Mathematical and Computer Modelling 55 (2012) 1418–1427

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Mathematical and Computer Modelling

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Fixed point theorems for Lipschitzian type mappings in CAT(0) spacesMujahid Abbas a, Zoran Kadelburg b,∗, D.R. Sahu c

a Department of Mathematics, Lahore University of Management Sciences, 54792-Lahore, Pakistanb Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Beograd, Serbiac Department of Mathematics, Banaras Hindu University, Varanasi-221005, India

a r t i c l e i n f o

Article history:Received 9 April 2011Received in revised form 18 October 2011Accepted 18 October 2011

Keywords:Iteration processNearly asymptotically quasi-nonexpansivemappings

Fixed pointCAT(0) space

a b s t r a c t

The purpose of this paper is to investigate the demiclosedness principle, the existence the-orem, and convergence theorems in CAT(0) spaces for a class of mappings which is essen-tially wider than that of asymptotically nonexpansive mappings. Our results generalize,extend and unify the corresponding results of Dhompongsa and Panyanak [S. Dhompongsa,B. Panyanak, On △-convergence theorems in CAT(0) spaces, Comput. Math. Appl., 56(2008) 2572–2579], Osilike and Aniagbosor [M.O. Osilike, S.C. Aniagbosor,Weak and strongconvergence theorems for fixed points of asymptotically nonexpansive mappings, Math.ComputerModel., 32 (2000), 1181–1191], Sahu and Beg [D.R. Sahu, I. Beg,Weak and strongconvergence for fixed points of nearly asymptotically non-expansive mappings, Internat. J.Modern Math., 3(2) (2008), 135–151].

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

One of the fundamental and celebrated results in the theory of nonexpansive mappings is Browder’s demiclosedprinciple [1] which states that if C is a nonempty closed convex subset of a uniformly convex Banach space X and T : C → Xis a nonexpansive mapping, then I − T is demiclosed at each y ∈ X , that is, for any sequence {xn} in C conditions xn → xweakly and (I − T )xn → y strongly imply that (I − T )x = y, where I is the identity mapping of X . It is well known that thedemiclosed principle plays an important role in studying the asymptotic behavior for nonexpansive mappings (for detailssee [2–4]). The demiclosed principle for the class ofmappingswhich is essentiallywider than that of nonexpansivemappingsin the setting of Banach spaces has been studied by several authors, see, e.g., [2,5,6].

There are numerous papers dealing with the approximation of fixed points of nonexpansive and asymptoticallynonexpansive mappings in uniformly convex Banach spaces (see, e.g., [2,7–11] and the references contained therein).The class of Lipschitzian mappings is larger than the classes of nonexpansive and asymptotically nonexpansive mappings.However, the theory of computation of fixed points of non-Lipschitzianmappings is equally important and interesting. Thereare a few results in this direction, see, e.g. [5,12–14].

In recent years, CAT(0) spaces have attracted the attention of many authors as they have played a very important rolein different aspects of geometry (see, e.g., [15,16]). Dhompongsa and Panyanak [17] studied the △-convergence of Picard,Mann and Ishikawa iterates in CAT(0) spaces (Theorems 3.1, 3.2 and 3.3, respectively in [17]).

Motivated by thework of Dhompongsa and Panyanak [17], the purpose of this paper is to investigate the demiclosednessprinciple, the existence theorem, and iterative approximation for fixed points of nearly asymptotically nonexpansivemappings in the framework of CAT(0) spaces. Precisely, the demiclosedness principle and the existence theorem are

∗ Corresponding author. Tel.: +381 11 3234967; fax: +381 11 3036819.E-mail addresses:mujahid@lums.edu.pk (M. Abbas), kadelbur@matf.bg.ac.rs (Z. Kadelburg), drsahudr@gmail.com (D.R. Sahu).

0895-7177/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.mcm.2011.10.019

M. Abbas et al. / Mathematical and Computer Modelling 55 (2012) 1418–1427 1419

established for nearly asymptotically nonexpansive mappings in CAT(0) spaces in Section 3. In Section 4, we prove△-convergence and strong convergence of the Ishikawa iteration scheme:

xn+1 = (1 − αn)xn ⊕ αnT nyn,yn = (1 − βn)xn ⊕ βnT nxn, n ∈ N,

(1.1)

where N stands for the set of natural numbers, {αn} and {βn} are real sequences in (0, 1) and T is a nearly asymptoticallynonexpansive self-mapping defined on a convex subset of a CAT(0) space. Our results are an improvement uponcorresponding results of Beg [18], Chang [19], Dhompongsa and Panyanak [17], Khan and Takahashi [20], Kirk andPanyanak [21], Osilike and Aniagbosor [9], and Sahu and Beg [5].

