Fixed Points and Common Fixed Points of Quasi-Contractive Mappings on Partially Ordered-Cone Metric Spaces ()
1. Introduction
Huang and Zhang [1] recently have introduced the concept of cone metric spaces and have established fixed point theorems for a contractive type map in a normal cone metric space. Subsequently, some authors [2] -[7] have generalized the results in [1] and have studied the existence of common fixed points of a finite self maps satisfying a contractive condition in the framework of normal or non-normal cone metric spaces. On the other hand, some authors discussed (common) fixed point problems for contractive maps defined on a partially ordered set with cone metric structure [8] -[13] . These results improved and generalized many corresponding (common) fixed point theorems of contractive maps on cone metric spaces. Here, we will obtain (common) fixed point theorems of maps with certain quasi-contractive conditions on a partially ordered set with cone metric structure.
Let E be a real Banach space. A subset P0 of E is called a cone if and only if:
i) P0 is closed, nonempty, and
;
ii)
,
and
implies
;
iii)
.
Given a cone
, we define a partial ordering ≤ on E with respect to P0 by
if and only if
. We will write
to indicate that
but
, while
will stand for
(interior of P0).
The cone P0 is called normal if there is a number
such that for all
,
.
The least positive number K satisfying the above is called the normal constant of P0. It is clear that
.
In the following we always suppose that E is a real Banach space, P0 is a cone in E with
and ≤ is a partial ordering with respect to P0.
Let X be a nonempty set. Suppose that the mapping
satisfies
d1)
for all
and
if and only if
;
d2)
for all
;
d3)
, for all
.
Then d is called a cone metric on X, and
is called a cone metric space.
Let
be a cone metric space. We say that a sequence
in X is
e) Cauchy sequence if for every
with
, there is an N such that for all n, m > N,
;
g) convergent sequence if for every
with
, there is an N such that for all
such that
for some
. Let
or
.
is said to be complete if every Cauchy sequence in X is convergent in X.
Let
be a cone metric space,
and
. f is said to be continuous [13] at x0 if for any sequence
, we have
.
Lemma 1 [14] Let
be a cone metric space. Then the following properties hold:
1) if
and
, then
; if
for all
, then
;
2) if
where
and
, then
.
Lemma 2 [15] Let
be a cone metric space,
a sequence in X and
a sequence in P0 and
. If
for any
, then
is Cauchy.
2. Main Results
At first, we give an example to show that there exists a self-map f on a partially ordered set
such that for each
there exists y satisfying
and
.
Example Let
be a real space. Define
by
![]()
Then obviously, for each
, there exists
satisfying
and
.
is said to be a partially order-cone metric space if
is a partially ordered set and
is a cone metric space.
Theorem 1 Let
be a complete partially ordered-cone metric space. Suppose that a map
is continuous and the following two assertions hold:
i) there exist A, B, C, D, E ≥ 0 with
and for
with
, such that
;
ii) for each
, there exists
such that
and
.
Then f has a fixed point
. Furthermore, if any two elements x and y in
are comparative and
, then f has a unique fixed point in X.
Proof Take any
, then by ii), we obtain a sequence
as follows:
for all
and
.
For any fixed
, since
, by i),
![]()
so
,
.
Let
, then
by i) and
.
Repeating this process,
.
Let
, then from the above,
.
Obviously,
and
as
since
So
for all
, hence
is a Cauchy sequence by Lemma 2 and there exists
such that
by the completeness of X. Since f is continuous and
, so
, i.e.,
is a fixed point of f.
If
and
are all fixed points of f and suppose that
, then by i),
![]()
Hence
by (2) in Lemma 1, so
is the unique fixed point of f.
Another version of Theorem 1 is following:
Theorem 2 Let
be a complete partially ordered-cone metric space. Suppose that
is continuous and the following two assertions hold:
i) there exist
with
and for all
with
,
;
ii) for each
, there exists
such that
and
.
Then f has a fixed point
. Furthermore, if x and y is comparative for all
, then f has an unique fixed point in X.
Proof Take
,
and
, then the conclusion is true by Theorem 1.
