Existence and approximation results for SKC mappings in CAT(0) spaces

EXISTENCE AND APPROXIMATION RESULTS FOR SKCMAPPINGS IN CAT(0) SPACES

MUJAHID ABBAS AND SAFEER HUSSAIN KHAN

Abstract. Karapinar and Tas [Computer and Mathematics with Applica-tions, 61 (2011) 3370-3380] extended the class of Suzuki-generalized nonex-pansive mappings to the class of so called SKC mappings. This new class ofmappings contains nonexpansive mappings and Suzuki-generalized nonexpan-sive mappings as proper subclasses. In this paper, existence and approximationof �xed points of these mappings is studied.We obtain approximation resultsusing Ishikawa-type iteration scheme for two mappings. Our results extendthe corresponding results of Nanjaras, Panyanak and Phuengrattana [Nonlin-ear Analysis: Hybrid system,4(2010) 25-31] and Khan nad Abbas [Computerand Mathematics with Applications, 61(2011), 109-116] to SKC mappings.Moreover, the results proved by Krapinar and Tas [Computer and Mathemat-ics with Applications, 61 (2011) 3370-3380 ] are extended to a special type ofmetric space called CAT(0) space.

1. Introduction and Preliminaries

Suzuki [19] introduced a class of single valued mappings called Suzuki-generalizednonexpansive mappings, satisfying a condition which is weaker than nonexpan-siveness and stronger than quasi-nonexpansiveness. Dhompongsa et al. [6] fur-ther improved the results presented in [19]. Nanjaras et al. [16] proved existenceand convergence results for this interesting class of mappings in the framework ofCAT(0) spaces. Khan and Abbas [6] considered some di¤erent iterative schemesfor nonexpansive mappings in CAT(0) spaces. See also [8]. Recently, Karapinarand Tas [11] proposed some new classes of mappings including the so called SKCmappings. These classes substantially generalize the notion of Suzuki-generalizednonexpansive mappings. In this paper, we prove an existence theorem for SKCmappings in the set up of CAT(0) spaces. We approximate common �xed pointsof two SKC mappings by an Ishikawa type iteration scheme. As a corollary, we getthe corresponding results for one mapping case.Let us recall some basics. A metric space X is called a CAT(0) space [10] if it is

geodesically connected and if every geodesic triangle in X is at least as "thin" asits comparison triangle in Euclidean plane. For a vigorous discussion , see Bridsonand Hae�iger [1] or Burago-Burago-Ivanov [3]. The complex Hilbert ball with ahyperbolic metric is a CAT(0) space, see [9] and [17].Let (X; d) be a metric space. A geodesic path joining x 2 X to y 2 X (or, more

brie�y, a geodesic from x to y) is a map c from a closed interval [0; l] � R to X suchthat c(0) = x; c(l) = y, and d(c(t); c(t0)) = jt� t0j for all t; t0 2 [0; l]. In particular,

2000 Mathematics Subject Classi�cation. 47H09,47H10,49M05 .Key words and phrases. Iterative Process, SKC Mappings, Existence, Common Fixed Points,

4� Convergence, Strong Convergence, CAT(0) Space.

1

2 MUJAHID ABBAS AND SAFEER HUSSAIN KHAN

c is an isometry and d(x; y) = l. The image of c is called a geodesic (or metric)segment joining x and y. When it is unique this geodesic segment is denoted by[x; y]. The space (X; d) is said to be a geodesic space if every two points of X arejoined by a geodesic, and X is said to be uniquely geodesic if there is exactly onegeodesic joining x and y for each x; y 2 X. A subset Y � X is said to be convex ifY includes every geodesic segment joining any two of its points. A geodesic triangle4(x1; x2; x3) in a geodesic metric space (X; d) consists of three points x1; x2; x3 inX (the vertices of 4) and a geodesic segment between each pair of vertices (theedges of 4). A comparison triangle for the geodesic triangle 4(x1; x2; x3) in (X; d)is a triangle 4(x1; x2; x3) := 4(�x1; �x2; �x3) in the Euclidean plane E2 such thatdE2(�xi; �xj) = d(xi; xj) for i; j 2 f1; 2; 3g: A geodesic space is said to be a CAT(0)space if all geodesic triangles of appropriate size satisfy the following comparisonaxiom.CAT(0) : Let 4 be a geodesic triangle in X and let 4 be a comparison triangle

for 4. Then 4 is said to satisfy the CAT(0) inequality if for all x; y 2 4 and allcomparison points �x; �y 2 4 ;

d(x; y) � dE2(�x; �y):

