- Research Article
- Open access
- Published:
Common Fixed Points of Weakly Contractive and Strongly Expansive Mappings in Topological Spaces
Journal of Inequalities and Applications volume 2010, Article number: 746045 (2010)
Abstract
Using the notion of weakly -contractive mappings, we prove several new common fixed point theorems for commuting as well as noncommuting mappings on a topological space X. By analogy, we obtain a common fixed point theorem of mappings which are strongly
-expansive on X.
1. Introduction
It is well known that if is a compact metric space and
is a weakly contractive mapping (see Section 2 for the definition), then
has a fixed point in
(see [1, p. 17]). In late sixties, Furi and Vignoli [2] extended this result to
-condensing mappings acting on a bounded complete metric space (see [3] for the definition). A generalized version of Furi-Vignoli's theorem using the notion of weakly
-contractive mappings acting on a topological space was proved in [4] (see also [5]).
On the other hand, in [6] while examining KKM maps, the authors introduced a new concept of lower (upper) semicontinuous function (see Definition 2.1, Section 2) which is more general than the classical one. In [7], the authors used this definition of lower semicontinuity to redefine weakly -contractive mappings and strongly
-expansive mappings (see Definition 2.6, Section 2 ) to formulate and prove several results for fixed points.
In this article, we have used the notions of weakly -contractive mappings (
where
is a topological space) to prove a version of the above-mentioned fixed point theorem [7, Theorem
] for common fixed points (see Theorem 3.1). We also prove a common fixed point theorem under the assumption that certain iteration of the mappings in question is weakly
-contractive. As a corollary to this fact, we get an extension (to common fixed points) of [7, Theorem
] for Banach spaces with a quasimodulus endowed with a suitable transitive binary relation. The most interesting result of this section is Theorem 3.8 wherein the strongly
-expansive condition on
(with some other conditions) implies that
and
have a unique common fixed point.
In Section 4, we define a new class of noncommuting self-maps and prove some common fixed point results for this new class of mappings.
2. Preliminaries
Definition 2.1 (see [6]).
Let be a topological space. A function
is said to be lower semi-continuous from above (lsca) at
if for any net
convergent to
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ1_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ2_HTML.gif)
A function is said to be lsca if it is lsca at every
Example 2.2.
-
(i)
Let
. Define
by
(2.3)
Let be a sequence of nonnegative terms such that
converges to
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ4_HTML.gif)
Similarly, if is a sequence in
of negative terms such that
converges to
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ5_HTML.gif)
Thus, is lsca at
-
(ii)
Every lower semi-continuous function is lsca but not conversely. One can check that the function
with
defined below is lsca at
but is not lower semi-continuous at
:
(2.6)
The following lemmas state some properties of lsca mappings. The first one is an analogue of Weierstrass boundedness theorem and the second one is about the composition of a continuous function and a function lsca.
Lemma 2.3 (see [6]).
Let be a compact topological space and
a function lsca. Then there exists
such that
Lemma 2.4 (see [7]).
Let be a topological space and
a continuous function. If
is a function lsca, then the composition function
is also lsca.
Proof.
Fix and consider a net
in
convergent to
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ7_HTML.gif)
Set and
Then since
is continuous,
and
lsca implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ8_HTML.gif)
with for
Thus
and
is lsca.
Remark 2.5 (see [6]).
Let be topological space. Let
be a continuous function and
lsca. Then
defined by
is also lsca. For this, let
be a net in
convergent to
Since
is continuous,
Suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ9_HTML.gif)
Then since is lsca, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ10_HTML.gif)
Definition 2.6 (see [7]).
Let be a topological space and
be lsca. The mapping
is said to be
(i)weakly -contractive if
for all
such that
(ii)strongly -expansive if
for all
such that
If is a metric space with metric
and
, then we call
respectively, weakly contractive and strongly expansive.
Let . The set of fixed points of
(resp.,
) is denoted by
(resp.,
). A point
is a coincidence point (common fixed point) of
and
if
  
. The set of coincidence points of
and
is denoted by
Maps
are called (1) commuting if
for all
, (2) weakly compatible [8] if they commute at their coincidence points, that is, if
whenever
, and (3) occasionally weakly compatible [9] if
for some
3. Common Fixed Point Theorems for Commuting Maps
In this section we extend some results in [7] to the setting of two mappings having a unique common fixed point.
Theorem 3.1.
Let be a topological space,
, and
self-mappings such that for every countable set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ11_HTML.gif)
and ,
commute on
If
(i) is continuous and weakly
-contractive or
(ii) is continuous and weakly
-contractive with
,
then and
have a unique common fixed point.