2. Preliminaries

A metric space X is called a CAT(0) space [22] if it is geodesically connected and if every geodesic triangle in X is at leastas ‘‘thin’’ as its comparison triangle in the Euclidean plane. For a rigorous discussion, see [23] or [24]. The complex Hilbertball with a hyperbolic metric is a CAT(0) space, see [4,25].

Let (X, d) be ametric space. A geodesic path joining x ∈ X to y ∈ X (or,more briefly, a geodesic from x to y) is amap c froma closed interval [0, l] ⊂ R to X such that c(0) = x, c(l) = y, and d(c(t), c(t ′)) = |t − t ′| for all t, t ′ ∈ [0, l]. In particular, cis 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 thisgeodesic segment is denoted by [x, y]. The space (X, d) is said to be a geodesic space if every two points of X are joined bya geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x, y ∈ X . A subsetY ⊆ X is said to be convex if Y includes every geodesic segment joining any two of its points. A geodesic triangle△(x1, x2, x3)in a geodesic metric space (X, d) consists of three points x1, x2, x3 in X (the vertices of △) and a geodesic segment betweeneach pair of vertices (the edges of △). A comparison triangle for the geodesic triangle △(x1, x2, x3) in (X, d) is a triangle△(x1, x2, x3) := △(x1, x2, x3) in the Euclidean plane E2 such that dE2(xi, xj) = d(xi, xj) for i, j ∈ {1, 2, 3}. A geodesic spaceis said to be a CAT(0) space if all geodesic triangles of appropriate size satisfy the following comparison axiom.

CAT(0): Let △ be a geodesic triangle in X and let △ be a comparison triangle for △. Then △ is said to satisfy the CAT(0)inequality if for all x, y ∈ △ and all comparison points x, y ∈ △,

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

d(x, y0)2 ≤12d(x, y1)2 +

12d(x, y2)2 −

14d(y1, y2)2. (CN)

This is the (CN) inequality of Bruhat and Tits [26]. In fact, a geodesic space is a CAT(0) space if and only if it satisfies the (CN)inequality (cf. [23, p. 163]).

Following are some elementary facts about CAT(0) spaces given in [17].

Lemma 2.1. Let (X, d) be a CAT(0) space. Then:

(i) (X, d) is uniquely geodesic.(ii) Let p, x, y be points of X and α ∈ [0, 1]. Let m1 and m2 denote, respectively, the points of [p, x] and [p, y] satisfying

d(p,m1) = αd(p, x) and d(p,m2) = αd(p, y). Then

d(m1,m2) ≤ αd(x, y).

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

d(x, z) = td(x, y) and d(y, z) = (1 − t)d(x, y). (2.1)

For convenience, from now on, we will use the notation (1− t)x⊕ ty for the unique point z satisfying (2.1). Nowwe givesome definitions and known results needed in the sequel.

Definition 2.2. Let C be a nonempty subset of a metric space X and T : C → C a mapping. A sequence {xn} in C is said to bean approximating fixed point sequence of T if limn→∞ d(xn, Txn) = 0.

Definition 2.3. Let C be a nonempty subset of a metric space X . The mapping T : C → C is said to be:

(a) uniformly L-Lipschitzian if for each n ∈ N, there exists a positive number L > 0 such that

d(T nx, T ny) ≤ Ld(x, y) for all x, y ∈ C;

(b) asymptotically nonexpansive if there exists a sequence {kn} in [0, ∞) with limn→∞ kn = 0 such that

d(T nx, T ny) ≤ (1 + kn)d(x, y) for all x, y ∈ C and n ∈ N;

1420 M. Abbas et al. / Mathematical and Computer Modelling 55 (2012) 1418–1427

(c) asymptotically quasi-nonexpansive if F(T ) = ∅ and there exists a sequence {kn} in [0, ∞) with limn→∞ kn = 0 such that

d(T nx, p) ≤ (1 + kn)d(x, p) for all x ∈ C, p ∈ F(T ) and n ∈ N.

The class of nearly Lipschitzian mappings is an important generalization of the class of Lipschitzian mappings and wasintroduced by Sahu [13]:

Let C be a nonempty subset of a metric space X and fix a sequence {an} in [0, ∞) with an → 0. A mapping T : C → C issaid to be nearly Lipschitzianwith respect to {an} if for each n ∈ N, there exists a constant kn ≥ 0 such that

d(T nx, T ny) ≤ kn(d(x, y) + an) for all x, y ∈ C . (2.2)

The infimum of constants kn for which (2.2) holds is denoted by η(T n) and is called the nearly Lipschitz constant of T n.A nearly Lipschitzian mapping T with sequence {(an, η(T n))} is said to be:

(d) nearly nonexpansive if η(T n) = 1 for all n ∈ N,(e) nearly asymptotically nonexpansive if η(T n) ≥ 1 for all n ∈ N and limn→∞ η(T n) = 1,(f) nearly uniformly k-Lipschitzian if η(T n) ≤ k for all n ∈ N,(g) nearly uniformly k-contraction if η(T n) ≤ k < 1 for all n ∈ N.

