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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
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
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
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 
:
Recall that the nonlinear scalarization function 
is defined by
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 
.
.
) 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
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
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 
, 
,
Definition 2.4 . (see [2, 4]).
Let 
be a nonempty set and 
a map. One calls an element 
a coupled fixed point of 
if
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
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
for all 
with 
. Hence 
for all 
. Let
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
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
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
Since 
and 
, it follows that
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
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 
,
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
Then 
is a family of seminorms on 
. For each 
, let
and let
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
By (3.10) and Theorem 2.1,
Similarly, by (3.11) and Theorem 2.1, we also have
Combining (3.12) and (3.13), we get
For each 
, let 
. Then 
. By induction, we can obtain the following. For each 
,
Since 
for all 
, the sequence 
is strictly decreasing in 
from (3.19). Since 
is a function of strong contractive factor, we have
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
Similarly, by (3.16), we obtain
From (3.21) and (3.22), we get
Hence it follows from (3.21), (3.22), and (3.23) that
Therefore, for 
with 
, we have
Given 
with 
(i.e., 
, there exists a neighborhood 
of 
such that 
. Therefore, there exists 
with 
such that 
, where
for some 
and 
, 
. Let
If 
, since each 
is a seminorm, we have 
and
for all 
and all 
. If 
, since 
, 
, and hence there exists 
such that 
for all 
. So, for each 
and any 
, we obtain
Therefore for any 
, 
for all 
, and hence 
. So we obtain
or
for all 
. For 
with 
, by (3.25), (3.26), (3.32), and Lemma 3.3, we obtain
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:
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
whenever 
with 
. Since 
and 
as 
, for 
there exists 
such that
So, for each 
with 
, by (3.39),
and hence we have from (3.38) that
Therefore
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 
,
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 
,
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).
] and some results in [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 
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
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
So, it follows from (3.47) that
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 
,
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 
,
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.
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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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