- Research Article
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Generalized Asymptotic Pointwise Contractions and Nonexpansive Mappings Involving Orbits
Fixed Point Theory and Applications volume 2010, Article number: 458265 (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
.
A mapping is called a pointwise contraction if there exists a function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ1_HTML.gif)
Let and for
let
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ3_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ4_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ5_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ6_HTML.gif)
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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ7_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ8_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ9_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ10_HTML.gif)
Taking the superior limit,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ11_HTML.gif)
Hence, (3.2) holds. Now let again . Then there exist
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ12_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ13_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ14_HTML.gif)
This means that , so
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ15_HTML.gif)
Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ16_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ17_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ18_HTML.gif)
This implies that and hence
Next we prove that the problem is well-posed. Let be an approximate fixed point sequence. We know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ19_HTML.gif)
Taking the superior limit and applying (3.2) of Proposition 3.1 for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ20_HTML.gif)
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 ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ21_HTML.gif)
and let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ22_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ23_HTML.gif)
(i)If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ24_HTML.gif)
The above is true because
(ii)If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ25_HTML.gif)
Theorem 3.4.
Let be a bounded metric space,
and suppose there exists a convexity structure
which is compact and
-stable. Assume
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ26_HTML.gif)
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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ27_HTML.gif)
When we obtain
which is false. Hence,
has at most one fixed point.
Let . We consider
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ28_HTML.gif)
Because is compact and
-stable there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ29_HTML.gif)
For ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ30_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ31_HTML.gif)
Taking the superior limit and applying (3.2) of Proposition 3.1,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ32_HTML.gif)
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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ33_HTML.gif)
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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ34_HTML.gif)
Since is a nonempty, closed, and convex subset of a complete
space there exists a unique
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ35_HTML.gif)
For ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ36_HTML.gif)
Let and let
denote the midpoint of the segment
. Using the (CN) inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ37_HTML.gif)
Letting and considering
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ38_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ39_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ40_HTML.gif)
Letting we obtain
. This yields
which is a fixed point since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ41_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ42_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ43_HTML.gif)
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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ44_HTML.gif)
Applying (3.2) of Proposition 3.1 for , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ45_HTML.gif)
Let and let
be the midpoint of
. As in the above proof, using the (CN) inequality we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ46_HTML.gif)
Since ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ47_HTML.gif)
Letting , we have
. This means
because
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ48_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ49_HTML.gif)
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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ50_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ51_HTML.gif)
Taking the supremum with respect to and
and then letting
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ52_HTML.gif)
For ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ53_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ54_HTML.gif)
Knowing that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ55_HTML.gif)
and taking the superior limit we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ56_HTML.gif)
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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ57_HTML.gif)
then is a Picard operator. Moreover, if additionally
satisfies
, then the fixed point problem is well-posed.
Proof.
Let . Denote
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ58_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ59_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ60_HTML.gif)
Hence, for every
and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ61_HTML.gif)
Therefore, for each
. This implies
which holds for every
. Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ62_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ63_HTML.gif)
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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ64_HTML.gif)
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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ65_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ66_HTML.gif)
Thus,
For ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ67_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ68_HTML.gif)
Because every subsequence of is also an approximate fixed point sequence, the conclusions of Proposition 3.1 still stand for
. This yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ69_HTML.gif)
But since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ70_HTML.gif)
by passing to the inferior limit follows
For
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ71_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ72_HTML.gif)
Then has a fixed point. Moreover,
is closed and convex.
Proof.
Let . Denote
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ73_HTML.gif)
Since is a complete
space there exists a unique
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ74_HTML.gif)
Supposing is not a fixed point of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ75_HTML.gif)
This is a contradiction and thus .
Let be a sequence of fixed points which converges to
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ76_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ77_HTML.gif)
Hence, is convex.
A simple example of a mapping which is not nonexpansive, but satisfies (5.1), is the following.
Example 5.2.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ78_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ79_HTML.gif)
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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ80_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ81_HTML.gif)
Then has a fixed point. Moreover,
is closed and convex.
Proof.
Let . Denote
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ82_HTML.gif)
Since is a complete
space there exists a unique
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ83_HTML.gif)
Let This limit exists since the sequence is decreasing and bounded below by
.
Suppose is not a fixed point of
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ84_HTML.gif)
This means that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ85_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ86_HTML.gif)
Inductively, it follows that for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ87_HTML.gif)
Let and let
. Obviously,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ88_HTML.gif)
since
Because of (5.9) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ89_HTML.gif)
Since , it is clear that
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ90_HTML.gif)
Let
Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ91_HTML.gif)
Taking into account (5.14), Now,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ92_HTML.gif)
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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ93_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ94_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ95_HTML.gif)
relation (5.9) is satisfied.
Also,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ96_HTML.gif)
Since we obtain
Hence, relation (5.10) is also accomplished.
Remark 5.6.
If we replace condition (5.9) of Theorem 5.3 with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ97_HTML.gif)
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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ98_HTML.gif)
Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ99_HTML.gif)
Let denote the midpoint of
. The (CN) inequality yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ100_HTML.gif)
Taking the superior limit, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ101_HTML.gif)
But since ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458265/MediaObjects/13663_2009_Article_1287_Equ102_HTML.gif)
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].
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Acknowledgment
The author wishes to thank the financial support provided from programs cofinanced by The Sectoral Operational Programme Human Resources Development, Contract POS DRU 61.5
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Nicolae, A. Generalized Asymptotic Pointwise Contractions and Nonexpansive Mappings Involving Orbits. Fixed Point Theory Appl 2010, 458265 (2009). https://doi.org/10.1155/2010/458265
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DOI: https://doi.org/10.1155/2010/458265