. DefineT:C→CbyTx= ( –x₎ _{for allx}∈_C. ThenT is uniformly equicontinuous, but it is neither uniformlyL-Lipschitzian nor uni-formly Hölder continuous.

In uniformly convex Banach spaces, the convergence of an Ishikawa-type algorithm and a Mann-type algorithm of nonexpansive maps, asymptotically nonexpansive maps and asymptotically nonexpansive maps in the intermediate sense to their fixed points have been studied by a number of researchers [, –]. For the iterative construction of fixed points of some other classes of nonlinear maps, see [–].

The sequence {kn} in definition (iv) satisfies the rate of convergence condition if _∞

n=(kn– ) <∞. This condition has been extensively used in iterative construction of

fixed points of asymptotically nonexpansive maps in uniformly convex Banach spaces and CAT() spaces (see,e.g., [, , , , ]).

Changet al.[] established strong convergence of an Ishikawa-type algorithm as well as a Mann-type algorithm to a fixed point of an asymptotically nonexpansive type map.

We shall follow the idea of a geodesic path, namely, there exists a unique pointαx⊕ ( –α)yfor anyx,y∈Candα∈[, ], to construct an Ishikawa-type algorithm of two asymptotically nonexpansive type maps on a nonempty subsetCof aCAT() space.

x∈C,

xn+= ( –αn)xn⊕αnSnyn,

yn= ( –βn)xn⊕βnTnxn, n≥,

(.)

where ≤αn,βn≤.

WhenT=I(the identity map) in (.), it reduces to the following Mann-type algorithm:

x∈C,

xn+= ( –αn)xn⊕αnTnyn, n≥,

(.)

where ≤αn≤.

work is a significant generalization of the corresponding results in [], and it provides analogues of the related results of Changet al.[] in uniformly convex Banach spaces. One of our results (Theorem .) gives an affirmative answer to a famous question of Tan and Xu [] on a nonlinear domain for common fixed points.

2 Fixed point approximation

We begin with the following asymptotic regularity result.

Lemma . Let C be a nonempty bounded closed convex subset of aCAT()space X.Let

S,T:C→C be uniformly equicontinuous.Then for the sequence{xn}in(.)satisfying

lim

n→∞d

xn,Snxn

=  = lim

n→∞d

xn,Tnxn

,

we have that

lim

n→∞d(xn,Sxn) =  =nlim→∞d(xn,Txn).

Proof SinceSis uniformly equicontinuous and

d(xn,yn) =d

xn, ( –βn)xn⊕βnTnxn

≤( –βn)d(xn,xn) +βnd

xn,Tnxn

=βnd

xn,Tnxn

→,

therefore,

dSnxn,Snyn

→.

Now

d(xn,xn+) =d

xn, ( –αn)xn⊕αnSnyn

≤αnd

xn,Snyn

≤dxn,Snxn

+dSnxn,Snyn

gives that

lim

n→∞d(xn,xn+) = . (.)

Clearly,

d(xn,Sxn)≤d(xn,xn+) +d

xn+,Sn+xn+

+dSn+xn+,Sn+xn

+dSn+xn,Sxn

, (.)

applyinglim supto both sides of (.), using the uniformly equicontinuous property ofS and (.), we get that

lim sup

n→∞

and hence

lim

n→∞d(xn,Sxn) = .

Similarly,

lim

n→∞d(xn,Txn) = .

That is,

lim

n→∞d(xn,Sxn) =  =nlim→∞d(xn,Txn).

Our main result is as follows.

Theorem . Let C be a nonempty, bounded, closed and convex subset of a CAT()

space X.Let S,T:C→C be uniformly equicontinuous and asymptotically nonexpansive

type maps such that F(S)∩F(T)=∅.Suppose that <δ≤αn,βn≤ –δfor someδ∈(, ),

where{αn}and{βn}are the control parameters of the iteration scheme{xn}in(.).If S or

T is semi-compact,then{xn}converges strongly to a common fixed point of S and T.

Proof For anyp∈F(S)∩F(T), by the (CN)-inequality, we have

d(xn+,p) =d

αnxn⊕αnSnyn,p 

≤( –αn)d(xn,p)+αnd

Snyn,p 

–αn( –αn)d

xn,Snyn 

=d(xn,p)+αn

dSnyn,p 

–d(yn,p)

+αn

d(yn,p)–d(xn,p)

–αn( –αn)d

xn,Snyn 

.

That is,

d(xn+,p)≤d(xn,p)+αn

dSnyn,p 

–d(yn,p)

+αn

d(yn,p)–d(xn,p)

–αn( –αn)d

xn,Snyn 

. (.)

