- Research Article
- Open access
- Published:
Fixed Point in Topological Vector Space-Valued Cone Metric Spaces
Fixed Point Theory and Applications volume 2010, Article number: 604084 (2010)
Abstract
We obtain common fixed points of a pair of mappings satisfying a generalized contractive type condition in TVS-valued cone metric spaces. Our results generalize some well-known recent results in the literature.
1. Introduction and Preliminaries
Many authors [1–16] studied fixed points results of mappings satisfying contractive type condition in Banach space-valued cone metric spaces. In a recent paper [17] the authors obtained common fixed points of a pair of mapping satisfying generalized contractive type conditions without the assumption of normality in a class of topological vector space-valued cone metric spaces which is bigger than that of studied in [1–16]. In this paper we continue to study fixed point results in topological vector space valued cone metric spaces.
Let be always a topological vector space (TVS) and
a subset of
. Then,
is called a cone whenever
(i) is closed, nonempty, and
,
(ii) for all
and nonnegative real numbers
,
(iii).
For a given cone , we can define a partial ordering
with respect to
by
if and only if
.
will stand for
and
, while
will stand for
, where
denotes the interior of
.
Definition 1.1.
Let be a nonempty set. Suppose the mapping
satisfies
()
for all
and
if and only if
,
()
for all
,
()
for all
.
Then is called a topological vector space-valued cone metric on
, and
is called a topological vector space-valued cone metric space.
If is a real Banach space then
is called (Banach space-valued) cone metric space [9].
Definition 1.2.
Let be a TVS-valued cone metric space,
and
a sequence in
. Then
(i) converges to
whenever for every
with
there is a natural number
such that
for all
. We denote this by
or
.
(ii) is a Cauchy sequence whenever for every
with
there is a natural number
such that
for all
.
(iii) is a complete cone metric space if every Cauchy sequence is convergent.
Lemma 1.3.
Let be a TVS-valued cone metric space,
be a cone. Let
be a sequence in
,and
be a sequence in
converging to
. If
for every
with
, then
is a Cauchy sequence.
Proof.
Fix take a symmetric neighborhood
of
such that
. Also, choose a natural number
such that
, for all
. Then
for every
. Therefore,
is a Cauchy sequence.
Remark 1.4.
Let be nonnegative real numbers with
, or
If
and
, then
. In fact, if
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ1_HTML.gif)
and if ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ2_HTML.gif)
2. Main Results
The following theorem improves/generalizes the results of [5, Theorems 1, 3, and 4] and [4, Theorems 2.3, 2.6, 2.7, and 2.8].
Theorem 2.1.
Let be a complete topological vector space-valued cone metric space,
be a cone and
be positive integers. If a mapping
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ3_HTML.gif)
for all , where
are non negative real numbers with
, or
Then
has a unique fixed point.
Proof.
For and
, define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ4_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ5_HTML.gif)
It implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ6_HTML.gif)
That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ7_HTML.gif)
where .
Similarly,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ8_HTML.gif)
which implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ9_HTML.gif)
with .
Now by induction, we obtain for each
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ10_HTML.gif)
By Remark 1.4, for we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ11_HTML.gif)
In analogous way, we deduced
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ12_HTML.gif)
Hence, for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ13_HTML.gif)
where with
the integer part of
Fix and choose a symmetric neighborhood
of
such that
. Since
as
, by Lemma 1.3, we deduce that
is a Cauchy sequence. Since
is a complete, there exists
such that
Fix
and choose
be such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ14_HTML.gif)
for all , where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ15_HTML.gif)
Now,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ16_HTML.gif)
So,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ17_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ18_HTML.gif)
for every . From
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ19_HTML.gif)
being closed, as
, we deduce
and so
. This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_IEq129_HTML.gif)
Similarly, by using the inequality,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ20_HTML.gif)
we can show that which in turn implies that
is a common fixed point of
and, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ21_HTML.gif)
Now using the fact that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ22_HTML.gif)
We obtain is a fixed point of
For uniqueness, assume that there exists another point
in
such that
for some
in
. From
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ23_HTML.gif)
we obtain that
Huang and Zhang [9] proved Theorem 2.1 by using the following additional assumptions.
(a) Banach Space.
(b) is normal (i.e., there is a number
such that for all
).
(c)
(d)One of the following is satisfied:
(i) with
[5, Theorem 1],
(ii)with
[5, Theorem 3],
(iii) with
[5, Theorem 4].
Azam and Arshad [4] improved these results of Huang and Zhang [5] by omitting the assumption (b).
Theorem 2.2.
Let be a complete topological vector space-valued cone metric space,
be a cone and
be positive integers. If a mapping
satisfies:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ24_HTML.gif)
for all , where
are non negative real numbers with
Then
has a unique fixed point.
Proof.
The symmetric property of and the above inequality imply that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ25_HTML.gif)
By substituting in the Theorem 2.1, we obtain the required result. Next we present an example to support Theorem 2.2.
