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Some Common Fixed Point Theorems for Weakly Compatible Mappings in Metric Spaces
Fixed Point Theory and Applications volume 2009, Article number: 804734 (2009)
Abstract
We establish a common fixed point theorem for weakly compatible mappings generalizing a result of Khan and Kubiaczyk (1988). Also, an example is given to support our generalization. We also prove common fixed point theorems for weakly compatible mappings in metric and compact metric spaces.
1. Introduction
In the last years, fixed point theorems have been applied to show the existence and uniqueness of the solutions of differential equations, integral equations and many other branches mathematics (see, e.g., [1–3]). Some common fixed point theorems for weakly commuting, compatible, -compatible and weakly compatible mappings under different contractive conditions in metric spaces have appeared in [4–15]. Throughout this paper,
is a metric space.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F804734/MediaObjects/13663_2008_Article_1180_Equ1_HTML.gif)
For all , we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F804734/MediaObjects/13663_2008_Article_1180_Equ2_HTML.gif)
where , for some
and
, for some
.
If for some
, we denote
,
and
for
,
and
, respectively. Also, if
, then one can deduce that
.
It follows immediately from the definition of that, for every
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F804734/MediaObjects/13663_2008_Article_1180_Equ3_HTML.gif)
We need the following definitions and lemmas.
Definition 1.1 (see [16]).
A sequence of nonempty subsets of
is said to be convergent to
if:
(i)each point in
is the limit of a convergent sequence
, where
is in
for
(
:= the set of all positive integers),
(ii)for arbitrary , there exists an integer
such that
for
, where
denotes the set of all points
in
for which there exists a point
in
, depending on
such that
.
is then said to be the limit of the sequence
.
Definition 1.2 (see [9]).
A set-valued function is said to be continuous if for any sequence
in
with
, it yields
.
Lemma 1.3 (see [16]).
If and
are sequences in
converging to
and
in
, respectively, then the sequence
converges to
.
Lemma 1.4 (see [16]).
Let be a sequence in
and let
be a point in
such that
. Then the sequence
converges to the set
in
.
Lemma 1.5 (see [9]).
For any , it yields that
.
Lemma 1.6 (see [17]).
Let be a right continuous function such that
for every
. Then,
for every
, where
denotes the
-times repeated composition of
with itself.
Definition 1.7 (see [15]).
The mappings and
are weakly commuting on
if
and
for all
.
Definition 1.8 (see [13]).
The mappings and
are said to be
-compatible if
whenever
is a sequence in
such that
,
and
for some
.
Definition 1.9 (see [13]).
The mappings and
are weakly compatible if they commute at coincidence points, that is, for each point
such that
, then
(note that the equation
implies that
is a singleton).
If is a single-valued mapping, then Definition 1.7 (resp., Definitions 1.8 and 1.9) reduces to the concept of weak commutativity (resp., compatibility and weak compatibility) for single-valued mappings due to Sessa [18] (resp., Jungck [11, 12]).
It can be seen that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F804734/MediaObjects/13663_2008_Article_1180_Equ4_HTML.gif)
but the converse of these implications may not be true (see, [13, 15]).
Throughtout this paper, we assume that is the set of all functions
satisfying the following conditions:
(i) is upper semi-continuous continuous at a point
from the right, and non-decreasing in each coodinate variable,
(ii)For each ,
.
Theorem 1.10 (see [19]).
Let be mappings of a complete metric space
into
and
be a mapping of
into itself such that
and
are continuous,
,
,
,
and for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F804734/MediaObjects/13663_2008_Article_1180_Equ5_HTML.gif)
where satisfies (i) and
for each
, and
,
with
. Then
and
have a unique common fixed point
such that
.
In the present paper, we are concerned with the following:
(1)replacing the commutativity of the mappings in Theorem 1.10 by the weak compatibility of a pair of mappings to obtain a common fixed point theorem metric spaces without the continuity assumption of the mappings,
(2)giving an example to support our generalization of Theorem 1.10,
(3)establishing another common fixed point theorem for two families of set-valued mappings and two single-valued mappings,
(4)proving a common fixed point theorem for weakly compatible mappings under a strict contractive condition on compact metric spaces.
2. Main Results
In this section, we establish a common fixed point theorem in metric spaces generalizing Theorems 1.10. Also, an example is introduced to support our generalization. We prove a common fixed point theorem for two families of set-valued mappings and two single-valued mappings. Finally, we establish a common fixed point theorem under a strict contractive condition on compact metric spaces.
First we state and prove the following.
Theorem 2.1.
Let be two sefmaps of a metric space
and let
be two set-valued mappings with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F804734/MediaObjects/13663_2008_Article_1180_Equ6_HTML.gif)
Suppose that one of and
is complete and the pairs
and
are weakly compatible. If there exists a function
such that for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F804734/MediaObjects/13663_2008_Article_1180_Equ7_HTML.gif)
then there is a point such that
.
