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Common Fixed Point and Approximation Results for Noncommuting Maps on Locally Convex Spaces
Fixed Point Theory and Applications volume 2009, Article number: 207503 (2009)
Abstract
Common fixed point results for some new classes of nonlinear noncommuting maps on a locally convex space are proved. As applications, related invariant approximation results are obtained. Our work includes improvements and extension of several recent developments of the existing literature on common fixed points. We also provide illustrative examples to demonstrate the generality of our results over the known ones.
1. Introduction and Preliminaries
In the sequel, will be a Hausdorff locally convex topological vector space. A family
of seminorms defined on
is said to be an associated family of seminorms for
if the family
where
and
, forms a base of neighborhoods of zero for
. A family
of seminorms defined on
is called an augmented associated family for
if
is an associated family with property that the seminorm
for any
. The associated and augmented associated families of seminorms will be denoted by
and
, respectively. It is well known that given a locally convex space
there always exists a family
of seminorms defined on
such that
(see [1, page 203]).
The following construction will be crucial. Suppose that is a
-bounded subset of
. For this set
we can select a number
for each
such that
where
Clearly,
is
-bounded,
-closed, absolutely convex and contains
. The linear span
of
in
is
The Minkowski functional of
is a norm
on
. Thus
is a normed space with
as its closed unit ball and
for each
(for details see [1–3]).
Let be a subset of a locally convex space
. Let
be mappings. A mapping
is called
-Lipschitz if there exists
such that
for any
and for all
. If
(resp.,
), then
is called an
-contraction (resp.,
-nonexpansive). A point
is a common fixed (coincidence) point of
and
if
(
). The set of coincidence points of
and
is denoted by
and the set of fixed points of
is denoted by
The pair
is called:
(1)commuting if for all
;
(2)-weakly commuting if for all
and for all
, there exists
such that
If
, then the maps are called weakly commuting [4];
(3)compatible [5] if for all ,
whenever
is a sequence such that
for some
in
;
(4)weakly compatible if they commute at their coincidence points, that is, whenever
.
Suppose that is
-starshaped with
and is both
- and
-invariant. Then
and
are called:
(5)-subcommuting on
if for all
and for all
, there exists a real number
such that
for each
. If
, then the maps are called
-subcommuting [6];
(6)-subweakly commuting on
(see [7]) if for all
and for all
, there exists a real number
such that
, where
and
;
(7)-commuting [8, 9] if
for all
, where
and
.
If then we define the set,
, of best
-approximations to
as
, for all
. A mapping
is called demiclosed at
if
converges weakly to
and
converges to
, then we have
. A locally convex space
satisfies Opial's condition if for every net
in
weakly convergent to
the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207503/MediaObjects/13663_2009_Article_1124_Equ1_HTML.gif)
holds for all and
.
In 1963, Meinardus [10] employed the Schauder fixed point theorem to prove a result regarding invariant approximation. Singh [11], Sahab et al. [12], and Jungck and Sessa [13] proved similar results in best approximation theory. Recently, Hussain and Khan [6] have proved more general invariant approximation results for 1-subcommuting maps which extend the work of Jungck and Sessa [13] and Al-Thagafi [14] to locally convex spaces. More recently, with the introduction of noncommuting maps to this area, Pant [15], Pathak et al. [16], Hussain and Jungck [7], and Jungck and Hussain [9] further extended and improved the above-mentioned results; details on the subject may be found in [17, 18]. For applications of fixed point results of nonlinear mappings in simultaneous best approximation theory and variational inequalities, we refer the reader to [19–21]. Fixed point theory of nonexpansive and noncommuting mappings is very rich in Banach spaces and metric spaces [13–17]. However, some partial results have been obtained for these mappings in the setup of locally convex spaces (see [22] and its references). It is remarked that the generalization of a known result in Banach space setting to the case of locally convex spaces is neither trivial nor easy (see, e.g., [2, 22]).
The following general common fixed point result is a consequence of Theorem 3.1 of Jungck [5], which will be needed in the sequel.
Theorem 1.1.
Let be a complete metric space, and let
be selfmaps of
. Suppose that
and
are continuous, the pairs
and
are compatible such that
. If there exists
such that for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207503/MediaObjects/13663_2009_Article_1124_Equ2_HTML.gif)
then there is a unique point in
such that
.
