- Research Article
- Open access
- Published:
Nonlinear Contractive Conditions for Coupled Cone Fixed Point Theorems
Fixed Point Theory and Applications volume 2010, Article number: 190606 (2010)
Abstract
We establish some new coupled fixed point theorems for various types of nonlinear contractive maps in the setting of quasiordered cone metric spaces which not only obtain several coupled fixed point theorems announced by many authors but also generalize them under weaker assumptions.
1. Introduction
The existence of fixed point in partially ordered sets has been studied and investigated recently in [1–13] and references therein. Since the various contractive conditions are important in metric fixed point theory, there is a trend to weaken the requirement on contractions. Nieto and RodrÃguez-López in [8, 10] 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 are presented in [2, 6, 9, 11, 12]. In [3, 13], the authors 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.
In 2006, Bhaskar and Lakshmikantham [2] first proved the following interesting coupled fixed point theorem in partially ordered metric spaces.
Theorem BL. (Bhaskar and Lakshmikantham).
Let be a partially ordered set and
a metric on
such that
is a complete metric space. Let
be a continuous mapping having the mixed monotone property on
. Assume that there exists a
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ1_HTML.gif)
If there exist such that
and
, then, there exist
, such that
and
.
Let be a topological vector space (t.v.s. for short) with its zero vector
. A nonempty subset
of
is called a convex cone if
and
for
. A convex cone
is said to be
if
. For a given proper, pointed, and convex cone
in
, we can define a partial ordering
with respect to
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ2_HTML.gif)
will stand for
and
while
will stand for
, where
denotes the interior of
.
In the following, unless otherwise specified, we always assume that is a locally convex Hausdorff t.v.s. with its zero vector
,
a proper, closed, convex, and pointed cone in
with
,
a partial ordering with respect to
, and
.
Very recently, Du [14] first introduced the concepts of -cone metric and
-cone metric space to improve and extend the concept of cone metric space in the sense of Huang and Zhang [15].
Definition 1.1 . (see [14]).
Let be a nonempty set. A vector-valued function
is said to be a
-cone metric if the following conditions hold:
(C1) for all
and
if and only if
;
(C2) for all
;
(C3) for all
.
The pair is then called a
-cone metric space.
Definition 1.2 . (see [14]).
Let be a
-cone metric space,
, and
a sequence in
.
(i) is said to
-cone converge to
if for every
with
there exists a natural number
such that
for all
. We denote this by cone-
or
as
and call
the
-cone limit of
.
(ii) is said to be a
-cone Cauchy sequence if for every
with
there is a natural number
such that
for all
,
.
(iii) is said to be
-cone complete if every
-cone Cauchy sequence in
is
-cone convergent in
.
In [14], the author proved the following important results.
Theorem 1.3 . (see [14]).
Let be a
-cone metric space. Then
defined by
is a metric, where
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ3_HTML.gif)
Theorem 1.4 . (see [14]).
Let be a
-cone metric space,
, and
a sequence in
. Then the following statements hold:
(a)if -cone converges to
(i.e.,
as
, then
as
(i.e.,
as
;
(b)if is a
-cone Cauchy sequence in
, then
is a Cauchy sequence (in usual sense) in
.
In this paper, we establish some new coupled fixed point theorems for various types of nonlinear contractive maps in the setting of quasiordered cone metric spaces. Our results generalize and improve some results in [2, 4, 9, 11] and references therein.
2. Preliminaries
Let be a nonempty set and "
" a quasiorder (preorder or pseudoorder, i.e., a reflexive and transitive relation) on
. Then
is called a quasiordered set. A sequence
is called
-nondecreasing (resp.,
-nonincreasing) if
(resp.,
) for each
. In this paper, we endow the product space
with the following quasiorder
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ4_HTML.gif)
Recall that the nonlinear scalarization function is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ5_HTML.gif)
Theorem 2.1 . (see [14, 16, 17]).
For each and
, the following statements are satisfied:
(i);
(ii);
(iii);
(iv);
(v) is positively homogeneous and continuous on
;
(vi)if , then
;
(vii) for all
.
