Generalized Asymptotic Pointwise Contractions and Nonexpansive Mappings Involving Orbits

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 .

A mapping is called a pointwise contraction if there exists a function such that

(2.1)

Let and for let such that

(2.2)

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

(2.3)

where for and . Denote also

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.

Let be a metric space and a compact convexity structure on which is -stable. Then for any type there is such that

(2.4)

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

(2.5)

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

(2.6)

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.

Let be a metric space and let be a mapping which satisfies . If is an approximate fixed point sequence, then for every and every ,

(3.1)

(3.2)

(3.3)

Proof.

Since satisfies and is an approximate fixed point sequence, it easily follows that (3.1) holds. To prove (3.2), let . Then there exists such that

(3.4)

Taking the superior limit,

(3.5)

Hence, (3.2) holds. Now let again . Then there exist such that

(3.6)

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

Theorem 3.2.

Let be a bounded metric space such that is compact. Also let for which there exists such that

(3.7)

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

Proof.

Because is compact, there exists a nonempty minimal -invariant for which . If then In a similar way as in the proof of Theorem  3.1 of [2] we show now that has a fixed point. Let . Then,

(3.8)

This means that , so Therefore,

(3.9)

Denote

(3.10)

since it is nonempty and

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 .

Now suppose are fixed points of . Then

(3.11)

This means that which is impossible.

Let denote the unique fixed point of , let and Observe that the sequence is decreasing and bounded below by so its limit exists and is precisely . Then

(3.12)

This implies that and hence

Next we prove that the problem is well-posed. Let be an approximate fixed point sequence. We know that

(3.13)

Taking the superior limit and applying (3.2) of Proposition 3.1 for ,

(3.14)

which implies .

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.

Let ,

(3.15)

and let

(3.16)

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

Proof.

is not continuous, so it is not nonexpansive and hence it cannot be a pointwise contraction. If or the conclusion is immediate. Suppose and . Then

(3.17)

(i)If , then

(3.18)

The above is true because

(ii)If , then

(3.19)

Theorem 3.4.

Let be a bounded metric space, and suppose there exists a convexity structure which is compact and -stable. Assume

(3.20)

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.

Assume has two fixed points . Then for each ,

(3.21)

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

Let . We consider ,

(3.22)

Because is compact and -stable there exists such that

(3.23)

For ,

(3.24)

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 .

Let be an approximate fixed point sequence and let . Then

(3.25)

Taking the superior limit and applying (3.2) of Proposition 3.1,

(3.26)

Letting we have .

Theorem 3.5.

Let be a complete space and let be nonempty, bounded, closed, and convex. Let and for , let be such that for all . If for all ,

(3.27)

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

Proof.

The idea of the proof follows to a certain extend the proof of Theorem  5.1 in [2]. Let . Denote ,

(3.28)

Since is a nonempty, closed, and convex subset of a complete space there exists a unique such that

(3.29)

For ,

(3.30)

Let and let denote the midpoint of the segment . Using the (CN) inequality, we have

(3.31)

Letting and considering , we have

(3.32)

Letting we obtain that is a Cauchy sequence which converges to . As in the proof of Theorem 3.4 we can show that is a fixed point for . To prove that is closed take a sequence of fixed points which converges to . Then

(3.33)

which shows that is a fixed point of .

The fact that is convex follows from the (CN) inequality. Let and let be the midpoint of . For we have

(3.34)

Letting we obtain . This yields which is a fixed point since

(3.35)

Hence, is convex.

We conclude this section by proving a demi-closed principle similarly to [2, Proposition ]. To this end, for , closed and convex and , as in [2], we introduce the following notation:

(3.36)

where the bounded sequence is contained in .

Theorem 3.6.

Let be a space and let , bounded, closed, and convex. Let satisfy and for , let be such that for all . Suppose that for

(3.37)

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

Proof.

Using (3.1) of Proposition 3.1 we obtain that for every and every ,

(3.38)

Applying (3.2) of Proposition 3.1 for , we have

(3.39)

Let and let be the midpoint of . As in the above proof, using the (CN) inequality we have

(3.40)

Since ,

(3.41)

Letting , we have . This means because

(3.42)

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.

Let be a complete metric space and let be a mapping with bounded orbits that is orbitally continuous. Also, for , let for which there exists such that for every , . If for each

(4.1)

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

Proof.

