# Generalized Asymptotic Pointwise Contractions and Nonexpansive Mappings Involving Orbits

- Adriana Nicolae
^{1}Email author

**2010**:458265

**DOI: **10.1155/2010/458265

© Adriana Nicolae. 2010

**Received: **30 September 2009

**Accepted: **25 November 2009

**Published: **2 December 2009

## Abstract

We give fixed point results for classes of mappings that generalize pointwise contractions, asymptotic contractions, asymptotic pointwise contractions, and nonexpansive and asymptotic nonexpansive mappings. We consider the case of metric spaces and, in particular, CAT spaces. We also study the well-posedness of these fixed point problems.

## 1. Introduction

Four recent papers [1–4] present simple and elegant proofs of fixed point results for pointwise contractions, asymptotic pointwise contractions, and asymptotic nonexpansive mappings. Kirk and Xu [1] study these mappings in the context of weakly compact convex subsets of Banach spaces, respectively, in uniformly convex Banach spaces. Hussain and Khamsi [2] consider these problems in the framework of metric spaces and spaces. In [3], the authors prove coincidence results for asymptotic pointwise nonexpansive mappings. Espínola et al. [4] examine the existence of fixed points and convergence of iterates for asymptotic pointwise contractions in uniformly convex metric spaces.

In this paper we do not consider more general spaces, but instead we formulate less restrictive conditions for the mappings and show that the conclusions of the theorems still stand even in such weaker settings.

## 2. Preliminaries

Let be a metric space. For and we denote the closed ball centered at with radius by .

Let and let . Throughout this paper we will denote the fixed point set of by . The mapping is called a Picard operator if it has a unique fixed point and converges to for each .

A sequence is said to be an approximate fixed point sequence for the mapping if

The fixed point problem for is well-posed (see [5, 6]) if has a unique fixed point and every approximate fixed point sequence converges to the unique fixed point of .

If the sequence converges pointwise to the function , then is called an asymptotic pointwise contraction.

If for every , , then is called an asymptotic pointwise nonexpansive mapping.

If there exists such that for every , , then is called a strongly asymptotic pointwise contraction.

For a mapping and we define the orbit starting at by

Given and , the number is called the radius of relative to . The diameter of is and the cover of is defined as

As in [2], we say that a family of subsets of defines a convexity structure on if it contains the closed balls and is stable by intersection. A subset of is admissible if it is a nonempty intersection of closed balls. The class of admissible subsets of denoted by defines a convexity structure on . A convexity structure is called compact if any family of elements of has nonempty intersection provided for any finite subset .

According to [2], for a convexity structure , a function is called -convex if for any . A type is defined as where is a bounded sequence in . A convexity structure is -stable if all types are -convex.

The following lemma is mentioned in [2].

Lemma 2.1.

A metric space is a geodesic space if every two points can be joined by a geodesic. A geodesic from to is a mapping , where , such that and for every . The image of forms a geodesic segment which joins and . A geodesic triangle consists of three points and in (the vertices of the triangle) and three geodesic segments corresponding to each pair of points (the edges of the triangle). For the geodesic traingle , a comparison triangle is the triangle in the Euclidean space such that for . A geodesic triangle satisfies the inequality if for every comparison triangle of and for every we have

where are the comparison points of and . A geodesic metric space is a space if every geodesic traingle satisfies the inequality. In a similar way we can define spaces for or using the model spaces .

A geodesic space is a space if and only if it satisfies the following inequality known as the (CN) inequality of Bruhat and Tits [7]. Let be points of a space and let be the midpoint of . Then

It is also known (see [8]) that in a complete space, respectively, in a closed convex subset of a complete space every type attains its infimum at a single point. For more details about spaces one can consult, for instance, the papers [9, 10].

In [2], the authors prove the following fixed point theorems.

Theorem 2.2.

Let be a bounded metric space. Assume that the convexity structure is compact. Let be a pointwise contraction. Then is a Picard operator.

Theorem 2.3.

Let be a bounded metric space. Assume that the convexity structure is compact. Let be a strongly asymptotic pointwise contraction. Then is a Picard operator.

Theorem 2.4.

Let be a bounded metric space. Assume that there exists a convexity structure that is compact and -stable. Let be an asymptotic pointwise contraction. Then is a Picard operator.

Theorem 2.5.

Let be a complete space and let be a nonempty, bounded, closed and convex subset of . Then any mapping that is asymptotic pointwise nonexpansive has a fixed point. Moreover, is closed and convex.

The purpose of this paper is to present fixed point theorems for mappings that satisfy more general conditions than the ones which appear in the classical definitions of pointwise contractions, asymptotic contractions, asymptotic pointwise contractions and asymptotic nonexpansive mappings. Besides this, we show that the fixed point problems are well-posed. Some generalizations of nonexpansive mappings are also considered. We work in the context of metric spaces and spaces.

## 3. Generalizations Using the Radius of the Orbit

In the sequel we extend the results obtained by Hussain and Khamsi [2] using the radius of the orbit. We also study the well-posedness of the fixed point problem. We start by introducing a property for a mapping , where is a metric space. Namely, we will say that satisfies property if

(*S*)for every approximate fixed point sequence
and for every
the sequence
converges to 0 uniformly with respect to *m*.

For instance, if for every , then property is fulfilled.

Proposition 3.1.

Proof.

