- Research Article
- Open access
- Published:
Quasicone Metric Spaces and Generalizations of Caristi Kirk's Theorem
Fixed Point Theory and Applications volume 2009, Article number: 574387 (2009)
Abstract
Cone-valued lower semicontinuous maps are used to generalize Cristi-Kirik's fixed point theorem to Cone metric spaces. The cone under consideration is assumed to be strongly minihedral and normal. First we prove such a type of fixed point theorem in compact cone metric spaces and then generalize to complete cone metric spaces. Some more general results are also obtained in quasicone metric spaces.
1. Introduction and Preliminaries
In 2007, Huang and Zhang [1] introduced the notion of cone metric spaces (CMSs) by replacing real numbers with an ordering Banach space. The authors there gave an example of a function which is contraction in the category of cone metric spaces but not contraction if considered over metric spaces and hence, by proving a fixed point theorem in cone metric spaces, ensured that this map must have a unique fixed point. After that series of articles about cone metric spaces started to appear. Some of those articles dealt with the extension of certain fixed point theorems to cone metric spaces (see, e.g., [2–5]), and some other with the structure of the spaces themselves (see, e.g., [3, 6]). Very recently, some authors have used regular cones to extend some fixed point theorems. For example, in [7] a result about Meir-Keeler type contraction mappings has been extended to regular cone metric spaces. In other works, some results about fixed points of multifunctions on cone metric spaces with normal cones have been obtained as well [8]. For the use of lower semicontinuous functions in obtaining fixed point theorems in cone metric spaces we refer to [9].
In this manuscript, we use cone-valued lower semicontinuous functions to extend some of the results in Caristi [10] and Ekeland [11] to CMS and quasicone metric space (QCMS). The cones under consideration are assumed to be strongly minihedral and normal and hence regular. In particular the cone in the real line
is strongly minihedral and normal; hence the results mentioned in the above references are recovered.
Throughout this paper stands for a real Banach space. Let
always be a closed subset of
.
is called cone if the following conditions are satisfied:
,
for all
and non-negative real numbers
,
and
.
For a given cone , one can define a partial ordering (denoted by
or
) with respect to
by
if and only if
. The notation
indicates that
and
while
will show
, where
denotes the interior of
. From now on, it is assumed that
The cone is called
()normal if there is a number such that for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F574387/MediaObjects/13663_2009_Article_1157_Equ1_HTML.gif)
()regular if every increasing sequence which is bounded from above is convergent. That is, if is a sequence such that
for some
, then there is
such that
.
In , the least positive integer
, satisfying (1.1), is called the normal constant of
. Note that, in [1, 2], normal constant
is stated a positive real number, (
). However, later on and in [2, Lemma
] it was proved that there is no normal cone with constant
.
Lemma 1.1.
-
(i)
Every regular cone is normal.
-
(ii)
For each
, there is a normal cone with normal constant
.
-
(iii)
The cone
is regular if every decreasing sequence which is bounded from below is convergent.
The proof of (i) and (ii) were given in [2] and the last one just follows from definition.
Example 1.2 (see [2]).
Let with the norm
, and consider the cone
For each , put
and
Then,
,
and
Since
,
is not normal constant of
and hence
is a nonnormal cone.
Definition 1.3.
Let be a nonempty set. Suppose that the mapping
satisfies the following:
() for all
,
() if and only if
,
(), for all
.
Then is said to be a quasicone metric on
, and the pair
is called a quasicone metric space (QCMS). Additionally, if
also satisfies
() for all
,
then is called a cone metric on
, and the pair
is called a cone metric space (CMS).
Example 1.4.
Let and
and
. Define
by
, where
are positive constants. Then
is a CMS. Note that the cone
is normal with the normal constant
Definition 1.5.
Let be a CMS,
, and let
be a sequence in
. Then
(i)converges to
if for every
with
there is a natural number
, such that
for all
. It is denoted by
or
;
(ii) is a Cauchy sequence if for every
with
there is a natural number
, such that
for all
;
(iii) is a complete cone metric space if every Cauchy sequence in
is convergent in
.
Lemma 1.6 (see [1]).
Let be a CMS, let
be a normal cone with normal constant
, and let
be a sequence in
. Then,
(i)the sequence converges to
if and only if d (
,
)
0
or equivalently
;
(ii)the sequence is Cauchy if and only if
(or equivalently
);
(iii)the sequence converges to
and the sequence
converges to
then
.
Let be a CMS over a cone
in
. Thenone has the following.
(1) and
,
.
(2)If , then there exists
such that
implies
.
(3)For any given and
there exists
such that
.
(4)If ,
are sequences in
such that
,
and
for all
, then
.
Definition 1.8 (see [12]).
is called minihedral cone if
exists for all
, and strongly minihedral if every subset of
which is bounded from above has a supremum.
It is easy to see that every strongly minihedral normal cone is regular.
Example 1.9.
Let with the supremum norm and
Then
is a cone with normal constant
which is not regular. This is clear, since the sequence
is monotonicly decreasing, but not uniformly convergent to
. Thus,
is not strongly minihedral. It is easy to see that the cone mentioned in Example 1.4 is strongly minihedral.