If x; y1; y2 are points in a CAT(0) space and if y0 is the midpoint of the segment[y1; y2], then the CAT(0) inequality implies

(CN) d(x; y0)2 � 1

2d(x; y1)

2 +1

2d(x; y2)

2 � 14d(y1; y2)

2

This is the (CN) inequality of Bruhat and Tits [2]. In fact (cf. [1], p. 163), ageodesic space is a CAT(0) space if and only if it satis�es the (CN) inequality.Following are some elementary facts about CAT(0) spaces, cf. Dhompongsa and

Panyanak [5].

Lemma 1. Let (X; d) be a CAT(0) space. Then(i) (X; d) is uniquely geodesic.(ii) Let p; x; y be points of X, let � 2 [0; 1], and let m1 and m2 denote, respec-

tively, the points of [p; x] and [p; y] satisfying d(p;m1) = �d(p; x) and d(p;m2) =�d(p; y). Then

(1.1) d(m1;m2) � �d(x; y):

(iii) Let x; y 2 X;x 6= y and z; w 2 [x; y] such that d(x; z) = d(x;w). Thenz = w.(iv) Let x; y 2 X. For each t 2 [0; 1], there exists a unique point z 2 [x; y] such

that

(1.2) d(x; z) = td(x; y) and d(y; z) = (1� t)d(x; y):

For convenience, from now on we will use the notation (1�t)x�ty for the uniquepoint z satisfying (1:2).The concept of 4� convergence in general metric spaces was coined by Lim [15].

Kirk and Panyanak [14] specialized this concept to CAT(0) spaces and showed thatmany Banach space results involving weak convergence have precise analogs in thissetting. Dhompongsa and Panyanak [5] continued to work in this direction.Let fxng be a bounded sequence in a CAT(0) space X. For x 2 X, we set

FIXED POINT AND CONVERGENCE RESULTS 3

r(x; fxng) = lim supn!1

d(x; xn):

The asymptotic radius r(fxng) of fxng is given by

r(fxng = inffr(x; fxng) : x 2 Xgand the asymptotic center A (fxng) of fxng is the set

A (fxng) = fx 2 X : r(x; fxng) = r(fxng)g:It is known (see, e.g., [4], Proposition 7) that in a CAT(0) space, A (fxng) consistsof exactly one point. A sequence fxng in X is said to 4� converge to x 2 X ifx is the unique asymptotic center of fung for every subsequence fung of fxng. Inthis case we write 4� limn xn = x and call x the 4� limit of fxng; see [14, 15].We denote ww(xn) :=

SfA(fung)g, where the union is taken over all subsequences

fung of fxng.The following lemmas can be found in Dhompongsa and Panyanak [5].

Lemma 2. ([5]; Lemma 2:4) Let X be a CAT(0) space. Then d((1� t)x� ty; z) �(1� t)d(x; z) + td(y; z) for all x; y; z 2 X and t 2 [0; 1].

Lemma 3. ([5]; Lemma 2:5) Let X be a CAT(0) space. Then

d((1� t)x� ty; z)2 � (1� t)d(x; z)2 + td(y; z)2 � t(1� t)d(x; y)2

for all x; y; z 2 X and t 2 [0; 1].

Lemma 4. ([5]; Lemma 2:7) (i) Every bounded sequence in X has a 4� convergentsubsequence.(ii) If C is a closed convex subset of X and if fxng is a bounded sequence in C,

then the asymptotic center of fxng is in C.