First, suppose that has two fixed points . Then for each ,

(4.2)

Letting we obtain that which is impossible. Hence, has at most one fixed point. Let . Notice that the sequence is decreasing and bounded below by so it converges to . For we have

(4.3)

Taking the supremum with respect to and and then letting we obtain

(4.4)

For ,

(4.5)

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 .

Next we prove that the problem is well-posed. Let be an approximate fixed point sequence. Taking into account (3.2) applied for and (3.3) of Proposition 3.1,

(4.6)

Knowing that

(4.7)

and taking the superior limit we obtain

(4.8)

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.

Let be a bounded metric space such that is compact and let be an orbitally continuous mapping. Also, for , let for which there exists such that for every , . If for each ,

(4.9)

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

Proof.

Let . Denote ,

(4.10)

As in the proof of Theorem 4.1 one can show that has at most one fixed point and for each , the sequence is Cauchy. This means that for each . Because is compact we can choose

(4.11)

Following the argument of [2, Theorem ] we can show that . For the sake of completeness we also include this part of the proof. The definition of yields that for and every there exists such that for any

(4.12)

Hence, for every and so

(4.13)

Therefore, for each . This implies which holds for every . Thus,

(4.14)

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]).

Let be a complete metric space and let be a mapping with bounded orbits. If there exists a continuous, increasing function for which for every and

(4.15)

then is a Picard operator.

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.

Let be a complete metric space and let be an orbitally continuous mapping with bounded orbits. Suppose there exist a continuous function satisfying for all and the functions such that the sequence converges pointwise to and for each ,

(4.16)

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.

We begin by supposing that has two fixed points . Then for each ,

(4.17)

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

Notice that for the sequence is decreasing and bounded below by so it converges to . For we have

(4.18)

Thus,

For ,

(4.19)

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

Let be the unique fixed point of and let be an approximate fixed point sequence. To show that the problem is well-posed, take a subsequence of such that

(4.20)

Because every subsequence of is also an approximate fixed point sequence, the conclusions of Proposition 3.1 still stand for . This yields

(4.21)

But since

(4.22)

by passing to the inferior limit follows

For

(4.23)

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.

Let be a bounded complete space and let be such that for every

(5.1)

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

Proof.

Let . Denote ,

(5.2)

Since is a complete space there exists a unique such that

(5.3)

Supposing is not a fixed point of , we have

(5.4)

This is a contradiction and thus .

Let be a sequence of fixed points which converges to . Then,

(5.5)

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

Now take . We show that the midpoint of denoted by is a fixed point of using the (CN) inequality. More precisely we have

(5.6)

Hence, is convex.

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

Example 5.2.

Let

(5.7)

Then is not nonexpansive but (5.1) is verified.

Proof.

is not continuous, so it cannot be nonexpansive. To show that (5.1) holds, we only consider the situation when and because in all other the condition is clearly satisfied. Then We can easily observe that

(5.8)

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

Theorem 5.3.

Let be a bounded complete space and let be such that for every ,

(5.9)

(5.10)

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

Proof.

Let . Denote ,

(5.11)

Since is a complete space there exists a unique such that

(5.12)

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

Suppose is not a fixed point of . Then

(5.13)

This means that

(5.14)

(5.15)

Inductively, it follows that for ,

(5.16)

Let and let . Obviously,

(5.17)

since

Because of (5.9) we have

(5.18)

Since , it is clear that

Hence,

(5.19)

Let

Then,

(5.20)

Taking into account (5.14), Now,

(5.21)

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.

The set with the usual metric is a space. Let us take ,

(5.22)

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

Proof.

To prove that does not verify (5.1) we take and . Then However,

(5.23)

Next we show that (5.9) and (5.10) hold. We only need to consider the case when and because in all the other situations this is evident. Then Since

(5.24)

relation (5.9) is satisfied.

Also,

(5.25)

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

Remark 5.6.

If we replace condition (5.9) of Theorem 5.3 with

(5.26)

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.

Using (3.1) of Proposition 3.1 we have Applying (3.2) and (3.3) of Proposition 3.1 for ,

(5.27)

Then,

(5.28)

Let denote the midpoint of . The (CN) inequality yields

(5.29)

Taking the superior limit, we have

(5.30)

But since ,

(5.31)

Hence, .

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].