By the definitions, we have the following implication:

contraction H⇒ nearly uniformly k-contraction.

Following is an example of a continuous nearly uniformly contraction mapping which is not a contraction.

Example 2.4. Let X = R, C = (−∞, 3] and T : C → C be a mapping defined by

Tx =

12x, if x ∈ (−∞, 2],

x − 1, if x ∈ (2, 3].

It is obvious that T is a continuous mapping but not a contraction. Also, note that for all x, y ∈ C , we have

d(T nx, T ny) ≤ knd(x, y) + an for n ∈ N,

where {kn} = {1, 12 ,

122

, 123

, . . .} and {an} = {5, 52 ,

522

, 523

, . . .}.

Following is an example of a continuous nearly uniformly Lipschitzian mapping.

Example 2.5. Let X = R, C = (0, ∞) and T : C → C be a mapping defined by

Tx =

1 +

√x, if x ∈ (0, 1],

2, if x ∈ (1, ∞).

Clearly, T is a continuous map. As

T nx = T ny = 2 for n ≥ 2,

it follows that

d(T nx, T ny) ≤ knd(x, y) + an, for n ≥ 2

holds for any sequence {an} in [0, ∞) with an → 0 and kn ≥ 0.

Definition 2.6. Let C be a nonempty subset of a metric space X and fix a sequence {an} in [0, ∞) with an → 0. A mappingT : C → C is said to be nearly asymptotically quasi-nonexpansive with respect to {an} if F(T ) = ∅ and there exists a sequence{un} in [0, ∞) with limn→∞ un = 0 such that

d(T nx, p) ≤ (1 + un)d(x, p) + an for all x ∈ C, p ∈ F(T ) and n ∈ N.

In Example 2.4 above, F(T ) = {0}. Also, T is a nearly asymptotically quasi-nonexpansive mapping with {un} = {1, 12 ,

122

, 123

, . . .} and {an} = {1, 12 ,

122

, 123

, . . .}.A nearly asymptotically quasi-nonexpansive mapping is called a nearly quasi-nonexpansive (asymptotically quasi-

nonexpansive mapping) if un = 0 for all n ∈ N(an = 0 for all n ∈ N). Notice that every nearly asymptotically quasi-nonexpansive mapping with bounded domain is nearly quasi-nonexpansive. Indeed, if C is a bounded subset of a metricspace and T : C → C a nearly asymptotically quasi-nonexpansive mapping with sequence {(an, un)}, then

d(T nx, p) ≤ (1 + un)d(x, p) + an ≤ d(x, p) +

un sup

x,y∈Cd(x, y) + an

for all x ∈ C, p ∈ F(T ) and n ∈ N.

We now give an example of a nearly quasi-nonexpansivemappingwhich is not Lipschitzian and not quasi-nonexpansive.

M. Abbas et al. / Mathematical and Computer Modelling 55 (2012) 1418–1427 1421

Example 2.7. Let X = R, C = [−1π, 1

π] and k ∈ (0, 1). Let T : C → C be a mapping defined by

Tx =

0, if x = 0,

kx sin1x, if x = 0.

Since T : C → C is obviously continuous, it easily follows that it is uniformly continuous. Note F(T ) = {0} and T nx → 0uniformly, but T is not Lipschitzian. For each fixed n ∈ N, define

fn(x) = |T nx| − |x| for all x ∈ C .

Fix a sequence {an} in R defined by

an = maxsupx∈C

fn(x), 0

for all n ∈ N.

It is clear that an ≥ 0 for all n ∈ N and an → 0, since T nx → 0 uniformly. By the definition of {an}, we have

|T nx| ≤ |x| + an for all x ∈ C and n ∈ N.

Clearly, T is a nearly quasi-nonexpansive mapping with respect to {an} and it is not Lipschitz and not quasi-nonexpansive.

Lemma 2.8 ([17, 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 ∈ X and t ∈ [0, 1].

Lemma 2.9 ([17, 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 ∈ X and t ∈ [0, 1].

Lemma 2.10 ([9]). Let {δn}, {βn} and {γn} be three sequences of nonnegative numbers such that

δn+1 ≤ βnδn + γn for all n ∈ N.

If βn ≥ 1 for all n ∈ N,∑

∞

n=1(βn − 1) < ∞ and∑

∞

n=1 γn < ∞, then limn→∞ δn exists.