Let K be a nonempty subset of a CAT(0) space X: A mapping T : K ! K issaid to satisfy condition (C) if

1

2d(x; Tx) � d(x; y) implies that d(Tx; Ty) < d(x; y) for all x; y 2 K:

Nanjaras et al. [16] proved that a self mapping satisfying condition (C) de�ned ona nonempty bounded and closed subset of a complete CAT(0) space has a �xedpoint.Following de�nitions are basically due to Karapinar and Tas [11] but here we

state them in the framework of CAT(0) spaces. Let K be a nonempty subset of aCAT(0) space X: A mapping T : K ! K is said to satisfy to be:

(1): Suzuki�Ciric mapping ( SCC ) if1

2d(x; Tx) � d(x; y) implies that

d(Tx; Ty) � maxfd(x; y); d(x; Tx); d(y; Ty); d(y; Tx); d(x; Ty)g for all x; y 2 K;(2): Suzuki-KC mapping (SKC) if

1

2d(x; Tx) � d(x; y) implies that

d(Tx; Ty) � maxfd(x; y); d(x; Tx) + d(y; Ty)2

;d(y; Tx) + d(x; Ty)

2g for all x; y 2 K;

4 MUJAHID ABBAS AND SAFEER HUSSAIN KHAN

(3): Kannan-Suzuki mapping (KSC) if1

2d(x; Tx) � d(x; y) implies that

d(Tx; Ty) � d(x; Tx) + d(y; Ty)

2for all x; y 2 K;

(4): Chatterjea-Suzuki mapping (CSC) if1

2d(x; Tx) � d(x; y) implies that

d(Tx; Ty) � d(y; Tx) + d(x; Ty)

2for all x; y 2 K:

For further details on these mappings and their implications, we refer to [11] andreferences therein. Followings are some basic properties of SKC mappings whoseproofs in the setup of CAT(0) spaces follow the same lines as those of Propositions11, 14 and 19 in [11] and therefore we omit them.

Proposition 1. Let K be a nonempty subset of a CAT(0) space X: An SKC map-ping T : K ! K is quasi-nonexpansive provided that the set of �xed point of T isnonempty.

Proposition 2. Let K be a nonempty closed subset of a CAT(0) space X andT : K ! K an SKC mapping then the set of �xed point of T is closed.

Proposition 3. Let K be a nonempty subset of a CAT(0) space X and T : K ! Kan SKC mapping then

d(x; Ty) � 5d(Tx; x) + d(x; y)holds for all x; y in K:

Propositions similar to above can be stated for the class of KSC and CSC map-pings in the framework of CAT(0) spaces.An Ishikawa type iteration process for two mappings S and T is de�ned as:

(1.3)

(xn+1 = (1� a)xn � aTyn;yn = (1� b)xn � bSxn

for all n 2 N;where �; b 2 [ 12 ; 1).When S = T , we have another Ishikawa iteration type process:

(1.4)

(xn+1 = (1� a)xn � aTyn;yn = (1� b)xn � bTxn

for all n 2 N;where a; b 2 [ 12 ; 1).When S = I, the identity mapping, we have Krasnoselkii type iteration process:

x1 2 C(1.5)

xn+1 = (1� a)xn � aTxnfor all n 2 N;where a 2 [ 12 ; 1).The purpose of this paper is:(i) To extend existence results given in [11] to the class of SKC mappings in

CAT(0) spaces. Consequently, corresponding results for KSC and CSC mappingsare also extended to CAT(0) spaces.

FIXED POINT AND CONVERGENCE RESULTS 5

(ii) To prove some strong and 4�convergence results for two SKC mappingsusing (1:3) in CAT(0) spaces.

2. Main Results

F (T ), in the sequel, denotes the set of �xed points of T and F the set of common�xed points of T and S: The next two theorems give the existence of �xed pointsof (SKC) mappings under di¤erent conditions on C.

