#### Abstract

We introduced a notion of topological vector space valued cone metric space and obtained some common fixed point results. Our results generalize some recent results in the literature.

#### 1. Introduction

Huang and Zhang [1] generalized the notion of metric space by replacing the set of real numbers by ordered Banach space, deffined a cone metric space, and established some fixed point theorems for contractive type mappings in a normal cone metric space. Subsequently, several other authors [2–5] studied the existence of common fixed point of mappings satisfying a contractive type condition in normal cone metric spaces. Afterwards, Rezapour and Hamlbarani [6] studied fixed point theorems of contractive type mappings by omitting the assumption of normality in cone metric spaces (see also [7–14]). In this paper we obtain common fixed points for a pair of self-mappings satisfying a generalized contractive type condition without the assumption of normality in a class of topological vector space valued cone metric spaces which is bigger than that introduced by Huang and Zhang [1].

Let be always a topological vector space 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 that the mapping satisfies() for all and if and only if ,() for all ,() for all .Then is called a cone metric on and is called a topological vector space valued cone metric space.

Note that Huang and Zhang [1] notion of cone metric space is a special case of our notion of topological vector space valued cone metric space.

*Example 1.2. *Let be the set of all real valued functions on which also have continuous derivatives onthen is a vector space over under the following operations:
for all Let be the strongest vector (locally convex) topology on then is a topological vector space which is not normable and is not even metrizable (see [15]). Define as follows:
Then is a topological vector space valued cone metric space.

Example 1.2 shows that this category of cone metric spaces is larger than that considered in [1–8] .

*Definition 1.3. *Let be a topological vector space valued cone metric space, and let and be 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 topological vector space valued cone metric space if every Cauchy sequence is convergent.

#### 2. Fixed Point

In this section, we shall give some results which generalize [6, Theorems 2.3, 2.6, 2.7, and 2.8] (and so [1, Theorems 1, 3, and 4]).

Theorem 2.1. *Let be a complete topological vector space valued cone metric space and let the self-mappings satisfy
**
for all , where with . Then and have a unique common fixed point.*

*Proof. *For and , define and . Then,
It implies that . Similarly,
Hence, . Thus,
for all , where . Now, for we have
Let . Take a symmetric neighborhood of such that . Also, choose a natural number such that , for all . Then, , for all . Thus,
for all . Therefore, is a Cauchy sequence in . Since is complete, there exists such that . Choose a natural number such that for all . Thus,
So,
for all . Therefore, for all . Hence, for all Since is closed, and so . Hence, is a fixed point of . Similarly, we can show that . Now, we show that and have a unique fixed point. For this, assume that there exists another point in such that . Then,
Since and so .

The following corollary generalizes [6, Theorems 2.3, 2.7, and 2.8] (and so [1, Theorems and ]).

Corollary 2.2. *Let be a complete topological vector space valued cone metric space and let the self-mapping satisfy for all , where with . Then has a unique fixed point.*

*Proof. *The symmetric property of and the above inequality imply that
By substituting and in Theorem 2.1, we obtain the required result.

Theorem 2.3. *Let be a complete topological vector space valued cone metric space and let the self-mappings satisfy
**
for all , where with . Then and have a unique common fixed point.*

*Proof. *For and , define and . Then,
It implies that . Similarly,
Hence, . Thus,
for all , where . Now, for we have
Let . Take a symmetric neighborhood of such that . Also, choose a natural number such that , for all . Then, , for all . Thus,
for all . Therefore, is a Cauchy sequence in . Since is complete, there exists such that . Choose a natural number such that for all . Thus,
So,
for all . Therefore, for all . Hence, for all Since is closed, and so . Hence, is a fixed point of . Similarly, we can show that . Now, we show that and have a unique fixed point. For this, assume that there exists another point in such that . Then,
Since ,?? and so .

The following corollary generalizes [6, Theorem ] (and so [1, Theorem ]).

Corollary 2.4. *Let be a complete topological vector space valued cone metric space and let the self-mapping satisfy for all , where with . Then has a unique fixed point.*

Proof is similar to the proof of Corollary 2.2.

*Example 2.5. *Let be a topological vector space valued cone metric space of Example 1.2. Define as follows:
Then,
if Hence all conditions of Theorem 2.3 are satisfied.

#### Acknowledgments

The present version of the paper owes much to the precise and kind remarks of the learned referees.