Lemma 2.11. Let C be a nonempty subset of a metric space (X, d) and T : C → C a quasi L-Lipschitzian, i.e., F(T ) = ∅ andthere exists a constant L > 0 such that

d(Tx, y) ≤ Ld(x, y) for all x ∈ C and y ∈ F(T ).

If {xn} is a sequence in C such that limn→∞ d(xn, F(T )) = 0 and limn→∞ xn = x ∈ C, where d(x, F(T )) = inf{d(x, p) : p ∈

F(T )}, then x is a fixed point of T .

Proof. The proof is obvious, indeed limn→∞ xn = x and limn→∞ d(xn, F(T )) = 0 give that d(x, F(T )) = 0. As T is a quasiL-Lipschitzian selfmap on C , therefore F(T ) is closed. Now, the closedness of F(T ) forces x to be in F(T ). �

3. Demiclosed principle and existence theorem

First, we give the concept of △-convergence and collect some of its basic properties.Let C be a nonempty subset of a metric space (X, d) and {xn} a bounded sequence in X . Let diam(C) denote the diameter

of C . Consider the function ra(·, {xn}) defined byra(x, {xn}) = lim sup

n→∞

d(xn, x), x ∈ X .

The infimum of ra(·, {xn}) over C is called the asymptotic radius of {xn} with respect to C and is denoted by ra(C, {xn}). Wedenote the asymptotic radius of {xn} with respect to X by ra({xn}). A point z ∈ C is said to be an asymptotic center of thesequence {xn} with respect to C if

ra(z, {xn}) = inf{ra(x, {xn}) : x ∈ C}.

The set of all asymptotic centers of {xn} with respect to C is denoted by Za(C, {xn}). The set of all asymptotic centers of {xn}with respect to X is the set

Za({xn}) = {z ∈ X : ra(z, {xn}) = ra{xn}}.It is known (see, e.g., [27], [28, Proposition 7]) that if C is a nonempty closed convex subset of a complete CAT(0) space,Za(C, {xn}) (and hence Za({xn})) consists of exactly one point.

Next, we define △-convergence of a sequence in a CAT(0) space.

1422 M. Abbas et al. / Mathematical and Computer Modelling 55 (2012) 1418–1427

Definition 3.1 ([21,29]). Let {xn} be a bounded sequence in a complete CAT(0) space X . Then {xn} is said to △-converge to xin X if x is the unique asymptotic center of {xm} for every subsequence {xm} of {xn}. In this case we write △- limn xn = x andcall x the △-limit of {xn}.

The following lemma can be found in [17].

Lemma 3.2 ([17, Lemma 2.7]). (i) Every bounded sequence in a complete CAT(0) space X has a△-convergent subsequence. (ii) IfC is a closed convex subset of a complete CAT(0) space X and if {xn} is a bounded sequence in C, then the asymptotic center of{xn} is in C.

Recall that a bounded sequence {xn} in a complete CAT(0) space X is said to be regular if ra(X, {xn}) = ra(X, {un}) forevery subsequence {un} of {xn}. It is known that every bounded sequence in a Banach space has a regular subsequence. Sinceevery regular sequence △-converges, we see immediately that every bounded sequence in a complete CAT(0) space has a△-convergent subsequence. Notice that (see, e.g., [28, Proposition 7]) given a bounded sequence {xn} in a complete CAT(0)space X such that {xn}△-converges to x and given y ∈ X with y = x,

lim supn→∞

d(xn, x) < lim supn→∞

d(xn, y).

Clearly, X satisfies a condition which is known in Banach space theory as the Opial property. We denote ww(xn) :={Za({un})}, where the union is taken over all subsequences {un} of {xn}.Before presenting the main results of this section, we give the following key result which extends Sahu and Beg

[5, Lemma 2.6] in the CAT(0) space setting:

Proposition 3.3. Let C be a nonempty closed convex subset of a complete CAT(0) space X and T : C → C a uniformly continuousnearly asymptotically nonexpansive mapping. If {yn} is a bounded sequence in C such that limn→∞ d(yn, Tyn) = 0, then T has afixed point.

Proof. Since C is a nonempty closed subset of a complete CAT(0) space X and {yn} is a bounded sequence in C , thereforeZa(C, {yn}) is in C and consists of exactly one point v (say). We now show that v is a fixed point of T . Suppose T is a nearlyasymptotically nonexpansive mapping with {(an, η(T n))}. By the uniform continuity of T ,

limn→∞

d(T iyn, T i+1yn) = 0 for i = 0, 1, 2, . . . . (3.1)

We define a sequence {zm} in C by zm = Tmv,m ∈ N. For integersm, n ∈ N, we have

d(zm, yn) ≤ d(Tmv, Tmyn) + d(Tmyn, Tm−1yn) + · · · + d(Tyn, yn)

≤ η(Tm)(d(v, yn) + am) +

m−1−i=0

d(T iyn, T i+1yn). (3.2)

Then, by (3.1) and (3.2), we have

ra(zm, {yn}) = lim supn→∞

d(yn, zm) ≤ η(Tm)(ra(v, {yn}) + am).

