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Common Fixed Points of Generalized Contractive Hybrid Pairs in Symmetric Spaces
Fixed Point Theory and Applications volume 2009, Article number: 869407 (2009)
Abstract
Several fixed point theorems for hybrid pairs of single-valued and multivalued occasionally weakly compatible maps satisfying generalized contractive conditions are established in a symmetric space.
1. Introduction and Preliminaries
In 1968, Kannan [1] proved a fixed point theorem for a map satisfying a contractive condition that did not require continuity at each point. This paper was a genesis for a multitude of fixed point papers over the next two decades. Sessa [2] coined the term weakly commuting maps. Jungck [3] generalized the notion of weak commutativity by introducing compatible maps and then weakly compatible maps [4]. Al-Thagafi and Shahzad [5] gave a definition which is proper generalization of nontrivial weakly compatible maps which have coincidence points. Jungck and Rhoades [6] studied fixed point results for occasionally weakly compatible (owc) maps. Recently, Zhang [7] obtained common fixed point theorems for some new generalized contractive type mappings. Abbas and Rhoades [8] obtained common fixed point theorems for hybrid pairs of single-valued and multivalued owc maps defined on a symmetric space (see also [9]). For other related fixed point results in symmetric spaces and their applications, we refer to [10–15]. The aim of this paper is to obtain fixed point theorems involving hybrid pairs of single-valued and multivalued owc maps satisfying a generalized contractive condition in the frame work of a symmetric space.
Definition 1.1.
A symmetric on a set is a mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ1_HTML.gif)
A set together with a symmetric
is called a symmetric space.
We will use the following notations, throughout this paper, where is a symmetric space,
and
, and
is the class of all nonempty bounded subsets of
The diameter of
is denoted and defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ2_HTML.gif)
Clearly, For
and
we write
and
respectively. We appeal to the fact that
if and only if
for
Recall that is called a coincidence point (resp., common fixed point) of
and
if
(resp.,
).
Definition 1.2.
Maps and
are said to becompatible if
for each
and
whenever
is a sequence in
such that
(
) and
for some
[21].
Definition 1.3.
Maps and
are said to be weakly compatible if
whenever
Definition 1.4.
Maps and
are said to be owc if and only if there exists some point
in
such that
and
Example 1.5.
Consider with usual metric.
-
(a)
Define
and
as:
and
(1.3)
then and
are weakly compatible.
-
(b)
Define
by
(1.4)
It can be easily verified that is coincidence point of
and
but
and
are not weakly compatible there, as
. Hence
and
are not compatible. However, the pair
is occasionally weakly compatible, since the pair
is weakly compatible at
Assume that satisfies the following.
(i) and
for each
.
(ii) is nondecreasing on
Define, satisfies
above
Let satisfy the following.
(iii) for each
.
(iv) is nondecreasing on
Define, satisfies
above
For some examples of mappings which satisfy
we refer to [7].
2. Common Fixed Point Theorems
In the sequel we shall consider, which is defined on
where
stands for a real number to the left of
and assume that the mapping
satisfies
above.
Theorem 2.1.
Let be self maps of a symmetric space
, and let
be maps from
into
such that the pairs
and
are
If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ5_HTML.gif)
for each for which
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ6_HTML.gif)
then , and
have a unique common fixed point.
Proof.
By hypothesis there exist points in
such that
, and
. Also,
Therefore by (2.2) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ7_HTML.gif)
Now we claim that . For, otherwise, by (2.1),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ8_HTML.gif)
a contradiction and hence Obviously,
Thus (2.2) gives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ9_HTML.gif)
Next we claim that If not, then (2.1) implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ10_HTML.gif)
which is a contradiction and the claim follows. Similarly, we obtain Thus
, and
have a common fixed point. Uniqueness follows from (2.1).
Corollary 2.2.
Let be self maps of a symmetric space
and let
be maps from
into
such that the pairs
and
are
If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ11_HTML.gif)
for each for which
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ12_HTML.gif)
and , then
have a unique common fixed point.
Proof.
Since (2.7) is a special case of (2.1), the result follows from Theorem 2.1.
Corollary 2.3.
Let be self maps of a symmetric space
and let
be maps from
into
such that the pairs
and
are
. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ13_HTML.gif)
for each for which
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ14_HTML.gif)
where and
Then
, and
have a unique common fixed point.
Proof.
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ15_HTML.gif)
So, (2.9) is a special case of (2.1) and hence the result follows from Theorem 2.1.
Corollary 2.4.
Let be a self map on a symmetric space
and let
be a map from
into
such that
and
are
. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ16_HTML.gif)
for each for which
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ17_HTML.gif)
Then and
have a unique common fixed point.
Proof.
Condition (2.12) is a special case of condition (2.1) with and
Therefore the result follows from Theorem 2.1.
Theorem 2.5.
Let be self maps of a symmetric space
and let
be maps from
into
such that the pairs
and
are
If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ18_HTML.gif)
for each for which
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ19_HTML.gif)
where , and
, then
and
have a unique common fixed point.