Proof.
Let and define the sequence
by setting
for
Let
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ12_HTML.gif)
so by hypothesis is compact. Define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ13_HTML.gif)
Now if or
is continuous and since
is lsca, then by Remark 2.5,
is lsca. So by Lemma 2.3,
has a minimum at, say,
(i   Suppose that
is continuous and weakly
-contractive. Then
as
is continuous. Now observe that if
is continuous, and
, then
We show that
Suppose that
; then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ14_HTML.gif)
a contradiction to the minimality of at
Having
one can see that
Indeed, if
then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ15_HTML.gif)
a contradiction.
-
(ii)
  Suppose that
  is continuous and weakly
-contractive with
. Then
as
is continuous. Put
; then
is continuous, and
implies that
. We claim that
for otherwise we will have
(3.6)
which is a contradiction. Hence the claim follows.
Now suppose that then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ17_HTML.gif)
a contradiction, hence
In both cases, uniqueness follows from the contractive conditions: suppose there exists such that
Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ18_HTML.gif)
which is false. Thus and
have a unique common fixed point.
If , then Theorem 3.1(i) reduces to [7, Theorem
].
Corollary 3.2 (see [7, Theorem ]).
Let be a topological space,
, and
continuous and weakly
-contractive. If the implication
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ19_HTML.gif)
holds for every countable set then
has a unique fixed point.
Example 3.3.
Let be the Banach space of all null real sequences. Define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ20_HTML.gif)
Let and
a sequence such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ21_HTML.gif)
with as
Define the mappings
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ22_HTML.gif)
where and
are such that for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ23_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ24_HTML.gif)
and for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ25_HTML.gif)
We verify the hypothesis of Theorem 3.1.
(i)Observe that and
are, clearly, continuous by their definition.
(ii)For we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ26_HTML.gif)
Since the sequences and
are null sequences, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ27_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ28_HTML.gif)
This implies that and
are weakly contractive. Thus
and
are continuous and weakly contractive. Next suppose that for any countable set
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ29_HTML.gif)
then by the definition of , we can consider
Hence closure of
being closed subset of a compact set is compact. Also
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ30_HTML.gif)
So by Theorem 3.1, and
have a unique common fixed point.
Corollary 3.4.
Let ( be a metric space,
, and
self-mappings such that for every countable set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ31_HTML.gif)
and ,
commute on
If
(i) is continuous and weakly contractive or
(ii) is continuous and weakly contractive with
,
then and
have a unique common fixed point.
Proof.
It is immediate from Theorem 3.1 with .
Corollary 3.5.
Let be a compact metric space,
, and
self-mappings such that for every countable set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ32_HTML.gif)
and ,
commute on
If
(i) is continuous and weakly contractive or
(ii) is continuous and weakly
-contractive with
then and
have a unique common fixed point.
Proof.
It is immediate from Theorem 3.1.
Theorem 3.6.
Let be a topological space,
, and
self-mappings such that for every countable set
-
(1)
is relatively compact;
-
(2)
is relatively compact for some
;
-
(3)
is relatively compact for some
.
And ,
commute on
Further, if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ33_HTML.gif)
then and
have a unique common fixed point.
Proof.
Part (3): we proceed as in Theorem 3.1. Let for some
and define the sequence
by setting
for
Let
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ34_HTML.gif)
so by hypothesis (3), is compact. Define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ35_HTML.gif)
Now since is lsca and if
or
is continuous, then by Remark 2.5  
would be lsca and hence by Lemma 2.3,
would have a minimum, say, at
(i)Suppose thatis continuous and
weakly
-contractive. Then
as
is continuous. Now observe that
is continuous, and
implies that
is continuous and
and so
for some
. We show that
. Suppose that
for any
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ36_HTML.gif)
a contradiction to the minimality of at
Therefore,
for some
. One can check that
. Suppose that
, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ37_HTML.gif)
a contradiction. Thus is a common fixed point of
and
and hence of
and
-
(ii)
Suppose that
  is continuous and
  weakly
-contractive with
. Then
as
is continuous. Put
Then
continuous and
imply that
. We claim that
for otherwise we will have
(3.28)
which is a contradiction. Hence the claim follows.
Now suppose that then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ39_HTML.gif)
a contradiction, hence Thus
is a common fixed point of
and
and hence of
and
Now we establish the uniqueness of Suppose there exists
such that
for some
. Now if
is continuous and
is weakly
- contractive, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ40_HTML.gif)
and if is continuous and
is weakly
- contractive, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ41_HTML.gif)
which is false. Thus and
have a unique common fixed point which obviously is a unique common fixed point of
and
.