Hence

lim supm→∞

ra(zm, {yn}) ≤ ra(v, {yn}). (3.3)

By (CN), we have

dyn,

12v ⊕

12zm

2

≤12d(yn, v)2 +

12d(yn, zm)2 −

14d(v, zm)2 for all m, n ∈ N. (3.4)

From (3.3) and (3.4) we obtain that

ra

12v ⊕

12zm, {yn}

2

≤12ra(v, {yn})2 +

12ra(zm, {yn})2 −

14d(v, zm)2.

Since Za(C, {yn}) = {v}, we have

ra(v, {yn})2 ≤ ra

12v ⊕

12zm, {yn}

2

≤12ra(v, {yn})2 +

12ra(zm, {yn})2 −

14d(v, zm)2,

which implies that

lim supm→∞

d(v, zm)2 ≤ 2 lim supm→∞

[ra(zm, {yn})2 − ra(v, {yn})2] = 0.

M. Abbas et al. / Mathematical and Computer Modelling 55 (2012) 1418–1427 1423

Thus, Tmv → v. By the continuity of T , we have

Tv = T

limm→∞

Tmv

= limm→∞

Tm+1v = v. �

We are now able to establish demiclosed principle and existence theorem for nearly asymptotically nonexpansivemappings in CAT(0) spaces.

Theorem 3.4. Let C be a nonempty closed convex subset of a complete CAT(0) space X and T : C → C a uniformlycontinuous nearly asymptotically nonexpansive mapping. If {xn} is a bounded sequence in C which △-converges to x andlimn→∞ d(xn, Txn) = 0, then x ∈ C and x = Tx.

Proof. Let {xn} be a bounded sequence in C which is an approximating fixed point sequence of T and △-converges to x. Onecan see by Lemma 3.2 that x ∈ C . Note that Za({xn}) = {x}, so we have ra(x, {xn}) = ra({xn}). By Proposition 3.3, we concludethat x = Tx. �

As Theorem 3.4 is true for mappings which are not necessarily Lipschitzian, therefore, it is an improvement uponProposition 3.7 of Kirk and Panyanak [21].

4. Convergence theorems

We begin with basic properties of the Ishikawa iterative sequence {xn} defined by (1.1) for nearly asymptotically quasi-nonexpansive mappings.

Lemma 4.1. Let C be a nonempty convex subset of a CAT(0) space X and T : C → C a nearly asymptotically quasi-nonexpansivemappingwith sequence {(an, un)} such that

∑∞

n=1 an < ∞ and∑

∞

n=1 un < ∞. Let {xn} be a sequence in C defined by (1.1), where{αn} and {βn} are sequences in (0, 1). Then limn→∞ d(xn, p) exists for each p ∈ F(T ) and

∑∞

n=1 αnβn(1−βn)d2(xn, T nxn) < ∞.

Proof. First, we show that limn→∞ d(xn, v) exists for each v ∈ F(T ). For p ∈ F(T ), we have

d(xn+1, p) = d((1 − αn)xn ⊕ αnT nyn, p)

≤ (1 − αn)d(xn, p) + αnd(T nyn, p)

≤ (1 − αn)d(xn, p) + αn((1 + un)d(yn, p) + an) (4.1)

and

d(yn, p) = d((1 − βn)xn ⊕ βnT nxn, p)

≤ (1 − βn)d(xn, p) + βnd(T nxn, p)

≤ (1 − βn)d(xn, p) + βn((1 + un)d(xn, p) + an)

= (1 + βnun)d(xn, p) + anβn. (4.2)

From (4.1) and (4.2), we obtain

d(xn+1, p) ≤ (1 + (1 + βn)un + βnu2n)d(xn, p) + (1 + βn(1 + un))an, n ∈ N.

It follows that

d(xn+1, p) ≤ (1 + Mun)d(xn, p) + M ′an (4.3)

for someM,M ′≥ 0, since {un} is bounded. By Lemma 2.10, we have that limn→∞ d(xn, p) exists. By Lemma 2.9, we have

d2(xn+1, p) = d2((1 − αn)xn ⊕ αnT nyn, p)

≤ (1 − αn)d2(xn, p) + αnd2(T nyn, p) − αn(1 − αn)d2(T nyn, xn)

≤ αn[(1 + un)d(yn, p) + an]2 + (1 − αn)d2(xn, p)

= αn[(1 + un)2d2(yn, p) + 2an(1 + un)d(yn, p) + a2n] + (1 − αn)d2(xn, p).

