Generalized Asymptotic Pointwise Contractions and Nonexpansive Mappings Involving Orbits
© Adriana Nicolae. 2010
Received: 30 September 2009
Accepted: 25 November 2009
Published: 2 December 2009
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.
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  study these mappings in the context of weakly compact convex subsets of Banach spaces, respectively, in uniformly convex Banach spaces. Hussain and Khamsi  consider these problems in the framework of metric spaces and spaces. In , the authors prove coincidence results for asymptotic pointwise nonexpansive mappings. Espínola et al.  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.
As in , 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 , 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 .
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
A geodesic space is a space if and only if it satisfies the following inequality known as the (CN) inequality of Bruhat and Tits . Let be points of a space and let be the midpoint of . Then
It is also known (see ) 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 , the authors prove the following fixed point theorems.
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  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
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 .
Next we give an example of a mapping which is not a pointwise contraction, but fulfills (3.7).
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.
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 .
A similar result can be given in a bounded metric space where the convexity structure defined by the class of admissible subsets is compact.
In connection with the use of the diameter of the orbit, Walter  obtained a fixed point theorem that may be stated as follows.
Theorem 4.3 (Walter ).
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 .
The proof follows closely the ideas presented in the proof of Theorem 4.1.
A simple example of a mapping which is not nonexpansive, but satisfies (5.1), is the following.
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.
relation (5.9) is satisfied.
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.
We conclude this paper with the following remarks.
All the above results obtained in the context of spaces also hold in the more general setting used in  of uniformly convex metric spaces with monotone modulus of convexity.
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 .
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