Remark 2.2.
-
(a)
Clearly,
.
-
(b)
The reverse statement of (vi) in Theorem 2.1 (i.e.,
) does not hold in general. For example, let
, and
. Then
is a proper, closed, convex, and pointed cone in
with
and
. For
, it is easy to see that
, and
. By applying (iii) and (iv) of Theorem 2.1, we have
but indeed
.
For any -cone metric space
, we can define the map
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ6_HTML.gif)
It is obvious that is also a
-cone metric on
, and if
and
as
, then
(i.e.,
TVS-cone converges to
.
By Theorem 1.3, we know that is a metric on
. Hence the function
:
, defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ7_HTML.gif)
is a metric on .
A map is said to be
-continuous at
if any sequence
with
implies that
.
is said to be
-continuous on
if
is continuous at every point of
.
Definition 2.3 . (see [2, 4]).
Let be a quasiordered set and
a map. one says 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%2F190606/MediaObjects/13663_2010_Article_1223_Equ8_HTML.gif)
Definition 2.4 . (see [2, 4]).
Let be a nonempty set and
a map. One calls an element
a coupled fixed point of
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ9_HTML.gif)
Definition 2.5.
Let be a
-cone metric space with a quasi-order
(
for short). A nonempty subset
of
is said to be
(i)-cone sequentially
-
if every
-nondecreasing
-cone Cauchy sequence in
converges,
(ii)-cone sequentially
-
if every
-nonincreasing
-cone Cauchy sequence in
converges,
(iii)-cone sequentially
-
if it is both
-cone sequentially
-complete and
-cone sequentially
-
.
Definition 2.6 . (see [4, 18]).
A function is said to be a
-
if it satisfies Mizoguchi-Takahashi's condition (i.e.,
for all
).
Clearly, 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
; for more detail, see [4, Remark
(iii)].
Very recently, Du and Wu [5] introduced and studied the concept of functions of contractive factor.
Definition 2.7 . (see [5]).
One says that is a function of contractive factor if for any strictly decreasing sequence
in
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ10_HTML.gif)
The following result tells us the relationship between -functions and functions of contractive factor.
Theorem 2.8.
Any -function is a function of contractive factor.
Proof.
Let be a
-function, and let
be a strictly decreasing sequence in
. Then
exists. Since
is a
-function, there exist
and
such that
for all
. On the other hand, there exists
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ11_HTML.gif)
for all with
. Hence
for all
. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ12_HTML.gif)
Then for all
, and hence
. Therefore
is a function of contractive factor.
3. Coupled Fixed Point Theorems for Various Types of Nonlinear Contractive Maps
Definition 3.1.
One says that is a function of strong contractive factor if for any strictly decreasing sequence
in
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ13_HTML.gif)
It is quite obvious that if is a function of strong contractive factor, then
is a function of contractive factor but the reverse is not always true.
The following results are crucial to our proofs in this paper.
Lemma 3.2.
A function of strong contractive factor can be structured by a function of contractive factor.
Proof.
Let be a function of contractive factor. Define
,
. We claim that
is a function of strong contractive factor. Clearly,
for all
. Let
be a strictly decreasing sequence in
. Since
is a function of contractive factor,
. Thus it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ14_HTML.gif)
Hence is a function of strong contractive factor.
Lemma 3.3.
Let be a t.v.s.,
a convex cone with
in
, and
. Then the following statements hold.
(i)If and
, then
;
(ii)If and
, then
;
(iii)If and
, then
.
Proof.
To see (i), since the set is open in
and
is a convex cone, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ15_HTML.gif)
Since and
, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ16_HTML.gif)
which means that . The proofs of conclusions (ii) and(iii) are similar to (i).
Lemma 3.4 (see [4]).
Let be a quasiordered set and
a multivalued 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%2F190606/MediaObjects/13663_2010_Article_1223_Equ17_HTML.gif)
for each . If
and
, then
is
-nondecreasing and
is
-nonincreasing.
