1. Introduction

In 2007, Huang and Zhang ^[5] introduced cone metric spaces replacing the real numbers by an ordered Banach space, and they have proved some fixed point theorems for self-mapping satisfying different types of contractive conditions in cone metric spaces. Later on, many authors have generalized and extended Huang and Zhang ^[5] fixed point theorems (see, e.g., ^{[1, 2, 3, 7, 8]}). In 1984, the concept of expanding mappings was introduced by Wang et. al. ^[9]. In 1992, Daffer and Kaneko ^[4] defined expanding mappings for pair of mappings in complete metric spaces and proved some fixed point theorems. In 2012, X. Huang, Ch. Zhu and Xi Wen ^[6] proved some fixed point theorems for expanding mappings cone metric spaces and they have also extended the results of Daffer and Kaneko ^[4]. The main aim of this paper is we proved a fixed point theorem for expanding mappings in cone metric spaces, our result extends and improves the results of ^[6].

The following definitions and properties are due to Huang and Zhang ^[5].

Definition 1.1. Let B be a real Banach space and θ is the zero element of B, P a subset of B. The set P is called a cone if and only if:

(i) P is closed, non–empty and ;

(ii) , implies ;

(iii) .

For a cone P in a Banach space B, define partial ordering with respect to P by if and only if . We shall write to indicate but , while will stand for , where Int P denotes the interior of the set P. This cone P is called an order cone.

Let B be a Banach space and be an order cone .The order cone P is called normal if there exists such that for all ,

The least positive number K satisfying the above inequality is called the normal constant of P.

Definition 1.2. Let X be a nonempty set of B. Suppose that the map d: satisfies:

(d1) for all and if and only if ;

(d2) for all ;

(d3) for all .

Then d is called a cone metric on X and (X, d) is called a cone metric space.

Definition 1.3. Let (X, d) be a cone metric space .We say that {x_n} is

(i) a Cauchy sequence if for every c in B with , there is N such that for all ;

(ii) a convergent sequence if for any , there is an N such that for all , for some fixed x in X. We write (as ).

The space (X, d) is called a complete cone metric space if every Cauchy sequence is convergent ^[5].

Definition 1.4. ^[5] Let (X, d) be a cone metric space and T: , then T is called a expanding mapping, if for every there exists a number such that .

2. Main Result

In this section, we prove a fixed point theorem for expanding mappings in complete cone metric spaces.

We prove a Lemma which is useful in the main theorem.

Lemma 2.1. Let (X, d) be a cone metric space and {x_n} be a sequence in X. If there exists a number such that

(1)

then {x_n} is a Cauchy sequence in X.

Proof. By the induction and the condition (1), we have

For n > m

Let be given. Choose such that , where . Also choose a natural number N₁ such that , for all . Thus

Hence, {x_n} is a Cauchy sequence in X.

The following theorem improved and extended the Theorem 2.1. of ^[6].

Theorem 2.2. Let (X, d) be a complete cone metric space and T: be a surjection. Suppose that there exists with such that

(2)

for all . Then T has a fixed point in X.

Proof. By our assumption, it is clear that T is injective. Let F be the inverse mapping of T.

Let , then ,

We assume that for all otherwise , for some , then is a fixed point of T.

From the condition (2) it follows that

If , then

Since .

Then the above inequality implies that a negative number is , which is not possible.

So, and .

Therefore,

where, .

By the Lemma 2.1, we get that {x_n} is a Cauchy sequence in X. Since (X, d) is complete, the sequence {x_n} converges to a point . Let , we have

Letting , we get that

That is, .

. That is, .

Therefore, .

Therefore, z is a fixed point of T.

Remark 2.3. If we choose in Theorem 2.1, then we get that Theorem 2.1. of ^[6].

Remark 2.4. If we choose and in Theorem 2.1, then we get that Corollary 2.1. of ^[6].

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