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

K. Prudhvi

Journal of Mathematical Sciences and Applications OPEN ACCESSPEER-REVIEWED

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.

Cite this article:

  • K. Prudhvi. A Fixed Point Result of Expanding Mappings in Complete Cone Metric Spaces. Journal of Mathematical Sciences and Applications. Vol. 3, No. 1, 2015, pp 1-2. http://pubs.sciepub.com/jmsa/3/1/1
  • Prudhvi, K.. "A Fixed Point Result of Expanding Mappings in Complete Cone Metric Spaces." Journal of Mathematical Sciences and Applications 3.1 (2015): 1-2.
  • Prudhvi, K. (2015). A Fixed Point Result of Expanding Mappings in Complete Cone Metric Spaces. Journal of Mathematical Sciences and Applications, 3(1), 1-2.
  • Prudhvi, K.. "A Fixed Point Result of Expanding Mappings in Complete Cone Metric Spaces." Journal of Mathematical Sciences and Applications 3, no. 1 (2015): 1-2.

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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

(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

(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.
In article      View Article
 
[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.
In article      View Article
 
[3]  A. Azam, M. Arshad, I. Beg, Banach contraction principle on cone rectangular metric spaces,Appl. Anal. Discrete Math., 3(2),(2009), 236-241.
In article      View Article
 
[4]  P. Z. Daffer and H. Kaneko, On expansive mappings, Math. Japon 37 (1992), 733-735.
In article      
 
[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.
In article      View Article
 
[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.
In article      
 
[7]  D. IIic, V. Rakocevic, Common fixed points for maps on cone metric space, J. Math. Anal. Appl. 341(2008) 876-886.
In article      View Article
 
[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.
In article      View Article
 
[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.
In article      
 
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