Part (2). The conclusion follows if we set in part (3).
Part (1). The conclusion follows if we set and
in part (3).
A nice consequence of Theorem 3.6 is the following theorem where is taken as a Banach space equipped with a transitive binary relation.
Theorem 3.7.
Let be a Banach space with a transitive binary relation
such that
for
with
Suppose, further, that the mappings
are such that the following conditions are satisfied:
(i) and
for all
(ii) then
(iii)A is bounded linear operator and for some
and for all
such that
with
If either
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ42_HTML.gif)
for all with
commuting on
and if one of the conditions, (1)–(3), of Theorem 3.6 holds, then
and
have a unique common fixed point.
Proof.
-
(a)
Suppose that
for all
with
commuting on
and
is contractive
Then we have
(3.33)
Next
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ44_HTML.gif)
Therefore, after -steps,
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ45_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ46_HTML.gif)
So is weakly contractive. Since
is continuous (as
is bounded and
contractive) by Theorem 3.6,
and
have a unique common fixed point.
b
Suppose that
and
is contractive for all
with
commuting on
and
being contractive
The proof now follows if we mutually interchange
in (a) above
Theorem 3.8.
Let be a topological space,
with
closed and
Let
be mappings such that for every countable set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ47_HTML.gif)
and ,
commute on
If
is a homeomorphism and strongly
-expansive, then
and
have a unique common fixed point.
Proof.
Suppose that is a homeomorphism and strongly
-expansive. Let
with
Then there exists
such that
and
or
and
Since
is strongly
-expansive, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ48_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ49_HTML.gif)
So is a weakly
-contractive mapping. Choose any countable subset
of
and set
Suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ50_HTML.gif)
Then for some
and we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ51_HTML.gif)
So by hypothesis is compact and since
is a homeomorphism,
is compact. Since
for every
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ52_HTML.gif)
for every Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ53_HTML.gif)
and for every
Since
is continuous and weakly
-contractive, by Theorem 3.1, the mappings
and
have a unique common fixed point, say,
. Since
implies that
so
is a unique common fixed point of
and
The following example illustrates Theorem 3.8.
Example 3.9.
Let with the River metric
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ54_HTML.gif)
where ,
, and
denotes the Euclidean metric on
Then
is a topological space with a topology induced by the metric
. Consider the sets
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ55_HTML.gif)
Let the mappings be defined by
and
for
Then
is clearly a homeomorphism and for an arbitrary countable subset
of
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ56_HTML.gif)
If and only if Indeed, if
such that
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ57_HTML.gif)
Further, for every
Set
where
is the Radial metric defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ58_HTML.gif)
and ;
Now for
since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ59_HTML.gif)
is strongly
-expansive. Also
is lower semi-continuous and hence lsca. Thus all the conditions of Theorem 3.8 are satisfied and
and
have a unique common fixed point.
4. Occasionally Banach Operator Pair and Weak F-Contractions
In this section, we define a new class of noncommuting self-maps and prove some common fixed point results for this new class of maps.
The pair is called a Banach operator pair [10] if the set
is
-invariant, namely,
. Obviously, commuting pair
is a Banach operator pair but converse is not true, in general; see [10–13]. If
is a Banach operator pair, then
need not be a Banach operator pair.
Definition 4.1.
The pair is called occasionally Banach operator pair if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ60_HTML.gif)
Clearly, Banach operator pair (BOP) is occasionally Banach operator pair (OBOP) but not conversely, in general.
Example 4.2.
Let with usual norm. Define
by
and
, for
and
.
and
. Obviously
is OBOP but not BOP as
. Further,
is not weakly compatible and hence not commuting.
Example 4.3.
Let with usual norm and
. Define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ61_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ62_HTML.gif)
Here and
implies that
is not Banach operator pair. Similarly,
is not Banach operator pair. Further,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ63_HTML.gif)
imply that is OBOP. Further, note that
and
. Hence
is not occasionally weakly compatible pair.
Definition 4.4.
Let be a nonempty set and
be a mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ64_HTML.gif)
For a space satisfying (4.5) and
the diameter of
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ65_HTML.gif)
Here we extend this concept to the space satisfying condition (4.5).
Definition 4.5.
Let be a space satisfying (4.5). The pair
is called occasionally Banach operator pair on
iff there is a point
in
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ66_HTML.gif)
Theorem 4.6.