The aim of this paper is to extend the above well-known result of Jungck to locally convex spaces and establish general common fixed point theorems for generalized -nonexpansive subcompatible maps in the setting of a locally convex space. We apply our theorems to derive some results on the existence of common fixed points from the set of best approximations. We also establish common fixed point and approximation results for the newly defined class of Banach operator pairs. Our results extend and unify the work of Al-Thagafi [14], Chen and Li [23], Hussain [24], Hussain and Berinde [25], Hussain and Jungck [7], Hussain and Khan [6], Hussain and Rhoades [8], Jungck and Sessa [13], Khan and Akbar [19, 20], Pathak and Hussain [21], Sahab et al. [12], Sahney et al. [26], Singh [11, 27], Tarafdar [3], and Taylor [28].
2. Subcompatible Maps in Locally Convex Spaces
Recently, Khan et al. [29] introduced the class of subcompatible mappings as follows:
Definition 2.1.
Let be a
-starshaped subset of a normed space
. For the selfmaps
and
of
with
we define
where
and
. Now
and
are subcompatible if
for all sequences
.
We can extend this definition to a locally convex space by replacing the norm with a family of seminorms.
Clearly, subcompatible maps are compatible but the converse does not hold, in general, as the following example shows.
Example 2.2 (see [29]).
Let with usual norm and
Let
and
for all
. Let
Then
is
-starshaped with
. Note that
and
are compatible. For any sequence
in
with
, we have,
. However,
. Thus
and
are not subcompatible maps.
Note that -subweakly commuting and
-subcommuting maps are subcompatible. The following simple example reveals that the converse is not true, in general.
Example 2.3 (see [29]).
Let with usual norm and
Let
if
and
if
, and
if
and
if
. Then
is
-starshaped with
and
. Note that
and
are subcompatible but not
-weakly commuting for all
. Thus
and
are neither
-subweakly commuting nor
-subcommuting maps.
We observe in the following example that the weak commutativity of a pair of selfmaps on a metric space depends on the choice of the metric; this is also true for compatibility, -weak commutativity, and other variants of commutativity of maps.
Example 2.4 (see [30]).
Let with usual metric and
Let
and
. Then
and
. Thus the pair
is not weakly commuting on
with respect to usual metric. But if
is endowed with the discrete metric
, then
for
. Thus the pair
is weakly commuting on
with respect to discrete metric.
Next we establish a positive result in this direction in the context of linear topologies utilizing Minkowski functional; it extends [6, Lemma 2.1].
Lemma 2.5.
Let and
be compatible selfmaps of a
-bounded subset
of a Hausdorff locally convex space
. Then
and
are compatible on
with respect to
Proof.
By hypothesis, for each
whenever
for some
. Taking supremum on both sides, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207503/MediaObjects/13663_2009_Article_1124_Equ3_HTML.gif)
whenever
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207503/MediaObjects/13663_2009_Article_1124_Equ4_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207503/MediaObjects/13663_2009_Article_1124_Equ5_HTML.gif)
whenever
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207503/MediaObjects/13663_2009_Article_1124_Equ6_HTML.gif)
Hence whenever
as desired.
There are plenty of spaces which are not normable (see [31, page 113]). So it is natural and essential to consider fixed point and approximation results in the context of a locally convex space. An application of Lemma 2.5 provides the following general common fixed point result.
Theorem 2.6.
Let be a nonempty
-bounded,
-complete subset of a Hausdorff locally convex space
and let
and
be selfmaps of
Suppose that
and
are nonexpansive, the pairs
and
are compatible such that
. If there exists
such that for all
and for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207503/MediaObjects/13663_2009_Article_1124_Equ7_HTML.gif)
then there is a unique point in
such that
.
Proof.
Since the norm topology on has a base of neighbourhoods of
consisting of
-closed sets and
is
-sequentially complete, therefore
is
- sequentially complete in
see [3, the proof of Theorem 1.2]. By Lemma 2.5, the pairs
and
are
compatible maps of
. From (2.5) we obtain for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207503/MediaObjects/13663_2009_Article_1124_Equ8_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207503/MediaObjects/13663_2009_Article_1124_Equ9_HTML.gif)
As and
are nonexpansive on
-bounded set
,
and
are also nonexpansive with respect to
and hence continuous (cf. [6]). A comparison of our hypothesis with that of Theorem 1.1 tells that we can apply Theorem 1.1 to
as a subset of
to conclude that there exists a unique
in
such that
.
