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

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

**2010**:604084

**DOI: **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.

## 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.

## 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 .

where with the integer part of

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

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.

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

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