- Research Article
- Open Access

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

- Akbar Azam
^{1}, - Ismat Beg
^{2}Email author and - Muhammad Arshad
^{3}

**2010**:604084

https://doi.org/10.1155/2010/604084

© Akbar Azam et al. 2010

**Received:**16 December 2009**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.

## Keywords

- Banach Space
- Positive Integer
- Real Number
- Vector Space
- Recent Result

## 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 ,

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 ,

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.

## 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.

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

Proof.

where with the integer part of

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

Huang and Zhang [9] proved Theorem 2.1 by using the following additional assumptions.

(b) is normal (i.e., there is a number such that for all ).

(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.

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

Proof.

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.

Then is a topological vector space-valued cone metric space. Define as , then all conditions of Theorem 2.2 are satisfied.

Corollary 2.4.

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.

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

## 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.2007.09.070MathSciNetView ArticleMATHGoogle Scholar - 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.2009.09.016MathSciNetView ArticleMATHGoogle Scholar - 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 - 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.MathSciNetMATHGoogle Scholar - 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/s12215-008-0032-5MathSciNetView ArticleMATHGoogle Scholar - 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/AADM0902236AMathSciNetView ArticleMATHGoogle Scholar - Çevik C, Altun I:
**Vector metric spaces and some properties.***Topological Methods in Nonlinear Analysis*2009,**34**(2):375–382.MathSciNetMATHGoogle Scholar - 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.2009.08.040MathSciNetView ArticleMATHGoogle Scholar - 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.2005.03.087MathSciNetView ArticleMATHGoogle Scholar - 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.2007.10.065MathSciNetView ArticleMATHGoogle Scholar - 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 - 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 - 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 - 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.2009.07.035MathSciNetView ArticleMATHGoogle Scholar - 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.2008.04.049MathSciNetView ArticleMATHGoogle Scholar - Vetro P:
**Common fixed points in cone metric spaces.***Rendiconti del Circolo Matematico di Palermo*2007,**56**(3):464–468. 10.1007/BF03032097MathSciNetView ArticleMATHGoogle Scholar - 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 - Rudin W:
*Functional Analysis, Higher Mathematic*. McGraw-Hill, New York, NY, USA; 1973:xiii+397.Google Scholar - Schaefer HH:
*Topological Vector Spaces, Graduate Texts in Mathematics*.*Volume 3*. 3rd edition. Springer, New York, NY, USA; 1971:xi+294.View ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.