Example 2.3.
be the set of all complex-valued functions on
then
is a vector space over
under the following operations:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ26_HTML.gif)
for all . Let
be the topology on
defined by the the family
of seminorms on
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ27_HTML.gif)
then is a topological vector space which is not normable and is not even metrizable (see [18, 19]). Define
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ28_HTML.gif)
Then is a topological vector space-valued cone metric space. Define
as
, then all conditions of Theorem 2.2 are satisfied.
Corollary 2.4.
Let be a complete Banach space-valued cone metric space,
be a cone, and
be positive integers. If a mapping
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ29_HTML.gif)
for all , where
are non negative real numbers with
, or
Then
has a unique fixed point.
Next we present an example to show that corollary 2.4 is a generalization of the results [9, Theorems 1, 3, and 4] and [15, Theorems 2.3, 2.6, 2.7, and 2.8].
Example 2.5.
Let , and
. Define
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ30_HTML.gif)
Define the mapping as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F604084/MediaObjects/13663_2009_Article_1312_Equ31_HTML.gif)
Note that the assumptions (d) of results [9, Theorems 1, 3, and 4] and [15, Theorems 2.3, 2.6, 2.7, and 2.8] are not satisfied to find a fixed point of In order to apply inequality (2.1) consider mapping
for each
then for
, and
satisfy all the conditions of Corollary 2.4 and we obtain
.
References
Abbas M, Jungck G: Common fixed point results for noncommuting mappings without continuity in cone metric spaces. Journal of Mathematical Analysis and Applications 2008,341(1):416–420. 10.1016/j.jmaa.2007.09.070
Altun I, Damjanović B, Djorić D: Fixed point and common fixed point theorems on ordered cone metric spaces. Applied Mathematics Letters 2010,23(3):310–316. 10.1016/j.aml.2009.09.016
Arshad M, Azam A, Vetro P: Some common fixed point results in cone metric spaces. Fixed Point Theory and Applications 2009, 2009:-11.
Azam A, Arshad M: Common fixed points of generalized contractive maps in cone metric spaces. Bulletin of the Iranian Mathematical Society 2009,35(2):255–264.
Azam A, Arshad M, Beg I: Common fixed points of two maps in cone metric spaces. Rendiconti del Circolo Matematico di Palermo 2008,57(3):433–441. 10.1007/s12215-008-0032-5
Azam A, Arshad M, Beg I: Banach contraction principle on cone rectangular metric spaces. Applicable Analysis and Discrete Mathematics 2009,3(2):236–241. 10.2298/AADM0902236A
Çevik C, Altun I: Vector metric spaces and some properties. Topological Methods in Nonlinear Analysis 2009,34(2):375–382.
Choudhury BS, Metiya N: Fixed points of weak contractions in cone metric spaces. Nonlinear Analysis: Theory, Methods & Applications 2010,72(3–4):1589–1593. 10.1016/j.na.2009.08.040
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
Ilić D, Rakočević V: Common fixed points for maps on cone metric space. Journal of Mathematical Analysis and Applications 2008,341(2):876–882. 10.1016/j.jmaa.2007.10.065
Janković S, Kadelburg Z, Radenović S, Rhoades BE: Assad-Kirk-type fixed point theorems for a pair of nonself mappings on cone metric spaces. Fixed Point Theory and Applications 2009, 2009:-16.
Kadelburg Z, Radenović S, Rosić B: Strict contractive conditions and common fixed point theorems in cone metric spaces. Fixed Point Theory and Applications 2009, 2009:-14.
Raja P, Vaezpour SM: Some extensions of Banach's contraction principle in complete cone metric spaces. Fixed Point Theory and Applications 2008, 2008:-11.
Radenović S: Common fixed points under contractive conditions in cone metric spaces. Computers & Mathematics with Applications 2009,58(6):1273–1278. 10.1016/j.camwa.2009.07.035
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
Vetro P: Common fixed points in cone metric spaces. Rendiconti del Circolo Matematico di Palermo 2007,56(3):464–468. 10.1007/BF03032097
Beg I, Azam A, Arshad M: Common fixed points for maps on topological vector space valued cone metric spaces. International Journal of Mathematics and Mathematical Sciences 2009, 2009:-8.
Rudin W: Functional Analysis, Higher Mathematic. McGraw-Hill, New York, NY, USA; 1973:xiii+397.
Schaefer HH: Topological Vector Spaces, Graduate Texts in Mathematics. Volume 3. 3rd edition. Springer, New York, NY, USA; 1971:xi+294.
Acknowledgment
The authors are thankful to referee for precise remarks to improve the presentation of the paper.
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
Azam, A., Beg, I. & Arshad, M. Fixed Point in Topological Vector Space-Valued Cone Metric Spaces. Fixed Point Theory Appl 2010, 604084 (2010). https://doi.org/10.1155/2010/604084
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/604084