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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
(C
_{n}(x)-1)＜
to answer in the affirma-tive to the open question posed by Tan and Xu [1] even in a general setup.

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 [

.

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 [

.

A subset of a space is convex if for any we have

In 2008, Kirk and Xu [

for all where

Their main result ([

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 c_{n} is a bounded function for every and all the functions c_{n} 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. [

Recently Kozlowski [

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. ([

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. [

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?