Since limn→∞ d(xn, p) exists, {xn} is bounded. It follows from (4.2) that {yn} is also bounded. Then, there exists a constantK ≥ 0 such that

d2(xn+1, p) ≤ αn(1 + un)2d2(yn, p) + anK + (1 − αn)d2(xn, p)

= αn(1 + un)2d2((1 − βn)xn ⊕ βnT nxn, p) + anK + (1 − αn)d2(xn, p).

1424 M. Abbas et al. / Mathematical and Computer Modelling 55 (2012) 1418–1427

Again using Lemma 2.9, we obtain

d2(xn+1, p) ≤ αn(1 + un)2[(1 − βn)d2(xn, p) + βnd2(T nxn, p)

− βn(1 − βn)d2(xn, T nxn)] + anK + (1 − αn)d2(xn, p)≤ αn(1 + un)

2[(1 − βn)d2(xn, p) + βn{(1 + un)d(xn, p) + an}2

− βn(1 − βn)d2(xn, T nxn)] + anK + (1 − αn)d2(xn, p)≤ αn(1 + un)

2[(1 − βn)d2(xn, p) + βn{(1 + un)

2d2(xn, p) + anK ′}

− βn(1 − βn)d2(xn, T nxn)] + anK + (1 − αn)d2(xn, p)≤ αn(1 + un)

2[(1 − βn)(1 + un)

2d2(xn, p) + βn(1 + un)2d2(xn, p)

+ βnanK ′− βn(1 − βn)d2(xn, T nxn)] + anK + (1 − αn)d2(xn, p)

= αn(1 + un)2[(1 + un)

2d2(xn, p) + βnanK ′

− βn(1 − βn)d2(xn, T nxn)] + anK + (1 − αn)d2(xn, p)≤ αn(1 + un)

4d2(xn, p) + anK ′′− αnβn(1 − βn)d2(xn, T nxn) + anK + (1 − αn)d2(xn, p)

≤ (1 + un)4d2(xn, p) + anK ′′

− αnβn(1 − βn)d2(xn, T nxn) + anK≤ (1 + M ′′un)d2(xn, p) + anK ′′′

− αnβn(1 − βn)d2(xn, T nxn)

for some K ′, K ′′, K ′′′,M ′′≥ 0. By the boundedness of {xn}, we have

d2(xn+1, p) ≤ d2(xn, p) + (an + un)M ′′′− αnβn(1 − βn)d2(xn, T nxn)

for all n ∈ N and for someM ′′′≥ 0. Thus,

αnβn(1 − βn)d2(xn, T nxn) ≤ d2(xn, p) − d2(xn+1, p) + (an + un)M ′′′.

This implies that

m−n=1

αnβn(1 − βn)d2(xn, T nxn) ≤

m−n=1

[d2(xn, p) − d2(xn+1, p)] + M ′′′

m−n=1

(an + un)

= d2(x1, p) − d2(xm+1, p) + M ′′′

m−n=1

(an + un).

Since∑

∞

n=1(an + un) < ∞, therefore,∑

∞

n=1 αnβn(1 − βn)d2(xn, T nxn) < ∞. �

First, we prove the △-convergence theorem.

Theorem 4.2. Let C be a nonempty closed convex subset of a complete CAT(0) space X and T : C → C a uniformly continuousnearly asymptotically nonexpansive mapping with F(T ) = ∅ and sequence {(an, η(T n))} such that

∑∞

n=1(η(T n) − 1) < ∞ and∑∞

n=1 an < ∞. For arbitrary x1 ∈ C, let {xn} be a sequence in C defined by (1.1), where {αn} and {βn} are sequences in (0, 1)such that limn→∞ αnβn(1 − βn) = 0. Then {xn} is △-convergent to an element of F(T ).

Proof. Assume that limn→∞ αnβn(1−βn) = 0. Then, by Lemma4.1,we have limn→∞ d(xn, T nxn) = 0. By uniform continuityof T , d(xn, T nxn) → 0 implies that d(Txn, T n+1xn) → 0. Observe that

d(xn+1, xn) = d((1 − αn)xn ⊕ αnT nyn, xn)≤ d(xn, T nyn)≤ d(xn, T nxn) + d(T nxn, T nyn)≤ d(xn, T nxn) + η(T n)(d(xn, yn) + an)≤ d(xn, T nxn) + η(T n)(d(xn, T nxn) + an) → 0 as n → ∞.

