We establish weak and strong convergence of Ishikawa type iterates of two pointwise asymptotic nonexpansive maps in a Hadamard space. For weak and strong convergence results, we drop “rate of convergence condition”, namely
We establish weak and strong convergence of Ishikawa type iterates of two pointwise asymptotic nonexpansive maps in a Hadamard space. For weak and strong convergence results, we drop “rate of convergence condition”, namely
A metric space 
is a length space if any two points of X are joined by a rectifiable path (that is, a path of finite length) and the distance between any two points of X is taken to be the infimum of the lengths of all rectifiable paths joining them. In this case, d is known as length metric (otherwise an inner metric or intrinsic metric). In case, no rectifiable path joins two points of the space, the distance between them is taken to be 
A geodesic path joining 
to 
(or, more briefly, a geodesic from x to y) is a map c from a closed interval 
to X such that 
and 
for all
. In particular, c is an isometry and 
The image 
of c is called a geodesic (or metric) segment joining x and y. We say X is: 1) a geodesic space if any two points of X are joined by a geodesic and 2) uniquely geodesic if there is exactly one geodesic joining x and y. for each
, which we will denote by 
called the segment joining x to y.
A geodesic triangle 
in a geodesic metric space 
consists of three points in X (the vertices of Δ) and a geodesic segment between each pair of vertices (the edges of Δ). A comparison triangle for geodesic triangle 
in 
is a triangle 
in 
such that
for 
and such a triangle always exists (see [
space if all geodesic triangles of appropriate size satisfy 
comparison axiom: Let Δ be a geodesic triangle in X and let 
be a comparison triangle for Δ. Then Δ is said to satisfy the 
inequality if for all 
and all comparison points 
we have
.
For any 
and 
Dhompongsa and Panyanak [
If 
then (1.1) reduces to the original (CN) inequality of Bruhat and Tits [
Let us recall that a geodesic metric space is a 
space if and only if it satisfies the (CN) inequality (see [2, p. 163]). Complete 
spaces are often called Hadamard spaces (see [
metric space and 
then there exists a unique point 
such that
.
A subset 
of a 
space 
is convex if for any 
we have 
In 2008, Kirk and Xu [
on 
defined by:
for all 
where 
Their main result ([
is a constant function. In 2009, the results of [
can be, for instance, expressed in terms of the derivatives of iterations of T for differentiable T.
T is said to be asymptotic pointwise nonexpansive map if there exists a sequence of maps
: 
such that 
for all x, 
, 
, where
. Denoting
Then note that without any loss of generality, T is an asymptotic pointwise nonexpansive map if 
for all x, 
, 
, where 
and 
Moreover, we recall that 
is uniformly LLipschitzian if for some 
we have that 
for 
and 
asymptotic nonexpansive if there is a sequence 
with 
such that
for all 
and
;
semi-compact (completely continuous) if for any bounded sequence 
in C with 
as 
there is a subsequence 
of 
such that 
as 
Let 
be asymptotic pointwise nonexpansive maps with function sequences 
and 
satisfying 
and
respectively. Set
Then
Therefore throughout the paper, we shall take 
as the class of all pointwise asymptotic nonexpansive self maps T on C with function sequence 
with 
for every 
Also F will stand for the set of common fixed points of the two maps 
We assume that cn is a bounded function for every 
and all the functions cn are not bounded by a common constant, therefore a pointwise asymptotic nonexpansive map is not uniformly Lipschitzian. However, an asymptotic nonexpansive map is a pointwise asymptotic nonexpansive as well as uniformly Lipschitzian.
A strictly increasing sequence 
of natural numbers is quasi-periodic if the sequence 
is bounded or equivalently if there exists a natural number q such that any block of q consecutive natural numbers must contain a term of the sequence 
The smallest of such numbers q will be called a quasi-period of
.
Hussain and Khamsi [
Several authors have studied the generalizations of known iterative fixed point construction processes like the Mann process (see e.g. [9,10]) or the Ishikawa process (see e.g. [
has remained in extensive use to prove both weak and strong convergence theorems to approximate fixed points of asymptotic nonexpansive maps in uniformly convex Banach spaces. Also Tan and Xu [
(the sequence associated with the asymptotic nonexpansive map T) is allowed to approach 1 slowly enough so that 
diverges.
Recently Kozlowski [
of two points x, y in geodesic space and define Ishikawa type iterative process of two pointwise asymptotic nonexpansive maps in a geodesic space.