Theorem 1. Let C be a nonempty closed bounded and convex subset of a CAT(0)space X ;T : C ! C an (SKC) mapping. De�ne a sequence fxng as in (1:5).Then T has a �xed point in C provided that fxng is an approximate �xed pointsequence, that is, limn!1 d (xn; Txn) = 0:

Proof. Since fxng is a bounded sequence in C; A (fxng) consists of exactly onepoint by ( [4], Proposition 7). Suppose that A (fxng) = fpg: Using Lemma 4, weobtain that fpg � C: Since T is an (SKC) mapping, therefore

d(xn; Tp) � 5d(Txn; xn) + d(xn; p)which on taking lim sup on both sides implies that

lim supn!1

d(xn; Tp) � lim supn!1

d(xn; p):

Hence, we haver(Tp; fxng) � r(p; fxng):

Uniqueness of asymptotic centers now implies that p = Tp: �

Theorem 2. Let C be a nonempty compact convex subset of a CAT(0) space X ,T : C ! C a (SKC) mapping; then F (T ) 6= � and fxng given by (1:5) convergestrongly to a �xed point of T provided that fxng is an approximate �xed pointsequence.

Proof. Since C is compact, we obtain a subsequence fxnkg of fxng and p in C suchthat d(xnk ; p)! 0 as k !1. By Proposition 3, we have

d(xnk ; Tp) � 5d(Txnk ; xnk) + d(xnk ; p)for all k 2 N. Now taking limit as k ! 1; we obtain that d(xnk ; Tp) ! 0; whichimplies that Tp = p. Now for such p,

d(xn+1; p) = d((1� a)xn � aTxn; p)� (1� a)d(xn; p) + ad(xn; p)= d(xn; p):

Thus limn!1

d(xn; p) exists and hence fxng converges strongly to p: �

To prepare for our approximation results, we start with the following usefullemma.

Lemma 5. Let C be a nonempty closed convex subset of a CAT(0) space X;T; S :C ! C be two (SKC)- mappings: De�ne a sequence fxng as in (1:3). If F 6= �,then(i) limn!1 d (xn; q) exists for all q 2 F:(ii) limn!1 d (xn; Txn) = 0 = limn!1 d (xn; Sxn) :

6 MUJAHID ABBAS AND SAFEER HUSSAIN KHAN

Proof. Let q 2 F: Then by Lemma 2;d(xn+1; q) = d((1� a)xn � aTyn; q)

� (1� a)d(xn; q) + ad(Tyn; q)� (1� a)d(xn; q) + ad(yn; q):(2.1)

But

d(yn; q) = d((1� b)xn � bSxn; q)� (1� b) d(xn; q) + bd(Sxn; q)� d(xn; q):(2.2)

Combining (2:1) and (2:2), we have

(2.3) d(xn+1; q) � d(xn; q)This shows that fd(xn; q)g is decreasing and this proves part (i): Let(2.4) lim

n!1d (xn; q) = c:

If c = 0, we obviously have

d(xn; Txn) � 2d(xn; q)and by taking lim sup on both sides of above inequality, we have limn!1 d (xn; Txn) =0. A similar argument works to give limn!1 d (xn; Sxn) = 0. Thus (ii) holds.If c 6= 0; then we proceed as follows:By (2:1), we have

d(xn+1; q) � (1� a)d(xn; q) + ad(yn; q)which gives

ad(xn; q) � d(xn; q) + ad(yn; q)� d(xn+1; q)so that

(2.5) c � lim infn!1

d(yn; q):

But (2:2) giveslim supn!1

d(yn; q) � c

so that

(2.6) limn!1

d (yn; q) = c:

Next, by Lemma 3;

d(yn; q)2 = d((1� b)xn � bSxn; q)2

� (1� b) d(xn; q)2 + bd(Sxn; q)2 � b (1� b) d(xn; Sxn)2

� d(xn; q)2 � b (1� b) d(xn; Sxn)2:

Now using (2:4) and (2:6) ; lim sup d(xn; Sxn) � 0 and we can conclude that(2.7) lim

n!1d (xn; Sxn) = 0:

Next, from

d(xn+1; q)2 = d((1� a)xn � aTyn; q)2

� (1� a) d(xn; q)2 + ad(Tyn; q)2 � a (1� a) d(xn; T yn)2

� d(xn; q)2 � a (1� a) d(xn; T yn)2;

FIXED POINT AND CONVERGENCE RESULTS 7

it follows that

(2.8) limn!1

d (xn; T yn) = 0:

Now using (1:3) ;

d (yn; xn) = d ((1� b)xn � bSxn; xn)� (1� b)d (xn; xn) + bd(Sxn; xn):

This implies by (2:7) that

(2.9) limn!1

d (yn; xn) = 0:

Finally,

d (xn; Txn) � d (xn; yn) + d (yn; Txn)

� d (xn; yn) + 5d(yn; T yn) + d(xn; yn)

� 2d (xn; yn) + 5d(yn; xn) + 5d(xn; T yn)

� 7d (xn; yn) + 5d(xn; T yn)

yields

limn!1

d (xn; Txn) = 0

as desired. �

We now give our 4� convergence results.