In this section, we first present the following new coupled fixed point theorem for functions of contractive factor in quasiordered cone metric spaces which is one of the main results of this paper.
Theorem 3.5.
Let be a
-cone sequentially
-complete metric space,
a map having the mixed monotone property on
, and
. Assume that there exists a function of contractive factor
such that for any
with
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ18_HTML.gif)
and there exist such that
and
. Define the iterative sequence
in
by
and
for
. Then the following statements hold.
(a)There exists a nonempty subset of
, such that
is a complete metric space.
(b)There exists a nonempty subset of
, such that
is a complete metric space, where
for any
. Moreover, if
is
-continuous on
, then
-cone converges to a coupled fixed point in
of
.
Proof.
Since is a locally convex Hausdorff t.v.s. with its zero vector
, let
denote the topology of
and let
be the base at
consisting of all absolutely convex neighborhood of
. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ19_HTML.gif)
Then is a family of seminorms on
. For each
, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ20_HTML.gif)
and let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ21_HTML.gif)
Then is a base at
, and the topology
generated by
is the weakest topology for
such that all seminorms in
are continuous and
. Moreover, given any neighborhood
of
, there exists
such that
(see, e.g., [19, Theorem
in II.12, Page 113]).
By Lemma 3.2, we can define a function of strong contractive factor by
. Then
for all
. For any
, let
and
. Then, by Lemma 3.4,
is
-nondecreasing and
is
-nonincreasing. So
and
for each
. By (3.6), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ22_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ23_HTML.gif)
By (3.10) and Theorem 2.1,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ24_HTML.gif)
Similarly, by (3.11) and Theorem 2.1, we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ25_HTML.gif)
Combining (3.12) and (3.13), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ26_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%2F190606/MediaObjects/13663_2010_Article_1223_Equ27_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ28_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ29_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ30_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ31_HTML.gif)
Since for all
, the sequence
is strictly decreasing in
from (3.19). Since
is a function of strong contractive factor, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ32_HTML.gif)
So for all
. We want to prove that
is a
-nondecreasing
-cone Cauchy sequence and
is a
-nonincreasing
-cone Cauchy sequence in
. For each
, by (3.15), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ33_HTML.gif)
Similarly, by (3.16), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ34_HTML.gif)
From (3.21) and (3.22), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ35_HTML.gif)
Hence it follows from (3.21), (3.22), and (3.23) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ36_HTML.gif)
Therefore, for with
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ37_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ38_HTML.gif)
Given with
(i.e.,
, there exists a neighborhood
of
such that
. Therefore, there exists
with
such that
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ39_HTML.gif)
for some and
,
. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ40_HTML.gif)
If , since each
is a seminorm, we have
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ41_HTML.gif)
for all and all
. If
, since
,
, and hence there exists
such that
for all
. So, for each
and any
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ42_HTML.gif)
Therefore for any ,
for all
, and hence
. So we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ43_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ44_HTML.gif)
for all . For
with
, by (3.25), (3.26), (3.32), and Lemma 3.3, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ45_HTML.gif)
Hence is a
-nondecreasing
-cone Cauchy sequence and
is a
-nonincreasing
-cone Cauchy sequence in
. By the
-cone sequential
-completeness of
, there exist
such that
-cone converges to
and
-cone converges to
. Therefore
-cone converges to
.
On the other hand, applying Theorem 1.4, we have the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ46_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ47_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ48_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ49_HTML.gif)
Since for all
, by (3.36) and (3.37), we have
as
. Let
,
, and
. Then
,
, and
are also complete metric spaces. Hence conclusion (a) holds.
Finally, in order to complete the proof of conclusion (b), we need to verify that is a coupled fixed point of
. Let
be given. Since
is
-continuous on
and
,
is
-continuous at
. So there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ50_HTML.gif)
whenever with
. Since
and
as
, for
there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ51_HTML.gif)
So, for each with
, by (3.39),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ52_HTML.gif)
and hence we have from (3.38) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ53_HTML.gif)
Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ54_HTML.gif)
Since is arbitrary,
or
. Similarly, we can also prove that
. So
is a coupled fixed point of
. The proof is finished.