We now prove the main result of this section.
Theorem 2.7.
Let be a nonempty
-bounded,
-sequentially complete,
-starshaped subset of a Hausdorff locally convex space
and let
and
be selfmaps of
Suppose that
and
are affine and nonexpansive with
, and
. If the pairs
and
are subcompatible and, for all
and for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207503/MediaObjects/13663_2009_Article_1124_Equ10_HTML.gif)
then provided that one of the following conditions holds:
(i) is
-sequentially compact, and
is continuous (
stands for closure);
(ii) is
-sequentially compact, and
is continuous;
(iii) is weakly compact in
and
is demiclosed at
.
Proof.
Define by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207503/MediaObjects/13663_2009_Article_1124_Equ11_HTML.gif)
for all and a fixed sequence of real numbers
) converging to
. Then, each
is a selfmap of
and for each
,
since
and
are affine and
As
is affine and the pair
is subcompatible, so for any
with
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207503/MediaObjects/13663_2009_Article_1124_Equ12_HTML.gif)
Thus the pair is compatible on
for each
. Similarly, the pair
is compatible for each
.
Also by (2.8),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207503/MediaObjects/13663_2009_Article_1124_Equ13_HTML.gif)
for each ,
and
. By Theorem 2.6, for each
, there exists
such that
.
-
(i)
The compactness of
implies that there exists a subsequence
of
and a
such that
as
. Since
,
also converges to
Since
,
and
are continuous, we have
Thus
-
(ii)
Proof follows from (i).
-
(iii)
Since
is weakly compact, there is a subsequence
of
converging weakly to some
. But,
and
being affine and continuous are weakly continuous, and the weak topology is Hausdorff, so we have
. The set
is bounded, so
as
Now the demiclosedness of
at
guarantees that
and hence
.
Theorem 2.7 extends and improves [14, Theorem 2.2], [7, Theorems  2.2-2.3, and Corollaries 2.4–2.7], [13, Theorem 6], and the main results of Tarafdar [3] and Taylor [28](see also [6, Remarks 2.4]).
Theorem 2.8.
Let be a nonempty
-bounded,
-sequentially complete,
-starshaped subset of a Hausdorff locally convex space
and let
and
be selfmaps of
Suppose that
and
are affine and nonexpansive with
, and
. If the pairs
and
are subcompatible and
is
-nonexpansive, then
provided that one of the following conditions holds
(i) is
-sequentially compact;
(ii) is
-sequentially compact;
(iii) is weakly compact in
,
is demiclosed at
.
(iv) is weakly compact in an Opial space
.
Proof.
(i)–(iii) follow from Theorem 2.7.
-
(iv)
As in (iii) we have
and
as
If
, then by the Opial's condition of
and
-nonexpansiveness of
we get,
(2.12)
which is a contradiction. Thus and hence
.
As -subcommuting maps are subcompatible, so by Theorem 2.8, we obtain the following recent result of Hussain and Khan [6] without the surjectivity of
. Note that a continuous and affine map is weakly continuous, so the weak continuity of
is not required as well.
Corollary 2.9 ([6, Theorem 2.2]).
Let be a nonempty
-bounded,
-sequentially complete,
-starshaped subset of a Hausdorff locally convex space
and let
be selfmaps of
Suppose that
is affine and nonexpansive with
, and
. If the pair
is
-subcommuting and
is
-nonexpansive, then
provided that one of the following conditions holds:
(i) is
-sequentially compact;
(ii) is
-sequentially compact;
(iii) is weakly compact in
,
is demiclosed at
.
(iv) is weakly compact in an Opial space
.
The following theorem improves and extends the corresponding approximation results in [6–8, 11–14, 25, 27].
Theorem 2.10.
Let be a nonempty subset of a Hausdorff locally convex space
and let
be mappings such that
for some
and
. Suppose that
and
are affine and nonexpansive on
with
is
-bounded,
-sequentially complete,
-starshaped and
. If the pairs
and
are subcompatible and, for all
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207503/MediaObjects/13663_2009_Article_1124_Equ15_HTML.gif)
then , provided that one of the following conditions holds
(i) is
-sequentially compact, and
is continuous;
(ii) is
-sequentially compact, and
is continuous;
(iii) is weakly compact, and
is demiclosed at
.