Theorem 3. Let X; C; T; S;and fxng be in Lemma 5. If F 6= �, then fxng 4�converges to a common �xed point of T and S:

Proof. Let q 2 F: Then by Lemma 5; limn!1 d (xn; q) exists for all q 2 F: Thusfxng is bounded. Also, Lemma 5, gives that limn!1 d(xn; Txn) = limn!1 d(xn; Sxn) =0: First, we show that ww(fxng) � F . Let u 2 ww(fxng), then there exists a sub-sequence fung of fxng such that A(fung) = fug. Since fung being a subsequenceof fxng is bounded, by Lemma 4 there exists a subsequence fvng of fung suchthat 4� limn vn = v for some v 2 C: Since lim

n!1d(vn; T vn) = 0 and T is (SKC)

mapping, therefore

d(vn; T v) � 5d(Tvn; vn) + d(vn; v)

which on taking lim sup on both sides implies that

lim supn!1

d(vn; T v) � lim supn!1

d(vn; v):

Hence, we have

r(Tv; fvng) � r(v; fvng):

Since fvng is 4 convergent to v; thus v is unique asymptotic centre for everysubsequence of fvng: Hence Uniqueness of asymptotic centers implies that v = Tv:That is, v 2 F (T ): A similar argument shows that v 2 F (S) and hence v 2 F: Wenow claim that u = v. Assume on contrary, that u 6= v. Then, by the uniqueness

8 MUJAHID ABBAS AND SAFEER HUSSAIN KHAN

of asymptotic centers, we have

lim supn!1

d(vn; v) < lim supn!1

d(vn; u)

� lim supn!1

d(un; u)

< lim supn!1

d(un; v)

= lim supn!1

d(xn; v)

= lim supn!1

d(vn; v);

a contradiction. Thus, u = v 2 F and hence ww(fxng) � F . To show that fxng4� converges to a common �xed point of T and S, it su¢ ces to show that ww(fxng)consists of exactly one point. Let fung be a subsequence of fxng. By Lemma 4,there exists a subsequence fvng of fung such that 4� limn vn = v for some v 2 C:Let A(fung) = fug and A(fxng) = fxg. We have already seen that u = v andv 2 F . Finally, we claim that x = v. If not, then by existence of lim

n!1d(xn; v) and

uniqueness of asymptotic centers, we have

lim supn!1

d(vn; v) < lim supn!1

d(vn; x)

� lim supn!1

d(xn; x)

< lim supn!1

d(xn; v)

= lim supn!1

d(vn; v);

a contradiction and hence x = v 2 F . Therefore, ww(fxng) = fxg. �

Remark 1. The above thoerem extends Theorem 4 of Khan and Abbas [13] toSKC-mappings.

Although the following is a corollary to our above theorem, yet it is new in itself.

Corollary 1. Let C be a nonempty closed convex subset of a CAT(0) space X;T :C ! C an (SKC)- mapping. Let fxng be as in (1:4). If F (T ) 6= �; then fxng 4�converges to a �xed point of T .

Proof. Take S = T in Theorem 3. �

Following corollary extends Theorem 30 of Karapinar and Tas [11] to the settingof a CAT(0) space.

Corollary 2. Let C be a nonempty closed convex subset of a CAT(0) space X ;T : C ! C an (SKC)- mapping. If F (T ) 6= �, then the sequence fxng de�ned in(1:5) 4� converges to a �xed point of T .