Next we consider the third term on the right side of (.):

d(yn,p)–d(xn,p) =d

( –βn)xn⊕βnTnxn,p 

–d(xn,p)

≤( –βn)d(xn,p)+βnd

Tnxn,p 

–d(xn,p)

–βn( –βn)d

xn,Tnxn 

=βn

dTnxn,p 

–d(xn,p)

–βn( –βn)d

xn,Tnxn 

That is,

αn

d(yn,p)–d(xn,p)

≤αnβn

dTnxn,p 

–d(xn,p)

–αnβn( –βn)d

xn,Tnxn 

. (.)

Substituting (.) into (.) and using  <δ≤αn,βn≤ –δ, we have

d(xn+,p)

≤d(xn,p)–

αn( –αn)

 d

Snyn,p 

–αnβn( –βn)

 d

xn,Tnxn 

+αn d

Snyn,p 

–d(yn,p)–

( –αn)

 d

Snyn,p 

+αnβn d

Tnxn,p 

–d(xn,p)–

( –βn)

 d

xn,Tnxn 

≤d(xn,p)–

δ d

Sn_y n,p



–δ



d

xn,Tnxn 

+ ( –δ) dSnyn,p 

–d(yn,p)–

δ d

Snyn,p 

+ ( –δ) dTnxn,p 

–d(xn,p)–

δ d

xn,Tnxn 

. (.)

Next we prove that

lim

n→∞d

xn,Snyn

=  = lim

n→∞d

xn,Tnxn

.

Assume thatlim sup_n→∞d(xn,Snyn) >  andlim supn→∞d(xn,Tnxn) > .

Then there exist subsequences (we use the same notation for a subsequence as well) of

{xn},{yn}andμ> ,μ>  such thatd(xn,Snyn)≥μ>  andd(xn,Tnxn)≥μ> .

Now from (.) it follows that

d(xn+,p)≤d(xn,p)–

δμ_

 –

δμ_ 

+ ( –δ) dSnyn,p 

–d(yn,p)–

δμ_ 

+ ( –δ) dTnxn,p 

–d(xn,p)–

δμ 



. (.)

For an asymptotically nonexpansive type mapT, we have that

lim sup

n→∞

sup

x∈C

dTnx,p–d(x,p)_≤.

That is,

lim

n→∞supm≥n

sup

x∈C

Hence, for givenδμ 

i

 >  (i= , ), there exists a positive integernsuch that

sup

n≥n

sup

x∈C

dTnx,p–d(x,p)<δμ



i

 .

Since{xn}and{yn}are sequences inC, therefore, forn≥n, it follows that

dSnyn,p 

–d(yn,p)<

δμ_ 

and

dTnxn,p 

–d(xn,p)<

δμ_  .

In the light of the two inequalities above, (.) reduces to

δμ_

 +

δμ_

 ≤d(xn,p)

_–d(x

n+,p) for alln≥n. (.)

Letm≥nbe any positive integer. Obtainm–ninequalities from (.) and then,

sum-ming up these inequalities, we get

δ_μ 

 +

δ_μ 



(m–n)≤d(xn,p)

_–d(x

m+,p)

≤d(xn,p)

_<∞.

Ifm→ ∞, then

∞=d(xn,p)

_<∞,

a contradiction.

This proves thatlim sup_n→∞d(xn,Snyn) =  =lim supn→∞d(xn,Tnxn).

That is,

lim

n→∞d

xn,Snyn

=  = lim

n→∞d

xn,Tnxn

.

As

dxn,Snxn

≤dxn,Snyn

+dSnxn,Snyn

,

d(xn,yn)→ andSis uniformly equicontinuous. So, by takinglim supon both sides, we

get

lim

n→∞d

xn,Snxn

= .

Now, Lemma . implies that

lim

SinceT is semi-compact, therefore there exists a subsequence{xni}of{xn}andq∈C

such that

xni→q. (.)

Now, by the uniform equicontinuity ofSandTand hence continuity, it follows from (.) that

d(q,Sq) =  =d(q,Tq).

This gives thatqis a common fixed point ofSandT. We now proceed to establish strong convergence of{xn}toq.

Since

dTni_x

ni,q

≤dTni_x

ni,xni

+d(xni,q),

therefore

Tni_x

ni→q asni→ ∞. (.)

Clearly,

d(yni,q) =d

( –βni)xni⊕βniT

ni_x

ni,q

≤( –βni)d(xni,q) +βnid

Tni_x

ni,q

.

Therefore, from (.) and (.), it follows that

yni→q asni→ ∞.

Next we prove thatSni_y

ni→qasni→ ∞.

SinceS:C→C is of asymptotically nonexpansive type and{yni}is a sequence inC,

therefore we have

lim sup

ni→∞

dSni_y

ni,q



–d(yni,q)



≤lim sup

ni→∞

sup

x∈C

dSni_x,q_–d(x,q) ≤lim sup

n→∞ supx∈C

dSnx,q–d(x,q)

≤. (.)

Asyni→qasni→ ∞, it follows from (.) that

lim sup

ni→∞

dSni_y

ni,q



≤.