The following conclusions are immediate from Theorems 2.8 and 3.5.
Theorem 3.6.
Let be a
-cone sequentially
-complete metric space,
a map having the mixed monotone property on
, and
. Assume that there exists a
-function
such that for any
with
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ55_HTML.gif)
and there exist such that
and
. Define the iterative sequence
in
by
and
for
. Then the following statements hold.
(a)There exists a nonempty subset of
, such that
is a complete metric space.
(b)There exists a nonempty subset of
, such that
is a complete metric space. Moreover, if
is
-continuous on
, then
-cone converges to a coupled fixed point in
of
.
Theorem 3.7.
Let be a
-cone sequentially
-complete metric space,
a map having the mixed monotone property on
, and
. Assume that there exists a nonnegative number
such that for any
with
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ56_HTML.gif)
and there exist such that
and
. Define the iterative sequence
in
by
and
for
. Then the following statements hold.
(a)There exists a nonempty subset of
, such that
is a complete metric space.
(b)There exists a nonempty subset of
, such that
is a complete metric space. Moreover, if
is
-continuous on
, then
-cone converges to a coupled fixed point in
of
.
Remark 3.8.
-
(a)
Theorems 3.5 and 3.6 all generalize and improve [4, Theorem
] and some results in [2, 9, 11].
-
(b)
Theorems 3.5–3.7 all generalize Bhaskar-Lakshmikantham's coupled fixed points theorem (i.e., Theorem BL).
Finally, we focus our research on -cone metric spaces.
Theorem 3.9.
Let be a
-cone complete metric space,
a map, and
. Assume that there exists a function of contractive factor
such that for any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ57_HTML.gif)
Let . Define the iterative sequence
in
by
and
for
. Then the following statements hold.
(a)There exists a nonempty subset of
, such that
is a complete metric space.
(b)There exists a nonempty subset of
, such that
is a complete metric space.
(c) has a unique coupled fixed point in
. Moreover,
-cone converges to the coupled fixed point of
.
Proof.
For any , by (3.45) and Theorem 2.1, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ58_HTML.gif)
From (3.46), we know that is
-continuous on
. Following the same argument as in the proof of Theorem 3.5, we can prove that conclusions (a) and (b) hold and there exists
, such that
-cone converges to
and
is a coupled fixed point of
. To complete the proof, it suffices to show the uniqueness of the coupled fixed point of
. On the contrary, suppose that there exists
, such that
and
. By (3.46), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ59_HTML.gif)
So, it follows from (3.47) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ60_HTML.gif)
which leads to a contradiction. The proof is completed.
The following results are immediate from Theorem 3.9.
Theorem 3.10.
Let be a
-cone complete metric space,
a map, and
. Assume that there exists a
-function
such that for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ61_HTML.gif)
Let . Define the iterative sequence
in
by
and
for
. Then the following statements hold.
(a)There exists a nonempty subset of
, such that
is a complete metric space.
(b)There exists a nonempty subset of
, such that
is a complete metric space.
(c) has a unique coupled fixed point in
. Moreover,
-cone converges to the coupled fixed point of
.
Theorem 3.11.
Let be a
-cone complete metric space and
a map. Assume that there exists a nonnegative number
such that for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190606/MediaObjects/13663_2010_Article_1223_Equ62_HTML.gif)
Let . Define the iterative sequence
in
by
and
for
. Then the following statements hold.
(a)There exists a nonempty subset of
, such that
is a complete metric space.
-
(b)
There exists a nonempty subset
of
, such that
is a complete metric space.
(c) has a unique coupled fixed point in
. Moreover,
-cone converges to the coupled fixed point of
.
Remark 3.12.
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Acknowledgment
This research was supported by the National Science Council of the Republic of China.
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Du, WS. Nonlinear Contractive Conditions for Coupled Cone Fixed Point Theorems. Fixed Point Theory Appl 2010, 190606 (2010). https://doi.org/10.1155/2010/190606
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DOI: https://doi.org/10.1155/2010/190606