Open Access

# Fixed Point in Topological Vector Space-Valued Cone Metric Spaces

Fixed Point Theory and Applications20102010:604084

DOI: 10.1155/2010/604084

Accepted: 2 June 2010

Published: 14 June 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 [116] 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 [116]. 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
(1.1)
and if ,
(1.2)

## 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
(2.1)

for all , where are non negative real numbers with , or Then has a unique fixed point.

Proof.

For and , define
(2.2)
Then
(2.3)
It implies that
(2.4)
That is,
(2.5)

where .

Similarly,
(2.6)
which implies
(2.7)

with .

Now by induction, we obtain for each
(2.8)
By Remark 1.4, for we have
(2.9)
In analogous way, we deduced
(2.10)
Hence, for
(2.11)

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
(2.12)
for all , where
(2.13)
Now,
(2.14)
So,
(2.15)
Hence
(2.16)
for every . From
(2.17)

being closed, as , we deduce and so . This implies that

Similarly, by using the inequality,
(2.18)
we can show that which in turn implies that is a common fixed point of and, that is,
(2.19)
Now using the fact that
(2.20)
We obtain is a fixed point of For uniqueness, assume that there exists another point in such that for some in . From
(2.21)

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:
(2.22)

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
(2.23)

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:
(2.24)
for all . Let be the topology on defined by the the family of seminorms on , where
(2.25)
then is a topological vector space which is not normable and is not even metrizable (see [18, 19]). Define as follows:
(2.26)

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
(2.27)

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:
(2.28)
Define the mapping as follows:
(2.29)

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 .

## Declarations

### Acknowledgment

The authors are thankful to referee for precise remarks to improve the presentation of the paper.

## Authors’ Affiliations

(1)
Department of Mathematics, COMSATS Institute of Information Technology
(2)
Department of Mathematics, Centre for Advanced Studies in Mathematics, Lahore University of Management Sciences
(3)
Department of Mathematics, International Islamic University

## References

1. 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
2. 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
3. Arshad M, Azam A, Vetro P: Some common fixed point results in cone metric spaces. Fixed Point Theory and Applications 2009, 2009:-11.Google Scholar
4. 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.
5. 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
6. 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
7. Çevik C, Altun I: Vector metric spaces and some properties. Topological Methods in Nonlinear Analysis 2009,34(2):375–382.
8. 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
9. 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
10. 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
11. 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.Google Scholar
12. 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.Google Scholar
13. Raja P, Vaezpour SM: Some extensions of Banach's contraction principle in complete cone metric spaces. Fixed Point Theory and Applications 2008, 2008:-11.Google Scholar
14. 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
15. 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
16. Vetro P: Common fixed points in cone metric spaces. Rendiconti del Circolo Matematico di Palermo 2007,56(3):464–468. 10.1007/BF03032097
17. 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.Google Scholar
18. Rudin W: Functional Analysis, Higher Mathematic. McGraw-Hill, New York, NY, USA; 1973:xiii+397.Google Scholar
19. Schaefer HH: Topological Vector Spaces, Graduate Texts in Mathematics. Volume 3. 3rd edition. Springer, New York, NY, USA; 1971:xi+294.