ABSTRACT

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 to answer in the affirmative to the open question posed by Tan and Xu [1] even in a general setup.

1. Introduction

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 [2]). A geodesic metric space is a 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 [3] modified the (CN) inequality of Bruhat and Tits [4] as

(1.1)

If then (1.1) reduces to the original (CN) inequality of Bruhat and Tits [4].

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 [5]). Moreover, if X is a 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 [6] studied (in Banach spaces) the existence of fixed points of asymptotic pointwise nonexpansive selfmap on defined by:

for all where

Their main result ([6], Theorem 3.5) states that every asymptotic pointwise nonexpansive selfmap of a nonempty closed bounded convex subset C of a uniformly convex Banach space has a fixed point. This result of Kirk and Xu is a generalization of Goebel and Kirk fixed point theorem [7] for a narrower class of maps, the class of asymptotic nonexpansive maps, where (using our notation) every function is a constant function. In 2009, the results of [6] were extended to the case of metric spaces by Hussain and Khamsi [8]. As pointed out by Kirk and Xu in [6], asymptotic pointwise maps seem to be a natural generalization of nonexpansive maps. The conditions on 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 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 [8] have shown that if X is a Hadamard space and C a nonempty bounded closed convex subset of X, then any pointwise asymptotic nonexpansive selfmap on C has a fixed point. Moreover, this fixed point set is closed and convex. The proof of this important theorem is of the existential nature and does not describe any algorithm for constructing a fixed point of an asymptotic pointwise nonexpansive map. This paper aims at complementing their paper. It is also well known that the fixed point construction iteration processes for generalized nonexpansive maps have been successfully used to develop efficient and powerful numerical methods for solving various nonlinear equations.

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. [11]) to the case of asymptotic (but not pointwise asymptotic) nonexpansive maps. There is huge literature on the iterative construction of fixed points for asymptotic nonexpansive maps in Hilbert, Banach and metric spaces, see e.g. [1,3,7,9-25,27-32] and the references therein. Schu [32] proved the weak convergence of the Mann iteration process to a fixed point of asymptotic nonexpansive maps in uniformly convex Banach spaces with the Opial property [28] and the strong convergence for compact asymptotic nonexpansive maps in uniformly convex Banach spaces. Tan and Xu [1] proved the weak convergence of Mann and Ishikawa iteration processes for asymptotic nonexpansive maps in uniformly convex Banach spaces either satisfying the Opial condition or possessing Fréchet differentiable norm. Moreover, the rate of convergence condition namely 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 [1] remarked: we do not know whether our Theorem 3.1 remains valid if (the sequence associated with the asymptotic nonexpansive map T) is allowed to approach 1 slowly enough so that diverges.

Recently Kozlowski [23] defined Mann type and Ishikawa type iterative processes to approximate fixed points of pointwise asymptotic nonexpansive maps in Banach spaces. We follow his idea and the concept of unique geodesic path denoted by 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:

(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?

REFERENCES