Common Fixed Point Results in Cone Metric Spaces Using Altering Distance Function

Tanmoy Som,Lokesh Kumar

doi:10.5923/j.ajms.20120206.09

Abstract

Cone metric space was introduced by Huang Long-Guang et al. (2007) which generalized the concept of metric space. Several fixed point results have been proved in such spaces which generalized and extended the analogous results in metric spaces by different authors. In the present paper two common fixed point results for a sequence of self maps of a complete cone metric space, using altering distance function between the points under a certain continuous control function, are obtained, which generalize the results of Sastry et al. (2001) and Pandhare et al. (1998). Two examples are given in support of our results.

Highlights

Keywords Co mplete Cone Metric Space, Altering Distance Function, Co mmon Fixed Po int Results concerning the existence and properties of fixed points are known as fixed point theorems
Let Ф be the set of all continuous self maps φφ of PP satisfying (i) φφ is monotone increasing (ii) φφ(tt) = 0 if and only if tt = 0 it is called an altering distance function on the cone PP
We obtain two fixed point results on a complete cone met ric space generalizing Theorem 2 of Sastry and Babu[9] and Pandhare and Waghmode[4] in turn

Summary

Introduction

Results concerning the existence and properties of fixed points are known as fixed point theorems. As the set of real numbers is well ordered but the concerned Banach space is only partially ordered, so it is a task to extend the existing results in metric space to cone metric spaces if possible. Definiti on 1.2[2] Let (XX, dd) be a cone metric space, {xxnn } a sequence in XX and xx εε XX. Tanmoy Som et al.: Common Fixed Point Results in Cone M etric Spaces UsingAltering Distance Function (XX, dd) is called a co mplete cone metric space if every Cauchy sequence in XX is convergent. Let Ф be the set of all continuous self maps φφ of PP satisfying (i) φφ is monotone increasing (ii) φφ(tt) = 0 if and only if tt = 0 it is called an altering distance function on the cone PP

Main Results

Conclusions

Examples