Also

d(xn, Txn) ≤ d(xn, xn+1) + d(xn+1, T n+1xn+1) + d(T n+1xn+1, T n+1xn) + d(T n+1xn, Txn)≤ (1 + η(T n+1))d(xn, xn+1) + d(xn+1, T n+1xn+1) + d(T n+1xn, Txn) + an+1,

and hence

limn→∞

d(xn, Txn) = 0. (4.4)

M. Abbas et al. / Mathematical and Computer Modelling 55 (2012) 1418–1427 1425

First, we show thatww({xn}) ⊆ F(T ). Let u ∈ ww({xn}). Then there exists a subsequence {un} of {xn} such that Za(C, {un}) =

{u}. By Lemma 3.2, there exists a subsequence {vn} of {un} such that △- limn vn = v for some v ∈ C . By Theorem 3.4,v ∈ F(T ). By Lemma 4.1, limn→∞ d(xn, v) exists. We now claim that u = v. Suppose, to the contrary, that u = v. Then, bythe uniqueness of asymptotic centers, we have

lim supn→∞

d(vn, v) < lim supn→∞

d(vn, u) ≤ lim supn→∞

d(un, u)

< lim supn→∞

d(un, v) = lim supn→∞

d(xn, v)

= lim supn→∞

d(vn, v),

a contradiction. Thus, u = v ∈ F(T ) and hence ww({xn}) ⊆ F(T ). To show that {xn}△-converges to a fixed point of T, itsuffices to show that ww({xn}) consists of exactly one point. Let {un} be a subsequence of {xn}. By Lemma 3.2, there existsa subsequence {vn} of {un} such that △- limn vn = v for some v ∈ C . Let Za(C, {un}) = {u} and Za(C, {xn}) = {x}. We havealready seen that u = v and v ∈ F(T ). Finally, we claim that x = v. Suppose not, then by the existence of limn→∞ d(xn, v)and uniqueness of asymptotic centers, we have that

lim supn→∞

d(vn, v) < lim supn→∞

d(vn, x) ≤ lim supn→∞

d(xn, x)

< lim supn→∞

d(xn, v) = lim supn→∞

d(vn, v),

a contradiction and hence x = v ∈ F(T ). Therefore, ww({xn}) = {x}. �

Theorem 4.2 extends Dhompongsa and Panyanak [17, Theorem 3.3] from the class of nonexpansivemappings to the classof mappings which are not necessarily Lipschitzian.

We now discuss the strong convergence Ishikawa iteration process defined by (1.1) for Lipschitzian type mappings inCAT(0) space setting.

Theorem 4.3. Let C be a nonempty closed convex subset of a complete CAT(0) space X, T : C → C a nearly asymptoticallyquasi-nonexpansive mapping with sequence {(an, un)} such that

∑∞

n=1 an < ∞ and∑

∞

n=1 un < ∞. Assume that F(T ) is aclosed set. Let {xn} be a sequence in C defined by (1.1), where {αn} and {βn} are sequences in (0, 1). Then {xn} converges stronglyto a fixed point of T if and only if lim infn→∞ d(xn, F(T )) = 0.

Proof. Necessity is obvious.Conversely, suppose that lim infn→∞ d(xn, F(T )) = 0. From (4.3), we have

d(xn+1, F(T )) ≤ (1 + Mun)d(xn, F(T )) + M ′an, n ∈ N,

so limn→∞ d(xn, F(T )) exists. It follows that limn→∞ d(xn, F(T )) = 0. Following arguments similar to those given in [3,Lemma 5], 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 ≥ 1, where L = eM(∑n+m−1

j=n uj) > 0 and bj = M ′aj. As,∑

∞

n=1 un < ∞ so

L∗= eM(

∑∞n=1 un) ≥ L = eM(

∑n+m−1j=n uj) > 0. Let ε > 0 be arbitrarily chosen. Since limn→∞ d(xn, F) = 0 and

∑∞

n=1 an < ∞,there exists a positive integer n0 such that

d(xn, F) <ε

4L∗and

∞−j=n0

bj <ε

6L∗, ∀n ≥ n0.

In particular, inf{d(xn0 , p) : p ∈ F} < ε4L∗ . Thus there must exist p∗

∈ F such that

d(xn0 , p∗) <

ε

3L∗.

Hence for n ≥ n0, we have

d(xn+m, xn) ≤ d(xn+m, p∗) + d(p∗, xn)

≤ 2L∗

d(xn0 , p

∗) +

∞−j=n0

bj

< 2L∗

ε

3L∗+

ε

6L∗

= ε.