Let be a topological space,
, and
self-mappings such that for every countable set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ67_HTML.gif)
If is continuous and weakly
-contractive,
satisfies condition (4.5), and the pair
is occasionally Banach operator pair, then
and
have a unique common fixed point.
Proof.
By Corollary 3.2, is a singleton. Let
. Then, by our hypothesis,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ68_HTML.gif)
Therefore, . That is,
is unique common fixed point of
and
.
Corollary 4.7.
Let ( be a metric space,
, and
self-mappings such that for every countable set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ69_HTML.gif)
If is continuous and weakly contractive and the pair
is occasionally Banach operator pair, then
and
have a unique common fixed point.
Proof.
It is immediate from Theorem 4.6 with .
Corollary 4.8.
Let be a compact metric space,
, and
self-mappings such that for every countable set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ70_HTML.gif)
If is continuous and weakly contractive and the pair
is occasionally Banach operator pair, then
and
have a unique common fixed point.
Proof.
It is immediate from Theorem 4.6.
Theorem 4.6 holds for a Banach operator pair without condition (4.5) as follows.
Theorem 4.9.
Let be a topological space,
, and
self-mappings such that for every countable set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ71_HTML.gif)
If is continuous and weakly
-contractive and the pair
is a Banach operator pair, then
and
have a unique common fixed point.
Proof.
By Corollary 3.2, is a singleton. Let
. As
is a Banach operator pair, by definition
. Thus
and hence
. That is,
is unique common fixed point of
and
.
Corollary 4.10.
Let ( be a metric space,
, and
self-mappings such that for every countable set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F746045/MediaObjects/13660_2010_Article_2236_Equ72_HTML.gif)
If is continuous and weakly contractive and the pair
is a Banach operator pair, then
and
have a unique common fixed point.
References
Dugundji J, Granas A: Fixed Point Theory, Vol. 1. PWN, Warsaw, Poland; 1982:209.
Furi M, Vignoli A: A fixed point theorem in complete metric spaces. Bolletino della Unione Matematica Italiana 1969, 4: 505–509.
Bugajewski D: Some remarks on Kuratowski's measure of noncompactness in vector spaces with a metric. Commentationes Mathematicae 1992, 32: 5–9.
Bugajewski D: Fixed point theorems in locally convex spaces. Acta Mathematica Hungarica 2003, 98(4):345–355. 10.1023/A:1022842429470
Ćirić LB: Coincidence and fixed points for maps on topological spaces. Topology and Its Applications 2007, 154(17):3100–3106. 10.1016/j.topol.2007.08.004
Chen YQ, Cho YJ, Kim JK, Lee BS: Note on KKM maps and applications. Fixed Point Theory and Applications 2006, 2006:-9.
Bugajewski D, Kasprzak P: Fixed point theorems for weakly
-contractive and strongly
-expansive mappings. Journal of Mathematical Analysis and Applications 2009, 359(1):126–134. 10.1016/j.jmaa.2009.05.024
Jungck G: Common fixed points for noncontinuous nonself maps on nonmetric spaces. Far East Journal of Mathematical Sciences 1996, 4(2):199–215.
Jungck G, Rhoades BE: Fixed point theorems for occasionally weakly compatible mappings. Fixed Point Theory 2006, 7(2):287–296.
Chen J, Li Z: Common fixed-points for Banach operator pairs in best approximation. Journal of Mathematical Analysis and Applications 2007, 336(2):1466–1475. 10.1016/j.jmaa.2007.01.064
Hussain N: Common fixed points in best approximation for Banach operator pairs with Ćirić type
-contractions. Journal of Mathematical Analysis and Applications 2008, 338(2):1351–1363. 10.1016/j.jmaa.2007.06.008
Hussain N, Cho YJ: Weak contractions, common fixed points, and invariant approximations. Journal of Inequalities and Applications 2009, 2009:-10.
Pathak HK, Hussain N: Common fixed points for Banach operator pairs with applications. Nonlinear Analysis 2008, 69: 2788–2802. 10.1016/j.na.2007.08.051
Acknowledgments
N. Hussain thanks the Deanship of Scientific Research, King Abdulaziz University for the support of the Research Project no. (3-74/430). A. R. Khan is grateful to the King Fahd University of Petroleum Minerals and SABIC for the support of the Research Project no. SB
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Shah, M.H., Hussain, N. & Khan, A.R. Common Fixed Points of Weakly Contractive and Strongly Expansive Mappings in Topological Spaces. J Inequal Appl 2010, 746045 (2010). https://doi.org/10.1155/2010/746045
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DOI: https://doi.org/10.1155/2010/746045