Proof.
By hypothesis there exist points in
such that
and
Therefore by (2.15) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ20_HTML.gif)
Now we show that . Suppose not. Then condition (2.14) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ21_HTML.gif)
which is a contradiction and hence Note that,
Thus (2.15) gives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ22_HTML.gif)
Now we claim that If not, then condition (2.14) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ23_HTML.gif)
which is a contradiction, and hence the claim follows. Similarly, we obtain Thus
, and
have a common fixed point. Uniqueness follows easily from (2.14).
Define such that
) is nondecreasing in the 4th and 5th variables,
) if is such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ24_HTML.gif)
then .
Theorem 2.6.
Let be self maps of a symmetric space
and let
be maps from
into
such that the pairs
and
are
. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ25_HTML.gif)
for all for which
where
then
, and
have a unique common fixed point.
Proof.
By hypothesis there exist points in
such that
, and
Also,
First we show that
. Suppose not. Then condition (2.21) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ26_HTML.gif)
which, from implies that
this further implies that,
a contradiction. Hence the claim follows. Also,
Next we claim that
If not, then condition (2.21) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ27_HTML.gif)
which, from and
implies that
this further implies that
Hence the claim follows. Similarly, it can be shown that
which proves that
is a common fixed point of
, and
. Uniqueness follows from (2.21) and (
).
A control function is a continuous monotonically increasing function that satisfies
and,
if and only if
Let be such that
for each
Theorem 2.7.
Let be self maps of symmetric space
and let
be maps from
into
such that the pairs
and
are
If for a control function
one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ28_HTML.gif)
for each for which right-hand side of (2.24) is not equal to zero
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ29_HTML.gif)
then , and
have a unique common fixed point.
Proof.
By hypothesis there exist points in
such that
, and
Also, using the triangle inequality, we obtain
Therefore by (2.25) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ30_HTML.gif)
Now we show that . Suppose not. Then condition (2.24) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ31_HTML.gif)
which is a contradiction. Therefore which further implies that,
Hence the claim follows. Again,
Therefore by (2.25) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ32_HTML.gif)
Next we claim that If not, then condition (2.24) implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ33_HTML.gif)
which is a contradiction. Therefore which further implies that
Hence the claim follows. Similarly, it can be shown that
which proves the result.
Set is continuous and nondecreasing mapping with
if and only if
The following theorem generalizes [16, Theorem ].
Theorem 2.8.
Let be self maps of a symmetric space
, and let
be maps from
into
such that the pairs
and
are
If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ34_HTML.gif)
for all , for which right-hand side of (2.30) is not equal to zero
where
then
, and
have a unique common fixed point.
Proof.
By hypothesis there exist points in
such that
, and
Also, using the triangle inequality, we obtain,
Now we claim that
. For, otherwise, by (2.30),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ35_HTML.gif)
which is a contradiction. Therefore Hence the claim follows. Again,
Now we claim that
If not, then condition (2.30) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ36_HTML.gif)
which is a contradiction, and hence the claim follows. Similarly, it can be shown that which, proves that
is a common fixed point of
, and
. Uniqueness follows easily from (2.30).
Example 2.9.
Let Define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ37_HTML.gif)
Note that is symmetric but not a metric on
.
Define by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ38_HTML.gif)
and as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F869407/MediaObjects/13663_2009_Article_1186_Equ39_HTML.gif)
Clearly, but
and
but
they show that
is not weakly compatible. On the other hand,
gives that
Hence
is occasionally weakly compatible. Note that
,
,
, and
they imply that
is not weakly compatible
Now
gives that
. Hence
is occasionally weakly compatible. As
and
so
is the unique common fixed point of
, and
Remark 2.10 s.
Weakly compatible maps are occasionally weakly compatible but converse is not true in general. The class of symmetric spaces is more general than that of metric spaces. Therefore the following results can be viewed as special cases of our results:
(a)([17, Theorem ] and [18, Theorem
]) are special cases of Theorem 2.7.
(b)[19, Theorem ], [20, Theorem
], [21, Theorem
], and [22, Theorem
] are special cases of Corollary 2.2. Moreover, [23, Theorem
] and [24, Theorem
] also become special cases of Corollary 2.2.
-
(c)
([25, Theorem
]) is a special case of Theorem 2.1. Theorem 2.1 also generalizes ([26, Theorem
]) and ( [27, Theorems
and
]).
(d)[28, Theorem ] becomes special case of Corollary 2.4.
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Acknowledgments
The authors are thankful to the referees for their critical remarks to improve this paper. The second author gratefully acknowledges the support provided by King Fahad University of Petroleum and Minerals during this research.
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Abbas, M., Khan, A.R. Common Fixed Points of Generalized Contractive Hybrid Pairs in Symmetric Spaces. Fixed Point Theory Appl 2009, 869407 (2009). https://doi.org/10.1155/2009/869407
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DOI: https://doi.org/10.1155/2009/869407