Proof. Take S = I; the identity mapping, in Theorem 3 . �

Following Senter and Dotson [18], Khan and Fukhar-ud-din [12] introduced theso-called condition (A0) for two mappings and gave an improved version of it in [7]as follows: Two mappings S; T : C ! C are said to satisfy the condition (A0) ifthere exists a nondecreasing function f : [0;1) ! [0;1) with f(0) = 0; f(r) > 0for all r 2 (0;1) such that either d(x; Tx) � f(d(x; F )) or d(x; Sx) � f(d(x; F ))

FIXED POINT AND CONVERGENCE RESULTS 9

for all x 2 C: This condition becomes condition (A) of Senter and Dotson [18] wheneitherS = T .Nanjaras [16] obtained a strong convergence result for a Suzuki-generalized non-

expansive mappings employing the Condition (A). We use Condition (A0) to studystrong convergence of fxng de�ned in Lemma 5:

Theorem 4. Let C be a nonempty closed convex subset of a CAT(0) space X ,T; S : C ! C be two (SKC) mappings satisfying condition (A

0). If F 6= ;, then

the sequence fxng given in (1:3) converges strongly to a common �xed point of Sand T:

Proof. By Lemma 5; limn!1

d(xn; x�) exists for all x� 2 F . Let this limit be c where

c � 0:If c = 0; there is nothing to prove.Suppose that c > 0: Now, d(xn+1; x�) � d(xn; x�) gives that

infx�2F

d(xn+1; x�) � inf

x�2Fd(xn; x

�);

which means that d(xn+1; F ) � d(xn; F ) and so limn!1

d(xn; F ) exists. By using the

condition (A0); either

limn!1

f(d(xn; F )) � limn!1

d(xn; Txn) = 0

orlimn!1

f(d(xn; F )) � limn!1

d(xn; Sxn) = 0:

In both the cases, we have

limn!1

f(d(xn; F )) = 0:

Since f is a nondecreasing function and f(0) = 0; it follows that limn!1

d(xn; F ) = 0:

Next, we show that fxng is a Cauchy sequence in C. Let � > 0 be arbitrarilychosen. Since lim

n!1d(xn; F ) = 0, there exists a positive integer n0 such that

d(xn; F ) <�

4; 8n � n0:

In particular, inffd(xn0 ; p) : p 2 Fg <�

4: Thus there must exist p� 2 F such that

d(xn0 ; p�) <

�

2:

Now, for all m;n � n0, we haved(xn+m; xn) � d(xn+m; p

�) + d(p�; xn)

� 2d(xn0 ; p�)

< 2� �2

�= �:

Hence fxng is a Cauchy sequence in a closed subset C of a complete CAT (0) spaceand so it must converge to a point p in C. Now, lim

n!1d(xn; F ) = 0 gives that

d(p; F ) = 0 and closedness of F forces p to be in F: �

Remark 2. The above thoerem extends Theorem 6 of Khan and Abbas [13] toSKC-mappings.

Although the following is a corollary to 4, yet it is new in itself.

10 MUJAHID ABBAS AND SAFEER HUSSAIN KHAN

Corollary 3. Let C be a nonempty closed convex subset of a CAT(0) space X;T :C ! C an (SKC)-mapping satisfying Condition (A). Let fxng be as in (1:4). IfF (T ) 6= �; then fxng converge strongly to a �xed point of T .

Proof. Take S = T in Theorem 4. �The following corollary extends Theorem 5.5 of Nanjaras [16] to SKC mappings

and, in turn, the results involving KSC and CSC mappings.

Corollary 4. Let C be a nonempty closed convex subset of a CAT(0) space X;T :C ! C an (SKC)-mapping satisfying Condition (A). Let fxng be as in (1:5). IfF (T ) 6= �; then fxng converge strongly to a �xed point of T .

Proof. Take S = I, the identity mapping, in Theorem 4. �Remark 3. (1) Theorem 4:4 of Nanjaras [16] about the existence of common �xedpoint of a countable family of commuting maps can now be extended to a countablefamily of SKC mappings.(2) Theorem 5 of Khan and Abbas [13] can also be extended to SKC mappings.(3) Theorems 25; 32 of Karapinar and Tas [11] and their corollaries can now be

extended to the setting of a CAT(0) space.(4) Results for KSC and CSC mappings or for mappings given in [11] satisfying

the so called conditions (A1) and (A2) in the setup of CAT(0) spaces can be alsoobtained from corresponding results proved in this paper. As a matter of fact, theseresults are special cases of our results presented here.