Proof.
Let . Then for each
,
. Note that for any
,
It follows that the line segment and the set
are disjoint. Thus
is not in the interior of
and so
. Since
,
must be in
. Also since
,
and
satisfy (2.13), we have for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207503/MediaObjects/13663_2009_Article_1124_Equ16_HTML.gif)
Thus Consequently,
. Now Theorem 2.7 guarantees that
.
Remark 2.11.
One can now easily prove on the lines of the proof of the above theorem that the approximation results are similar to those of Theorems 2.11-2.12 due to Hussain and Jungck [7] in the setting of a Hausdorff locally convex space.
We define and denote by
the class of closed convex subsets of
containing
. For
, we define
for each
. It is clear that
.
The following result extends [14, Theorem 4.1] and [7, Theorem 2.14].
Theorem 2.12.
Let be selfmaps of a Hausdorff locally convex space
with
and
such that
. Suppose that
and
for all
and for each
where
is compact. Then
(i) is nonempty, closed, and convex,
(ii)
(iii) provided
and
are subcompatible, affine, and nonexpansive on
, and, for some
and for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207503/MediaObjects/13663_2009_Article_1124_Equ17_HTML.gif)
is continuous, the pairs
and
are subcompatible on
and satisfy for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207503/MediaObjects/13663_2009_Article_1124_Equ18_HTML.gif)
for all and for each
.
Proof.
Then there is a minimizing sequence in
such that
As
is compact so
has a convergent subsequence
with
(say) in
Now by using
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207503/MediaObjects/13663_2009_Article_1124_Equ19_HTML.gif)
we get for each ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207503/MediaObjects/13663_2009_Article_1124_Equ20_HTML.gif)
Hence Thus
is nonempty closed and convex.
(i)Follows from [7, Theorem 2.14].
(ii)By Theorem 2.7(i), so it follows that there exists
such that
Hence (iii) follows from Theorem 2.7(i).
3. Banach Operator Pair in Locally Convex Spaces
Utilizing similar arguments as above, the following result can be proved which extends recent common fixed point results due to Hussain and Rhoades [8, Theorem 2.1] and Jungck and Hussain [9, Theorem 2.1] to the setup of a Hausdorff locally convex space which is not necessarily metrizable.
Theorem 3.1.
Let be a
-bounded subset of a Hausdorff locally convex space
, and let
and let
be weakly compatible self-maps of
. Assume that
,
is
-sequentially complete, and
and
satisfy, for all
,
and for some
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207503/MediaObjects/13663_2009_Article_1124_Equ21_HTML.gif)
Then is a singleton.
As an application of Theorem 3.1, the analogue of all the results due to Hussain and Berinde [25], and Hussain and Rhoades [8] can be established for -commuting maps
and
defined on a
-bounded subset
of a Hausdorff locally convex space. We leave details to the reader.
Recently, Chen and Li [23] introduced the class of Banach operator pairs, as a new class of noncommuting maps and it has been further studied by Hussain [24], Ciric et al. [32], Khan and Akbar [19, 20], and Pathak and Hussain [21]. The pair is called a Banach operator pair, if the set
is
-invariant, namely,
. Obviously, commuting pair
is a Banach operator pair but converse is not true, in general; see [21, 23]. If
is a Banach operator pair, then
need not be a Banach operator pair (cf. [23, Example 1]).
Chen and Li [23] proved the following.
Theorem 3.2 ([23, Theorems 3.2-3.3]).
Let be a
-starshaped subset of a normed space
and let
,
be self-mappings of
Suppose that
is
-starshaped and
is continuous on
. If
is compact (resp.,
is weakly continuous,
is complete,
is weakly compact, and either
is demiclosed at
or
satisfies Opial's condition),
is a Banach operator pair, and
is
-nonexpansive on
, then
.
In this section, we extend and improve the above-mentioned common fixed point results of Chen and Li [23] in the setup of a Hausdorff locally convex space.
Lemma 3.3.
Let be a nonempty
-bounded subset of Hausdorff locally convex space
, and let
and
be self-maps of
If
is nonempty,
,
is
-sequentially complete, and
,
and
satisfy for all
and for some
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207503/MediaObjects/13663_2009_Article_1124_Equ22_HTML.gif)
then is singleton.