We only need to let in the above relation to prove (3.3).

Theorem 3.2.

Then is a Picard operator. Moreover, if additionally satisfies , then the fixed point problem is well-posed.

Proof.

Let . As above we have and hence Because is minimal -invariant it follows that . This yields for every . In particular, and using (3.9) we obtain which implies that consists of exactly one point which will be fixed under .

This means that which is impossible.

We remark that if in the above result is, in particular, a pointwise contraction then the fixed point problem is well-posed without additional assumptions for .

Next we give an example of a mapping which is not a pointwise contraction, but fulfills (3.7).

Example 3.3.

Then is not a pointwise contraction, but (3.7) is verified.

Proof.

Theorem 3.4.

where for each and the sequence converges pointwise to a function . Then is a Picard operator. Moreover, if additionally satisfies , then the fixed point problem is well-posed.

Proof.

When we obtain which is false. Hence, has at most one fixed point.

Letting in the above relation yields so converges to which will be the unique fixed point of because and Thus, all the Picard iterates will converge to .

Theorem 3.5.

then has a fixed point. Moreover, is closed and convex.

Proof.

which shows that is a fixed point of .

where the bounded sequence is contained in .

Theorem 3.6.

Let also be an approximate fixed point sequence such that Then .

Proof.

## 4. Generalized Strongly Asymptotic Pointwise Contractions

In this section we generalize the strongly asymptotic pointwise contraction condition, by using the diameter of the orbit. We begin with a fixed point result that holds in a complete metric space.

Theorem 4.1.

then is a Picard operator. Moreover, if additionally satisfies , then the fixed point problem is well-posed.

Proof.

Letting in the above relation we have which implies that . This means that the sequence is Cauchy so it converges to a point . Because is orbitally continuous it follows that is a fixed point, which is unique. Therefore, all Picard iterates converge to .

If we let here it is clear that converges to .

A similar result can be given in a bounded metric space where the convexity structure defined by the class of admissible subsets is compact.

Theorem 4.2.

then is a Picard operator. Moreover, if additionally satisfies , then the fixed point problem is well-posed.

Proof.

Now it is clear that converges to . Because is orbitally continuous, will be the unique fixed point and all the Picard iterates will converge to this unique fixed point.

The fact that every approximate fixed point sequence converges to can be proved identically as in Theorem 4.1.

In connection with the use of the diameter of the orbit, Walter [11] obtained a fixed point theorem that may be stated as follows.

Theorem 4.3 (Walter [11]).

We conclude this section by proving an asymptotic version of this result. In this way we extend the notion of asymptotic contraction introduced by Kirk in [12].

Theorem 4.4.

then is a Picard operator. Moreover, if additionally satisfies and is continuous for each , then the fixed point problem is well-posed.

Proof.

The proof follows closely the ideas presented in the proof of Theorem 4.1.

Letting we obtain that which is impossible. Hence, has at most one fixed point.

Hence, which implies that and the proof may be continued as in Theorem 4.1 in order to conclude that is a Picard operator.

by passing to the inferior limit follows

If we let here , we have Passing here to the limit with respect to implies and this means Because of (4.20) it follows that converges to .

## 5. Some Generalized Nonexpansive Mappings in Spaces

In this section we give fixed point results in spaces for two classes of mappings which are more general than the nonexpansive ones.

Theorem 5.1.

Then has a fixed point. Moreover, is closed and convex.

Proof.

This is a contradiction and thus .

which proves that is a fixed point of so is closed.

A simple example of a mapping which is not nonexpansive, but satisfies (5.1), is the following.

Example 5.2.

Then is not nonexpansive but (5.1) is verified.

Proof.

If then Otherwise, In this way we have shown that (5.1) is accomplished.

Theorem 5.3.

Then has a fixed point. Moreover, is closed and convex.

Proof.

Let This limit exists since the sequence is decreasing and bounded below by .

which is a contradiction. Hence, .

The fact that is closed and convex follows as in the previous proof.

Remark 5.4.

It is clear that nonexpansive mappings and mappings for which (5.1) holds satisfy (5.9) and (5.10). However, there are mappings which satisfy these two conditions without verifying (5.1) as the following example shows.

Example 5.5.

Then does not satisfy (5.1) but conditions (5.9), (5.10) hold.

Proof.

relation (5.9) is satisfied.

Since we obtain Hence, relation (5.10) is also accomplished.

Remark 5.6.

where , then we may conclude that has s unique fixed point.

It is also clear that a pointwise contraction satisfies these conditions so we can apply this result to prove that it has a unique fixed point.

We next prove a demi-closed principle. We will use the notations introduced at the end of Section 3.

Theorem 5.7.

Let be a space, , bounded, closed, and convex. Let be a mapping that safisfies and (5.9) for each and let be an approximate fixed point sequence such that Then .

Proof.

We conclude this paper with the following remarks.

Remark 5.8.

All the above results obtained in the context of spaces also hold in the more general setting used in [4] of uniformly convex metric spaces with monotone modulus of convexity.

Remark 5.9.

In a similar way as for nonexpansive mappings, one can develop a theory for the classes of mappings introduced in this section. An interesting idea would be to study the approximate fixed point property of such mappings. A nice synthesis in the case of nonexpansive mappings can be found in the recent paper of Kirk [13].

## Declarations

## Authors’ Affiliations

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