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Coupled Fixed Point Theorems for Nonlinear Contractions Satisfied Mizoguchi-Takahashi's Condition in Quasiordered Metric Spaces
Fixed Point Theory and Applications volume 2010, Article number: 876372 (2010)
Abstract
The main aim of this paper is to study and establish some new coupled fixed point theorems for nonlinear contractive maps that satisfied Mizoguchi-Takahashi's condition in the setting of quasiordered metric spaces or usual metric spaces.
1. Introduction
Let be a metric space. For each
and
, let
. Denote by
the class of all nonempty subsets of
and
the family of all nonempty closed and bounded subsets of
. A function
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ1_HTML.gif)
is said to be the Hausdorff metric on induced by the metric
on
. A point
in
is a fixed point of a map
if
(when
is a single-valued map) or
(when
is a multivalued map). Throughout this paper we denote by
and
the set of positive integers and real numbers, respectively.
The existence of fixed point in partially ordered sets has been investigated recently in [1–11] and references therein. In [6, 8], Nieto and RodrÃguez-López used Tarski's theorem to show the existence of solutions for fuzzy equations and fuzzy differential equations, respectively. The existence of solutions for matrix equations or ordinary differential equations by applying fixed point theorems is presented in [2, 4, 7, 9, 10]. The authors in [3, 11] proved some fixed point theorems for a mixed monotone mapping in a metric space endowed with partial order and applied their results to problems of existence and uniqueness of solutions for some boundary value problems.
The various contractive conditions are important to find the existence of fixed point. There is a trend to weaken the requirement on the contraction. In 1989, Mizoguchi and Takahashi [12] proved the following interesting fixed point theorem for a weak contraction which is a partial answer of Problem 9 in Reich [13] (see also [14–16] and references therein).
Theorem MT. (Mizoguchi and Takahashi [12]).
Let be a complete metric space and
a map from
into
. Assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ2_HTML.gif)
for all ,
, where
is a function from
into
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ3_HTML.gif)
Then there exists such that
.
In fact, Mizoguchi-Takahashi's fixed point theorem is a generalization of Nadler's fixed point theorem [17, 18] which extended the Banach contraction principle (see, e.g., [18]) to multivalued maps, but its primitive proof is different. Recently, Suzuki [19] gave a very simple proof of Theorem MT.
The purpose of this paper is to present some new coupled fixed point theorems for weakly contractive maps that satisfied Mizoguchi-Takahashi's condition (i.e., for all
) in the setting of quasiordered metric spaces or usual metric spaces. Our results generalize and improve some results in [2, 7, 9] and references therein.
2. Generalized Bhaskar-Lakshmikantham's Coupled Fixed Point Theorems and Others
Let be a nonempty set and "
" a quasiorder (preorder or pseudo-order, i.e., a reflexive and transitive relation) on
. Then
is called a quasiordered set. A sequence
is called
-
(resp.,
-
) if
(resp.,
) for each
. Let
be a metric space with a quasi-order
(
for short). We endow the product space
with the metric
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ4_HTML.gif)
A map is said to be continuous at
if any sequence
with
implies
.
is said to be continuous on
if
is continuous at every point of
.
In this paper, we also endow the product space with the following quasi-order
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ5_HTML.gif)
Definition 2.1 . (see [2]).
Let be a quasiordered set and
a map. We say that
has the mixed monotone property on
if
is monotone nondecreasing in
and is monotone nonincreasing in
that is, for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ6_HTML.gif)
It is quite obvious that if has the mixed monotone property on
, then for any
,
with
(i.e.,
and
),
.
Definition 2.2 . (see [2]).
Let be a nonempty set and
a map. We call an element
a coupled fixed point of
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ7_HTML.gif)
Definition 2.3.
Let be a metric space with a quasi-order
A nonempty subset
of
is said to be
-
if every
-nondecreasing Cauchy sequence in
converges;
-
if every
-nonincreasing Cauchy sequence in
converges;
-
if it is both
-complete and
-
.
Definition 2.4 . (see [20]).
A function is said to be a
-
if it satisfies Mizoguchi-Takahashi's condition (i.e.,
for all
).
Remark 2.5.
Obviously, if
is defined by
where
, then
is a
-function.
If
is a nondecreasing function, then
is a
-function.
Notice that
is a
-function if and only if for each
there exist
and
such that
for all
Indeed, if
is a
-function, then
for all
So for each
there exists
such that
. Therefore we can find
such that
, and hence
for all
. The converse part is obvious.
The following lemmas are crucial to our proofs.
Lemma 2.6 . (see [20]).