Definition 1.10 (see [1]).
Let be a CMS and
.
is said to be sequentially compact if for any sequence
in
there is a subsequence
of
such that
is convergent in
.
Remark 1.11 (see [6]).
Every cone metric space is a topological space which is denoted by
. Moreover, a subset
is sequentially compact if and only if
is compact.
2. Main Results
Let be a CMS,
, and
a function on
. Then, the function
is called a lower semicontinuous (l.s.c) on
whenever
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F574387/MediaObjects/13663_2009_Article_1157_Equ2_HTML.gif)
Also, let be an arbitrary selfmapping on
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F574387/MediaObjects/13663_2009_Article_1157_Equ3_HTML.gif)
Then, is called a Caristi map on
.
The following Lemma will be used to prove the next results.
Lemma 2.1.
If is a decreasing sequence (via the partial ordering obtained by the closed cone
) such that
, then
Proof.
Since is an increasing sequence,
, for
and
for all
. Then closeness of
implies that
for all
. To see that
is the greatest lower bound of
, assume that some
satisfies
for all
. From
and the closeness of
we get
or
which shows that
Proposition 2.2.
Let be a compact CMS,
a strongly minihedral cone, and
a lower semicontinuous
function. Then,
attains a minimum on
.
Proof.
Let which exists by strong minihedrality. For each
, there is an
such that
, where
. Since
is compact, then
has a convergent subsequence. Let
be this sequence and let
.
From the definition of lower semicontinuity and Lemma 2.1 it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F574387/MediaObjects/13663_2009_Article_1157_Equ4_HTML.gif)
But then, by the definition of ,
for all
. This completes the proof.
Theorem 2.3.
Let be a CMS,
a compact subset of
,
a strongly minihedral normal cone, and
a lower semicontinuous
function. Then, each selfmap
satisfying (2.2) has a fixed point in
.
Proof.
By Proposition 2.2, attains its minimum at some point of
, say
. Since
is the minimum point of
, we have
. By (2.2),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F574387/MediaObjects/13663_2009_Article_1157_Equ5_HTML.gif)
Thus, and so
The following theorem is an extension of the result of Caristi ([10, Theorem ]).
Theorem 2.4.
Let be a complete CMS,
a strongly minihedral normal cone, and
a lower semicontinuous
function. Then, each selmap
satisfying (2.2) has a fixed point in
.
Proof.
Let have the normal constant
. Let
and
for all
. Since
,
and so
.
For , set
and construct a sequence
in the following way: let
be such that
, where
. Thus, one can observe that
(i),
(ii)
for all . Note that, (i) implies that the sequence
is a decreasing sequence in
and
is regular cone. So, the sequence
is convergent. Thus, for each
, there exists
such that
for all
,
. For
, the triangular inequality implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F574387/MediaObjects/13663_2009_Article_1157_Equ6_HTML.gif)
Hence, . By Lemma 1.6,
yields that the sequence
is a Cauchy in
. Completeness of
implies that the sequence
is convergent to some point in
, say
By (2.5), and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F574387/MediaObjects/13663_2009_Article_1157_Equ7_HTML.gif)
for all . By regarding (2.6), Lemma 1.6, and lower semicontinuity of the function
, one can obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F574387/MediaObjects/13663_2009_Article_1157_Equ8_HTML.gif)
for all . Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F574387/MediaObjects/13663_2009_Article_1157_Equ9_HTML.gif)
for all . Hence,
and it is trivial that
for all
. Note that (ii) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F574387/MediaObjects/13663_2009_Article_1157_Equ10_HTML.gif)
Thus, for all
. On the other hand, by lower semicontinuity of
and (2.9), one can obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F574387/MediaObjects/13663_2009_Article_1157_Equ11_HTML.gif)
Therefore, .
Since for each
and
, the following inequalities are obtained:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F574387/MediaObjects/13663_2009_Article_1157_Equ12_HTML.gif)
Hence, for all
. This implies that
for all
.
By (2.9), is obtained. As
is observed by (2.2) and that
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F574387/MediaObjects/13663_2009_Article_1157_Equ13_HTML.gif)
is achieved. Hence, . Finally, by (2.2) we have
.
The following theorem is a generalization of the result in [11].
Theorem 2.5.
Let be a
function on a complete CMS, where
is a strongly minihedral normal cone. If
is bounded below, then there exits
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F574387/MediaObjects/13663_2009_Article_1157_Equ14_HTML.gif)
Proof.
It is enough to show that the point , obtained in Theorem 2.4, satisfies the statement of the theorem. Following the same notation in the proof of Theorem 2.4, it is needed to show that
for
. Assume the contrary that for some
, we have
. Then,
implies
. By triangular inequality,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F574387/MediaObjects/13663_2009_Article_1157_Equ15_HTML.gif)
which implies that and thus
for all
. Taking the limit when
tends to infinity, one can obtain
, which is in contradiction with
. Thus, for any
,
implies
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F574387/MediaObjects/13663_2009_Article_1157_Equ16_HTML.gif)
Let be defined by
.