That is,

Sni_y

Replacepbyqin (.) to get

d(xni+,q)

_≤d(x

ni,q)

_–δ

d

Sni_y

ni,q



–δ



d

xni,T

ni_x

ni



+ ( –δ) dSni_y

ni,q



–d(yni,q)

_–δ

d

Sni_y

ni,q



+ ( –δ) dTni_x

ni,q



–d(xni,q)

_–δ

d

xni,T

ni_x

ni



,

which gives thatxni+→qasni→ ∞.

Continuing in this way, by induction, we can prove that for anym≥,

xni+m→q asni→ ∞.

By induction, one can prove that∞_m={xni+m}converges toqasi→ ∞; in fact{xn}∞n=n=

_∞

m={xni+m}∞i=gives thatxn→qasn→ ∞.

We need the following lemma to approximate a common fixed point of two asymptoti-cally nonexpansive maps.

Lemma . Every asymptotically nonexpansive selfmap T on a nonempty bounded subset C of a metric space X is uniformly equicontinuous and of asymptotically nonexpansive type.

Proof LetT :C→C be an asymptotically nonexpansive map with a sequence {kn} ⊆

[,∞) such thatlimn→∞kn= . Letε> . Then, for eachγ > , there exists a positive

integer n such that kn–  <γ for alln≥n. Put s=max{ +γ,k,k, . . . ,kn}. Then d(Tnx,Tny)≤knd(x,y)≤sd(x,y) forx,y∈C,n≥. Chooseδ=ε_s. Thend(Tnx,Tny)≤ε

wheneverd(x,y)≤δforx,y∈C,n≥, proving thatT is uniformly equicontinuous. The second part of the lemma follows from

lim sup

n→∞

sup

x∈C

dTnx,Tny–d(x,y)

≤ lim

n→∞(kn– )sup_x∈Cd

_(x,y)

= .sup

x∈C

d(x,y)

= .

By Theorem . and Lemma ., we have the following result which is new in the litera-ture and sets an analogue of Theorem  in [] without the rate of convergence condition.

Theorem . Let C be a nonempty, bounded, closed and convex subset of a CAT()

space X.Let S,T:C→C be asymptotically nonexpansive maps with sequences{sn},{tn} ⊆

[,∞),respectively and F(S)∩F(T)=∅.Suppose that  <δ ≤αn,βn≤ –δ for some

δ∈(, ),where{αn}and{βn}are the control parameters of the sequence{xn}in(.).If

As every uniformly equicontinuous map is uniformlyL-Lipschitzian, so the following result is immediate and it unifies Theorem . and Theorem . of Changet al.[] in Hadamard spaces.

Theorem . Let C be a nonempty, bounded, closed and convex subset of a CAT()

space X.Let S,T :C→C be uniformly L-Lipschitzian and asymptotically nonexpansive

type maps such that F(S)∩F(T)=∅.Suppose that <δ≤αn,βn≤ –δfor someδ∈(, ),

where{αn}and{βn}are the control parameters of the sequence{xn}in(.).If S or T is

semi-compact,then{xn}converges strongly to a common fixed point of S and T.

ForS=T, Theorem . sets an analogue of Theorem . in [].

Theorem . Let C be a nonempty, bounded, closed and convex subset of a CAT()

space X.Let T:C→C be a uniformly L-Lipschitzian and asymptotically nonexpansive

type map such that F(T)=∅.Suppose that <δ≤αn,βn≤ –δfor someδ∈(, ),where

{αn}and{βn}are the control parameters of the sequence{xn}in(.)with S=T.If T is

semi-compact,then{xn}converges strongly to a fixed point of T.

On takingS=I(the identity map) in Theorem ., we obtain an analogue of Theorem . in [].

Theorem . Let C be a nonempty, bounded, closed and convex subset of a CAT()

space X.Let T:C→C be a uniformly L-Lipschitzian and asymptotically nonexpansive

type map such that F(T)=∅.Suppose that <δ≤αn,βn≤ –δfor someδ∈(, ),where

{αn}and{βn}are the control parameters of the sequence{xn}in(.).If T is semi-compact,

then{xn}converges strongly to a fixed point of T.

Remark . () Tan and Xu [] obtained only weak convergence theorems for asymp-totically nonexpansive maps satisfying the rate of convergence condition and remarked, ‘We do not know whether our weak convergence Theorem . remains valid ifknis

al-lowed to approach  slowly enough so that∞_n=(kn– ) diverges’. Our Theorem . gives

an affirmative answer to their question inCAT() spaces.

() Our results are generalizations inCAT() spaces of the corresponding basic results in [, , , ].

() Theorem . improves and generalizes Theorems .-. in [].

Competing interests

The author did not provide this information.

Acknowledgements

The author is grateful to King Fahd University of Petroleum & Minerals for supporting research project IN 121023.

Received: 7 May 2013 Accepted: 8 July 2013 Published: 5 August 2013 References

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