Proof.
Note that being a subset of
is
-sequentially complete. Further, for all
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207503/MediaObjects/13663_2009_Article_1124_Equ23_HTML.gif)
Hence is a generalized contraction on
and
. By Theorem 3.1 (with
= identity map),
has a unique fixed point
in
and consequently,
is singleton.
The following result generalizes [19, Theorem 2.3], [24, Theorem 2.11], and [21, Theorem 2.2] and improves [14, Theorem 2.2] and [13, Theorem 6].
Theorem 3.4.
Let be a nonempty
-bounded subset of Hausdorff locally convex (resp., complete) space
and let
and
be self-maps of
Suppose that
is
-starshaped,
(resp.,
),
is compact (resp.,
is weakly compact),
is continuous on
(resp.,
is demiclosed at
, where
stands for identity map) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207503/MediaObjects/13663_2009_Article_1124_Equ24_HTML.gif)
For all then
.
Proof.
Define by
for all
and a fixed sequence of real numbers
) converging to
. Since
is
-starshaped and
(resp.,
), so
) (resp.,
) for each
. Also by (3.4)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207503/MediaObjects/13663_2009_Article_1124_Equ25_HTML.gif)
for each and some
If is compact, for each
,
is
-compact and hence
-sequentially complete. By Lemma 3.3, for each
there exists
such that
The compactness of
implies that there exists a subsequence
of
such that
as
. Since
is a sequence in
and
, therefore
. Further,
. By the continuity of
, we obtain
. Thus,
proves the first case.
The weak compactness of implies that
is weakly compact and hence
-sequentially complete due to completeness of
. From Lemma 3.3, for each
there exists
such that
Moreover, we have
as
. The weak compactness of
implies that there is a subsequence
of
converging weakly to
as
. Since
is a sequence in
, therefore
. Also we have,
as
. If
is demiclosed at
, then
. Thus
Corollary 3.5.
Let be a nonempty
-bounded subset of Hausdorff locally convex (resp., complete) space
and let
and
be self-maps of
Suppose that
is
-starshaped, and
-closed (resp.,
-weakly closed),
is compact (resp.,
is weakly compact),
is continuous on
(resp.,
is demiclosed at
),
and
are Banach operator pairs and satisfy (3.4) for all
then
.
Let where
and
It is important to note here that
is always bounded.
Corollary 3.6.
Let be a Hausdorff locally convex (resp., complete) space and
and
be self-maps of
If
,
,
is
-starshaped,
(resp.,
],
is compact (resp.,
is weakly compact),
is continuous on
(resp.,
is demiclosed at
), and (3.4) holds for all
then
.
Corollary 3.7.
Let be a Hausdorff locally convex (resp., complete) space and
and
be self-maps of
If
,
,
is
-starshaped,
(resp.,
),
is compact (resp.,
is weakly compact),
is continuous on
(resp.,
is demiclosed at
), and (3.4) holds for all
then
.
Remark 3.8.
Khan and Akbar [19, Corollaries 2.4–2.8] and Chen and Li [23, Theorems 4.1 and 4.2] are particular cases of Corollaries 3.5 and 3.6.
The following result extends [14, Theorem 4.1], [7, Theorem 2.14], [19, Theorem 2.9], and [21, Theorems 2.7–2.11].
Theorem 3.9.
Let be self-maps of a Hausdorff locally convex space
. If
and
such that
,
is compact and
for all
, then
is nonempty, closed, and convex with
. If, in addition,
,
is
-starshaped,
,
is continuous on
and (3.4) holds for all
then
.
Proof.
We utilize Corollary 3.5 instead of Theorem 2.7 in the proof of Theorem 2.12.
Remark 3.10.
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The author A. R. Khan gratefully acknowledges the support provided by the King Fahd University of Petroleum & Minerals during this research. The authors would like to thank the referees for their valuable suggestions to improve the presentation of the paper.
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Akbar, F., Khan, A.R. Common Fixed Point and Approximation Results for Noncommuting Maps on Locally Convex Spaces. Fixed Point Theory Appl 2009, 207503 (2009). https://doi.org/10.1155/2009/207503
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DOI: https://doi.org/10.1155/2009/207503