Theorem 2.6.
Let be a sequentially complete QCMS and let
be a strongly minihedral normal cone. Assume that for each
, the function
defined above is continuous on
and
is a family of mappings
. If there exists a l.s.c function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F574387/MediaObjects/13663_2009_Article_1157_Equ17_HTML.gif)
then for each there is a common fixed point
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F574387/MediaObjects/13663_2009_Article_1157_Equ18_HTML.gif)
Proof.
Let be strongly minihedral normal cone with normal constant
. First note that strong minihedrality of
guarantees that
exists. Let
and
for all
. Note that
, so
and also
.
For , set
and construct a sequence
as in the proof of Theorem 2.4:
such that
,
. Thus, one can observe that for each
,
(i),
(ii).
Similar to the proof of Theorem 2.4, (ii) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F574387/MediaObjects/13663_2009_Article_1157_Equ19_HTML.gif)
Also, by using the same method in the proof of Theorem 2.4, it can be shown that is a Cauchy sequence and converges to some
and
.
We shall show that for all
. Assume the contrary that there is
such that
. Then (2.16) with
implies that
Thus, by definition of
, there is
such that
. Since
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F574387/MediaObjects/13663_2009_Article_1157_Equ20_HTML.gif)
which implies that . Hence
which is in a contradiction with
Thus,
for all
. Since
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F574387/MediaObjects/13663_2009_Article_1157_Equ21_HTML.gif)
is obtained.
The following theorem is a generalization of [13, Theorem ].
Theorem 2.7.
Let be a set,
as in Theorem 2.6,
a surjective mapping, and
a family of arbitrary mappings
. If there exists a
function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F574387/MediaObjects/13663_2009_Article_1157_Equ22_HTML.gif)
and each , then
and
have a common coincidence point, that is, for some
,
for all
.
Proof.
Let be arbitrary and
as in Theorem 2.6. Since
is surjective, for each
there is some
such that
. Let
be a fixed mapping. Define by
a mapping
of
into itself such that
, where
, that is,
. Let
be a family of all mappings
. Then, (2.21) yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F574387/MediaObjects/13663_2009_Article_1157_Equ23_HTML.gif)
Thus, by Theorem 2.6, for all
. Hence
for all
, where
is such that
.
References
Huang L-G, Zhang X: Cone metric spaces and fixed point theorems of contractive mappings. Journal of Mathematical Analysis and Applications 2007,332(2):1468–1476. 10.1016/j.jmaa.2005.03.087
Rezapour Sh, Hamlbarani R: Some notes on the paper: "cone metric spaces and fixed point theorems of contractive mappings". Journal of Mathematical Analysis and Applications 2008,345(2):719–724. 10.1016/j.jmaa.2008.04.049
Turkoglu D, Abuloha M: Cone metric spaces and fixed point theorems in diametrically contractive mappings. Acta Mathematica Sinica, English Series, submitted
Turkoglu D, Abuloha M, Abdeljawad T: KKM mappings in cone metric spaces and some fixed point theorems. Nonlinear Analysis: Theory, Methods and Applications 2010,72(1):348–353. 10.1016/j.na.2009.06.058
Şahin İ, Telci M: Fixed points of contractive mappings on complete cone metric spaces. Hacettepe Journal of Mathematics and Statistics 2008, 345: 719–724.
Rezapour Sh, Derafshpour M, Hamlbarani R: A Review on Topological Properties of Cone Metric Spaces. Analysis, Topology and Applications 2008 (ATA2008), the 30th of May to the 4th of June, 2008, Technical Faculty, Cacak, University of Kragujevac Vrnjacka Banja, Serbia
Haghi RH, Rezapour Sh: Fixed points of multifunctions on regular cone metric spaces. Expositiones Mathematicae. In press
Rezapour Sh, Haghi RH: Fixed point of multifunctions on cone metric spaces. Numerical Functional Analysis and Optimization 2009,30(7–8):825–832. 10.1080/01630560903123346
Klim D, Wardowski D: Dynamic processes and fixed points of set-valued nonlinear contractions in cone metric spaces. Nonlinear Analysis: Theory, Methods and Applications 2009,71(11):5170–5175. 10.1016/j.na.2009.04.001
Caristi J: Fixed point theorems for mappings satisfying inwardness conditions. Transactions of the American Mathematical Society 1976, 215: 241–251.
Ekeland I: Sur les problèmes variationnels. Comptes Rendus de l'Académie des Sciences 1972, 275: A1057-A1059.
Deimling K: Nonlinear Functional Analysis. Springer, Berlin, Germany; 1985:xiv+450.
Ćirić LB: On a common fixed point theorem of a Greguš type. Publications de l' Institut Mathématique 1991,49(63):174–178.
Acknowledgment
This work is partially supported by the Scientific and Technical Research Council of Turkey.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Abdeljawad, T., Karapinar, E. Quasicone Metric Spaces and Generalizations of Caristi Kirk's Theorem. Fixed Point Theory Appl 2009, 574387 (2009). https://doi.org/10.1155/2009/574387
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2009/574387