(1.2)

We say that is well-defined if

2. Fixed Point Approximation

Following the investigations of Hussain and Khamsi [8], the existence of the fixed point of pointwise asymptotic nonexpansive map can not be achieved without its bounded domain. We shall follow them for the purpose. We start with proving the following lemma.

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

(2.1)

then

Proof. Set and

From

we have

(2.2)

Also

(2.3)

Using (2.1) and (2.2) in (2.3), we have

(2.4)

Since

(2.5)

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 [22] from Lipschitzian to non-Lipschitzian maps.

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

(2.6)

From (2.6), the following two inequalities are obtained

(2.7)

and

(2.8)

Now, we prove that

(2.9)

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

(2.10)

Let be any positive integer. Then from (2.10), we have

(2.11)

Letting in (2.11), we get

a contradiction.

Hence

Consequently, we have

(2.12)

Following the similar procedure of proof with (2.8), we conclude

(2.12.1)

Since

therefore with the help of (2.2) and (2.12), we get

Finally, Lemma 2.1 appeals that

(2.13)

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 [26].

Lemma 2.4. ([26], Proposition 3.12). Let 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 [8] needed in the next convergence theorem.

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. [14]. For any there exists a subsequence of such that

(2.14)

Hence from (2.12) and (2.14), it follows that

(2.15)

Now from (1.2), (2.14) and (2.15), we get that

(2.16)

Also from (2.12) and (2.14), we have that

(2.17)

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 [26] proved in Hadamard spaces in which rate of convergence condition is necessary for Δ-convergence of the sequence.

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?

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