Let C be a nonempty and convex subset of a geodesic space X Let 
be pointwise asymptotic nonexpansive maps and let 
be an increasing sequence of natural numbers and
, 
Then the Ishikawa iteration process denoted by 
in a geodesic space X is as under:
We say that 
is well-defined if 
Following the investigations of Hussain and Khamsi [
Lemma 2.1. Let C be a nonempty, bounded, closed and convex set in a geodesic space X and let 
Let 
be such that the sequence 
in (1.2) is well defined. If the set 
is quasiperiodic and
then
Proof. Set 
and 
From
we have
Also
Using (2.1) and (2.2) in (2.3), we have
Since
therefore taking 
on both the sides of inequality (2.5) and using (2.1) and (2.4), we get
and hence
Similarly
That is,
Remark 2.2. Lemma 2.1 extends the corresponding Lemma 3 of Khan and Takahashi [
Lemma 2.3. Let 
be a nonempty, bounded, closed and convex subset of a Hadamard space 
and let
. Let 
for some 
and 
be such that the sequence 
in 
is well-defined. If the set 
is quasiperiodic and 
then
Proof. Let 
Then use (CN) inequality (1.1) for the scheme (1.2) to have
Since 
is bounded, there exists 
such that 
for some 
Therefore the above inequality becomes
From (2.6), the following two inequalities are obtained
and
Now, we prove that
First assume 
Then there exists a subsequence(use the same notation for subsequence as for the sequence) of 
and 
such that
.
From (2.7), it follows that
Since 
and 
so there exists 
such that 
for all 
Hence the above inequality reduces to
Let 
be any positive integer. Then from (2.10), we have
Letting 
in (2.11), we get
a contradiction.
Hence
Consequently, we have
Following the similar procedure of proof with (2.8), we conclude
Since
therefore with the help of (2.2) and (2.12), we get
Finally, Lemma 2.1 appeals that
Let 
be a bounded sequence in a metric space X. For 
define 
The asymptotic radius 
of 
is given by:
A bounded sequence 
in 
is regular if 
for every subsequence 
of 
The asymptotic center of a bounded sequence 
with respect to 
is defined
If the asymptotic center is taken with respect to 
then it is simply denoted by 
A bounded sequence 
in X. is said to be regular if 
for every subsequence 
of 
Recall that a sequence 
converges weakly to w (written as
) if and only if 
where C is a closed and convex subset containing the bounded sequence 
Moreover, a sequence 
(in X.) Δ-converges to 
if x is the unique asymptotic center of 
for every subsequence 
of 
In this case, we write 
and x is called Δ-limit of 
In a Banach space setting, Δ-convergence coincides with weak convergence. A connection between weak convergence and Δ-convergence in geodesic spaces is characterized in the following lemma due to Nanjaras and Panyanak [
Lemma 2.4. ([
be a bounded sequence in a 
space 
and let 
be a closed and convex subset of 
and contains
. Then 1) 
implies that 
2) the converse of (1) is true if 
is regular.
Next, we state the demiclosed principle in 
spaces due to Hussain and Khamsi [
Lemma 2.5. Let 
be a nonempty, bounded, closed and convex set in a 
space X. and 
be a pointwise asymptotic nonexpansive map. Let 
be a sequence in 
such that 
and 
Then 
Next, we prove our weak convergence theorem.
Theorem 2.6. Let C be a nonempty, bounded, closed and convex set in a Hadamard space X. and let
. Let 
for some 
and 
be such that the sequence 
in 
is well defined. If the set 
is quasiperiodic and 
then 
converges weakly to a point in 
Proof. Let 
be the weak 
-limit set of 
given by
Since C is a nonempty bounded closed convex subset of a Hadamard space, there exists a subsequence 
of 
such that 
as 
and vice versa. This shows that 
As 
and 
(by Lemma 2.1), therefore by Lemma 2.5, 
That is, 
Next, we follow the idea of Chang et al. [
there exists a subsequence 
of 
such that
Hence from (2.12) and (2.14), it follows that
Now from (1.2), (2.14) and (2.15), we get that
Also from (2.12) and (2.14), we have that
Again from (1.2), (2.14) and (2.17), we conclude that
Continuing in this way, by induction, we can prove that, for any 
By induction, one can prove that 
converges weakly to 
as 
in fact 
gives that 
as 
Remark 2.7. If 
is regular in a geodesic space, then 
is Δ-convergent.
Our strong convergence theorem is as follows. We do not use the rate of convergence condition namely
in its proof.
Theorem 2.8. Let C be a nonempty, bounded, closed and convex set in a Hadamard space X and let
. Let 
for some 
and 
be such that the sequence 
in 
is well defined. If the set 
is quasiperiodic, 
and either 
or 
is semi-compact (completely continuous), then 
converges strongly to a point in F.
Proof. Let S be semi-compact. As 
, there exists a subsequence 
of 
such that
Using 
in (2.13) and continuity of 
and
, we obtain that 
The rest of the proof follows by replacing 
with 
in Theorem 2.6 and we, therefore, omit the details.
Finally, we state a theorem due to Nanjaras and Panyanak [
Theorem 2.9. Let C be a nonempty, bounded, closed and convex set in a Hadamard space X and let 
with a sequence 
for which 
Suppose that 
and 
is a sequence in 
for some
. Then the sequence
, Δ-converges to a fixed point of T.
We pose the following open question.
Open question: Does Theorem 2.6 hold if we replace weak convergence by Δ-convergence?