From now, we give common fixed point theorems for a pare of maps.
Theorem 3 Let
be a complete partially ordered-cone metric space. If
are two maps such that f or g is continuous and the following two assertions hold:
i) there exist A, B, C, D, E ≥ 0 with
,
and
such that for all comparative
;
ii) for each
, there exist
such that
,
and
,
.
Then f and g have a common fixed point
. Furthermore, if x and y in
are comparative and
, then
is singleton.
Proof Take any element
, then using ii), we can construct a sequence
satisfying the following condition
,
for all
, and
.
For any
, by i), we have
![]()
hence
![]()
where
. And
![]()
hence
![]()
where
.
Let
, then
by i), and by induction, for any ![]()
![]()
![]()
For any
with
,
![]()
where
. Similarly,
![]()
![]()
![]()
So for any
with m > n > 0, there exists
with
, that is,
such that
.
Obviously,
and
as
since
. So
for all
, hence
is Cauchy by Lemma 2 and there exists
such that
.
Suppose that f is continuous, then
since
. For
there exists
such that
and
by ii). By i),
![]()
So
by (2) in Lemma 1, hence
. Therefore
. Similarly, we can give the same result for the case of g being continuous.
If
then
and
are comparative, hence by i),
![]()
so
by (2) in Lemma 1. Hence
.
Modifying the idea of Zhang [16] , we obtain next three corollaries.
Corollary 1 The conditions of A, B, C, D, E in i) of Theorem 3 can be replaced by the following:
i') there exist A, B, C, D, E ≥ 0 and
such that
,
,
,
,
.
Proof Since
so
,
hence
therefore
.
Corollary 2 The conditions of A, B, C, D, E in i) of Theorem 3 can be replaced by the following:
i'') there exist A, B, C, D, E ≥ 0 such that A + B + C + D + E = 1, C > B and D > E or C < B and D < E.
Proof Take
such that
and
, and let
. Then the following holds: for all comparative elements
,
.
Obviously, A', B, C, D, E satisfy i') in Corollary 1.
Corollary 3 The conditions of
in 1) of Theorem 3 can be replaced by the following:
i''') there exist A, B, C, D, E ≥ 0 such that
and
or
.
Proof Since
, so
, hence
,
or
,
which implies that
,
or
.
If
or
, then the above two relations reduce
.
The following is a non-continuous version of Theorem 3.
Theorem 4 Let
be a complete partially ordered-cone metric space. If
are maps such that i) and ii) in Theorem 3 hold and iii) or iv) holds
iii) if an increasing sequence
converges to
, then
and
for all
and
;
iv) if an increasing sequence
converges to
, then
and
for all
and
.
Then f and g have a common fixed point
. Furthermore, if x and y in
are comparative and
, then
is singleton.
Proof By i) and ii) in Theorem 3, we construct a sequence
such that
,
, for all
, and
and
.
Case I: Suppose iv) holds, then
and
for all
and
. By i),
![]()
so we obtain
,
where
and
Since
, for any
there exist enough large
such that
and
for all
, hence
,.
So
by (1) in Lemma 1, hence
.
For
there exists
such that
and
by ii). Hence by i),
![]()
so
by (2) in Lemma 1. Hence
, i.e.,
.
Case II: Suppose iii) holds, then
and
for all
and
. By (1),
![]()
so we obtain
,
where
and
Since
, for any
there exist enough large
such that
and
for all
, hence
,
. So
by (1) in Lemma 1, that is,
.
For
there exists
such that
and
by ii). Hence by i),
![]()
so
by (1) in Lemma 1. Hence
, i.e.,
.
So in any case,
The uniqueness is obvious.
Remark 1 We can also modify Corollary 1 - 3 to give the corresponding corollaries of Theorem 4, but we omit the part.
Remark 2 In this paper, we discuss the common fixed point problems for mappings with quasi-contractive type (i.e., expansive type) on partially ordered cone metric spaces, but some authors in references discussed the same problems for contractive or Lipschitz type. So our results improve and generalize the corresponding conclusions.