Let be a
-function. Then
defined by
is also a
-function.
Proof.
Clearly, and
for all
. Let
be fixed. Since
is a
-function, there exist
and
such that
for all
Let
. Then
for all
and hence
is a
-function.
Lemma 2.7.
Let be a quasiordered set and
a map having the mixed monotone property on
. Let
,
. Define two sequences
and
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ8_HTML.gif)
for each . If
and
, then
is
-nondecreasing and
is
-nonincreasing.
Proof.
Since and
, by (2.5), and the mixed monotone property of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ9_HTML.gif)
Let and assume that
and
is already known. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ10_HTML.gif)
Hence, by induction, we prove that is
-nondecreasing and
is
-nonincreasing.
Theorem 2.8.
Let be a sequentially
-complete metric space and
a continuous map having the mixed monotone property on
. Assume that there exists a
-function
such that for any
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ11_HTML.gif)
If there exist such that
and
, then there exist
, such that
and
Proof.
By Lemma 2.6, we can define a -function
by
. Then
and
for all
. For any
, let
and
be defined as in Lemma 2.7. Then, by Lemma 2.7,
is
-nondecreasing and
is
-nonincreasing. So
and
for each
. By (2.8), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ12_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ13_HTML.gif)
For each , let
Then
By induction, we can obtain the following: for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ14_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ15_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ16_HTML.gif)
Since for all
, the sequence
is strictly decreasing in
from (2.13). Let
. Since
is a
-function, there exists
and
such that
for all
. Also, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ17_HTML.gif)
for all with
. So
for each
. Let
and
,
. We claim that
is a
-nondecreasing Cauchy sequence in
and
is a
-nonincreasing Cauchy sequence in
. Indeed, from our hypothesis, for each
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ18_HTML.gif)
Similarly,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ19_HTML.gif)
Hence we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ20_HTML.gif)
So it follows from (2.17) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ21_HTML.gif)
Let For
,
with
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ22_HTML.gif)
Since and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ23_HTML.gif)
So is a
-nondecreasing Cauchy sequence in
and
is a
-nonincreasing Cauchy sequence in
. By the sequentially
-completeness of
, there exist
,
such that
and
as
. Hence
and
as
.
Let be given. Since
is continuous at
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ24_HTML.gif)
whenever with
. Since
and
as
, for
, there exist
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ25_HTML.gif)
So, for each with
, by (2.22),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ26_HTML.gif)
and hence we have from (2.21) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ27_HTML.gif)
Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ28_HTML.gif)
Since is arbitrary,
or
. Similarly, we can also prove that
. The proof is completed.
Remark 2.9.
Theorem 2.8 generalizes and improves Bhaskar-Lakshmikantham's coupled fixed points theorem [2, Theorem  2.1] and some results in [7, 9].
Following a similar argument as in the proof of [2, Theorem  2.2] and applying Theorem 2.8, one can verify the following result where is not necessarily continuous.
Theorem 2.10.
Let be a sequentially
-complete metric space and
a map having the mixed monotone property on
. Assume that
any
-nondecreasing sequence
with
implies
for each
;
any
-nonincreasing sequence
with
implies
for each
;
there exists a
-function
such that for any
,
with
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ29_HTML.gif)
If there exist ,
such that
and
, then there exist
,
, such that
and
Remark 2.11.
[2, Theorem  2.2] is a special case of Theorem 2.10.
Similarly, we can obtain the generalizations of Theorems 2.4–2.6 in [2] for
-functions.
Finally, we discuss the following coupled fixed point theorem in (usual) complete metric spaces.
Theorem 2.12.
Let be a complete metric space and
a map. Assume that there exists a
-function
such that for any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ30_HTML.gif)
Then has a unique coupled fixed point in
; that is, there exists unique
such that
and
.
Proof.
Let be given. For any
define
and
. By our hypothesis, we know that
is continuous. Following the same argument as in the proof of Theorem 2.8, there exists
, such that
and
. We prove the uniqueness of the coupled fixed point of
. On the contrary, suppose that there exists
, such that
and
. Then we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ31_HTML.gif)
It follows from (2.28) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ32_HTML.gif)
a contradiction. The proof is completed.
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Acknowledgment
This paper is dedicated to Professor Wataru Takahashi in celebration of his retirement. This research was supported by the National Science Council of the Republic of China.
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Du, WS. Coupled Fixed Point Theorems for Nonlinear Contractions Satisfied Mizoguchi-Takahashi's Condition in Quasiordered Metric Spaces. Fixed Point Theory Appl 2010, 876372 (2010). https://doi.org/10.1155/2010/876372
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DOI: https://doi.org/10.1155/2010/876372