1426 M. Abbas et al. / Mathematical and Computer Modelling 55 (2012) 1418–1427

Hence {xn} is a Cauchy sequence in a closed subset C of a complete CAT(0) space and so it must converge strongly to a pointq in C . Now, limn→∞ d(xn, F(T )) = 0 gives that d(q, F(T )) = 0. Since F(T ) is closed, so we have q ∈ F(T ). �

In the next result, the closedness assumption on F(T ) is not required.

Theorem 4.4. Let C be a nonempty closed convex subset of a complete CAT(0) space X and T : C → C an asymptotically quasi-nonexpansive mapping with sequence {un} such that

∑∞

n=1 un < ∞. Let {xn} be a sequence in C defined by (1.1), where {αn} and{βn} are sequences in (0, 1). Then {xn} converges strongly to a fixed point of T if and only if lim infn→∞ d(xn, F(T )) = 0.

Proof. Following arguments similar to those of Theorem 4.3, we obtain that {xn} is a Cauchy sequence in C . Let limn→∞ xn =

x. Since an asymptotically quasi-nonexpansive mapping is quasi L-Lipschitzian, it follows from Lemma 2.11 that x is a fixedpoint of T . �

Theorem 4.5. Let C be a nonempty closed convex subset of a complete CAT(0) space X and T : C → C a uniformly continuousnearly asymptotically nonexpansive mapping with F(T ) = ∅ and sequence {(an, η(T n))} such that

∑∞

n=1(η(T n) − 1) < ∞ and∑∞

n=1 an < ∞. For arbitrary x1 ∈ C, let {xn} be a sequence in C defined by (1.1), where {αn} and {βn} are sequences in (0, 1).If limn→∞ αnβn(1 − βn) = 0, then limn→∞ d(xn, T nxn) = 0. Further, if T is uniformly continuous and Tm is demicompact forsome m ∈ N, it follows that {xn} converges strongly to a fixed point of T .

Proof. By (4.4), we have limn→∞ d(xn, Txn) = 0. By the uniform continuity of T , we have

d(xn, Txn) → 0 ⇒ d(Txn, T 2xn) → 0 ⇒ · · · ⇒ d(T ixn, T i+1xn) → 0

for all i ∈ N. It follows that

d(xn, Tmxn) ≤

m−1−i=0

d(T ixn, T i+1xn) → 0 as n → ∞.

Since d(xn, Tmxn) → 0, and Tm is demicompact, there exists a subsequence {xnj} of {xn} such that limj→∞ Tmxnj = x ∈ C .Note that

d(xnj , x) ≤ d(xnj , Tmxnj) + d(Tmxnj , x) → 0 as ȷ → ∞.

Since limn→∞ d(xn, Txn) = 0, we get x ∈ F(T ). Since limn→∞ d(xn, x) exists by Lemma 4.1, and limj→∞ d(xnj , x) = 0, weconclude that xn → x. �

Theorem 4.5 extends corresponding results of Beg [18], Chang [19], Khan and Takahashi [20] and Osilike andAniagbosor [9] for a more general class of non-Lipschitzian mappings in the framework of CAT(0) spaces. It also extendscorresponding results of Dhompongsa and Panyanak [17] from the class of nonexpansive mappings to a more general classof non-Lipschitzian mappings in the same space setting.

Recall that a mapping T from a subset of a metric space (X, d) into itself with F(T ) = ∅ is said to satisfy condition (A)(see [30]) if there exists a nondecreasing function f : [0, ∞) → [0, ∞) with f (0) = 0, f (t) > 0 for t ∈ (0, ∞) such that

d(x, Tx) ≥ f (d(x, F(T ))) for all x ∈ C .

Theorem 4.6. Let C be a nonempty closed convex subset of a complete CAT(0) space X and T : C → C a uniformly continuousnearly asymptotically nonexpansive mapping with F(T ) = ∅ and sequence {(an, η(T n))} such that

∑∞

n=1(η(T n) − 1) < ∞ and∑∞

n=1 an < ∞. For arbitrary x1 ∈ C, let {xn} be a sequence in C defined by (1.1), where {αn} and {βn} are sequences in (0, 1)such that limn→∞ βn(1 − βn) = 0. Suppose that T satisfies condition (A). Then {xn} converges strongly to a fixed point of T .

Proof. By (4.4), we have limn→∞ d(xn, Txn) = 0. Further, by condition (A),

limn→∞

d(xn, Txn) ≥ limn→∞

f (d(xn, F(T ))).

It follows that limn→∞ d(xn, F(T )) = 0. Therefore, the result follows from Theorem 4.3. �

Theorem 4.6 extends Sahu and Beg [5, Theorem 4.4] from Banach space to CAT(0) space.

Acknowledgments

The authors are grateful to the referees for their critical remarks that helped us to improve this paper.The second author thanks the Ministry of Science and Technological Development of Serbia.

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