References

[1] M. Bridson, A. Hae�iger, Metric Spaces of Non-Positive Curvature, Springer-Verlag, Berlin,Heidelberg, 1999.

[2] F. Bruhat, J. Tits, Groupes réductifs sur un corps local. I. Données radicielles valuées, Inst.Hautes Études Sci. Publ. Math. 41 (1972) 5�251.

[3] D. Burago, Y. Burago, S. Ivanov, A Course in Metric Geometry, in: Graduate Studies inMath., vol. 33, Amer. Math. Soc., Providence, RI, 2001.

[4] S. Dhompongsa, W.A. Kirk, B. Sims, Fixed points of uniformly lipschitzian mappings,Nonlinear Anal. 65 (2006) 762�772.

[5] S. Dhompongsa and B.Panyanak, On 4� Convergence Theorems in CAT(0) Spaces, Com-put. Math.Appl.56 (2008) 2572�2579.

[6] S. Dhompongsa, W. Inthakon, A. Kaewkhao, Edelstein�s method and �xed point theoremsfor some generalized nonexpansive mappings, J. Math. Anal. Appl., 350 (2009) 12-17.

[7] H. Fukhar-ud-din and S. H. Khan, Convergence of iterates with errors of asymptoticallyquasi-nonexpansive mappings and applications, J. Math. Anal. Appl. 328 (2007), 821�829.

[8] H. Fukhar-ud-din, A. Domolo and A. R. Khan,Strong Convergence of an Implicit Algorithmin CAT(0) Spaces,Fixed Point Theory and Applications, Volume 2011, Article ID 173621, 11pages doi:10.1155/2011/173621

[9] K. Goebel, S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings,Marcel Dekker, Inc., New York, 1984.

[10] M. Gromov, Metric structure for Riemannian and Non-Riemannian spaces, Progr. Math.,vol. 152, Birkhauser, Boston,1984.

[11] E. Karapinar and K. Tas, Generalized (C)-conditions and related �xed point theorems, Comp.Math. Appl., 61(2011) 3370-3380.

[12] S. H. Khan, H. Fukhar-ud-din, Weak and strong convergence of a scheme with errors for twononexpansive mappings, Nonlinear Anal. 8 (2005), 1295�1301

[13] S. H. Khan and M. Abbas, Strong and 4� convergence of some iterative schemes in CAT(0) spaces, Computer and Mathematics with Applications, 61(2011), 109-116.

[14] W.A. Kirk, B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal. 68(2008) 3689�3696.

FIXED POINT AND CONVERGENCE RESULTS 11

[15] T. C. Lim, Remarks on some �xed point theorems, Proc. Amer. Math. Soc. 60 (1976), 179-182.

[16] B. Nanjaras, B. Panyanak, and W. Phuengrattana, Fixed point theorems and convergencetheorems for Suzuki-generalized nonexpansive mappings in CAT(0) spaces, Nonlinear Anal.Hybrid Sys.4(2010), 25-31.

[17] S. Reich, I. Shafrir, Nonexpansive iterations in hyperbolic space, Nonlinear Anal. 15 (1990)537�558.

[18] H.F. Senter, W.G. Dotson, Approximating �xed points of nonexpansive mappings, Proc.Amer. Math. Soc. 44 (1974) 375�380.

[19] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansivemappings, J. Math. Anal. Appl., 340(2008), 1088-1095.

Mujahid Abbas, Department of Mathematics, Lahore University of Management Sci-ences, 54792- Lahore, Pakistan.

E-mail address : [email protected]

(Current Address) The University of Birmingham, School of Mathematics, The WatsonBuilding, Edgbaston B15 2TT Birmingham, U. K.

Safeer Hussain Khan, Department of Mathematics, Statistics and Physics, Qatar Uni-versity, Doha 2713, Qatar.

E-mail address : [email protected]; [email protected]