Proof.
Let be an arbitrary point in
. By (2.1), we choose a point
in
such that
and for this point
there exists a point
in
such that
. Continuing this manner we can define a sequence
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F804734/MediaObjects/13663_2008_Article_1180_Equ8_HTML.gif)
for . For simplicity, we put
for
. By (2.2) and (2.3), we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F804734/MediaObjects/13663_2008_Article_1180_Equ9_HTML.gif)
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F804734/MediaObjects/13663_2008_Article_1180_Equ10_HTML.gif)
This contradiction demands that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F804734/MediaObjects/13663_2008_Article_1180_Equ11_HTML.gif)
Similarly, one can deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F804734/MediaObjects/13663_2008_Article_1180_Equ12_HTML.gif)
So, for each , we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F804734/MediaObjects/13663_2008_Article_1180_Equ13_HTML.gif)
where . By (2.8) and Lemma 1.6, we obtain that
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F804734/MediaObjects/13663_2008_Article_1180_Equ14_HTML.gif)
then . Therefore,
is a Cauchy sequence.
Let be an arbitrary point in
for
. Then
and
is a Cauchy sequence. We assume without loss of generality that
is complete. Let
be the sequence defined by (2.3). But
for all
. Hence, we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F804734/MediaObjects/13663_2008_Article_1180_Equ15_HTML.gif)
as . So,
is a Cauchy sequence. Hence,
for some
. But
by (2.3), so that
. Consequently,
. Moreover, we have, for
, that
. Therefore,
. So, we have by Lemma 1.4 that
. In like manner it follows that
and
.
Since, for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F804734/MediaObjects/13663_2008_Article_1180_Equ16_HTML.gif)
and as
, we get from Lemma 1.3 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F804734/MediaObjects/13663_2008_Article_1180_Equ17_HTML.gif)
This is absurd. So, . But
, so
such that
. If
,
, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F804734/MediaObjects/13663_2008_Article_1180_Equ18_HTML.gif)
We must conclude that .
Since and the pair
is weakly compatible, so
. Using the inequality (2.2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F804734/MediaObjects/13663_2008_Article_1180_Equ19_HTML.gif)
This contradiction demands that . Similarly, if the pair
is weakly compatible, one can deduce that
. Therefore, we get that
.
The proof, assuming the completeness of , is similar to the above.
To see that is unique, suppose that
. If
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F804734/MediaObjects/13663_2008_Article_1180_Equ20_HTML.gif)
which is inadmissible. So, .
Now, we give an example to show the greater generality of Theorem 2.1 over Theorem 1.10.
Example 2.2.
Let endowed with the Euclidean metric
. Assume that
for every
. Define
and
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F804734/MediaObjects/13663_2008_Article_1180_Equ21_HTML.gif)
We have that and
. Moreover,
if
. If
, then
and
. So, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F804734/MediaObjects/13663_2008_Article_1180_Equ22_HTML.gif)
for all . It is clear that
is a complete metric space. Since
is a closed subset of
, so
is complete. We note that
is a
-compatible pair and therefore a weakly compatible pair. Also,
and
, that is,
and
are weakly compatible. On the other hand, if
, so that
even though
, that is,
is not a
-compatible pair. We know that
is the unique common fixed point of
and
. Hence the hypotheses of Theorem 2.1 are satisfied. Theorem 1.10 is not applicable because
for all
, and the maps I, J and G are not continuous at
.
In Theorem 2.1, if the mappings and
are replaced by
and
,
where
is an index set, we obtain the following.
Theorem 2.3.
Let be a metric space, and let
be selfmaps of
, and for
,
be set-valued mappings with
and
. Suppose that one of
and
is complete and for
the pairs
and
are weakly compatible. If there exists a function
such that, for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F804734/MediaObjects/13663_2008_Article_1180_Equ23_HTML.gif)
then there is a point such that
for each
.
Proof.
Using Theorem 2.1, we obtain for any , there is a unique point
such that
and
. For all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F804734/MediaObjects/13663_2008_Article_1180_Equ24_HTML.gif)
This yields that .
Inspired by the work of Chang [9], we state the following theorem on compact metric spaces.
Theorem 2.4.
Let be a compact metric space,
selfmaps of
set-valued functions with
and
Suppose that the pairs
,
are weakly compatible and the functions
,
are continuous. If there exists a function
, and for all
, the following inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F804734/MediaObjects/13663_2008_Article_1180_Equ25_HTML.gif)
holds whenever the right-hand side of (2.20) is positive, then there is a unique point in
such that
.
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Ahmed, M.A. Some Common Fixed Point Theorems for Weakly Compatible Mappings in Metric Spaces. Fixed Point Theory Appl 2009, 804734 (2009). https://doi.org/10.1155/2009/804734
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DOI: https://doi.org/10.1155/2009/804734