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American Journal of Public Health Research
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Anne Lanham, Lillian Mwanri

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A Fixed Point Result of Expanding Mappings in Complete Cone Metric Spaces

K. Prudhvi

Department of Mathematics, University College of Science, Saifabad, Osmania University, Hyderabad, Telangana State, India

Abstract: In this paper, we prove a fixed point theorem for expanding onto self-mappings in complete cone metric spaces. Our results improve and extend some comparable results in the literature.

Keywords: cone metric space, fixed point, expanding mapping

Received April 28, 2015; Revised May 20, 2015; Accepted June 01, 2015

Citation: K. Prudhvi, A Fixed Point Result of Expanding Mappings in Complete Cone Metric Spaces. Journal of Mathematical Sciences and Applications, 3 (1). 1-2. January 2015. doi:10.12691/jmsa-3-1-1

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 {xn} 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 {xn} be a sequence in X. If there exists a number such that

[picture] | (1)

then {xn} 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, {xn} 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

[picture] | (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 {xn} is a Cauchy sequence in X. Since (X, d) is complete, the sequence {xn} 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].

References

[1]   | M. Abbas and G. Jungck, Common fixed point results for non commuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl. 341(2008) 416-420.
[2]   | M. Arshad, A. Azam, P. Vetro, Some common fixed point results on cone metric spaces, Fixed Point Theory Appl. (2009)11, Article ID 493965.
[3]   | A. Azam, M. Arshad, I. Beg, Banach contraction principle on cone rectangular metric spaces,Appl. Anal. Discrete Math., 3(2),(2009), 236-241.
[4]   | P. Z. Daffer and H. Kaneko, On expansive mappings, Math. Japon 37 (1992), 733-735.
[5]   | L. G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332(2) (2007)1468-1476.
[6]   | X. Huang, Ch. Zhu and Xi Wen, Fixed point theorems for expanding mappings in cone metric spaces, Math Reports 14(64),2(2012), 141-148.
[7]   | D. IIic, V. Rakocevic, Common fixed points for maps on cone metric space, J. Math. Anal. Appl. 341(2008) 876-886.
[8]   | Sh. Rezapour and R. Hamlbarani, Some notes on the paper cone metric spaces and fixed point theorem of contractive mappings, J. Math. Anal. Appl. 345(2008), 719-724.
[9]   | S. Z. Wang, B. Y. Li, Z. M. Gao and K. Iseki, Some fixed point theorems for expansion mappings, Math. Japon 29(1984), 631-636.