Fixed and Periodic Point Theorems on Symmetric Spaces

Priyanka Kumari ,J Madhusudan Rao

doi:10.13189/ms.2014.020106

Abstract

This paper proves the existence of periodic and fixed points for self maps satisfying some contractive conditions in symmetric space and also we prove coincidence and fixed points without continuity requirement satisfying a slightly more general Seghal’s contractive conditions with suitable example.

Highlights

( ) In [1], the authors gave a notion E.A which generalizes the concept of noncompatible mappings in metric spaces, and they proved some common fixed-point theorems for noncompatible mappings under strict contractive conditions
Some common fixed point theorems due to Aamri and El Moutawakil [1],Pant and Pant [4] proved for strict contractive mappings in metric spaces are extended to symmetric spaces under tight conditions. we present a few results that establish the existence of common periodic points for a pair of maps on a symmetric space when the maps have a unique common fixed point
These results are supported by suitable examples

Summary

Consider the sequences xn

Corollary 2.4: Let ( X , d ) be a symmetric (semi-metric) space that enjoys (W 3) (the Hausdorff separation axiom).Let g be a self map of X such that for all x ≠ y ∈ X d (g x , g y) < max{d (x , g x) , d ( y , g y) , d (x , y)} g has a unique fixed point. Theorem 2.6: Let g be a self map on a symmetric space ( X , d ) satisfying d (gx , gy) < max{d (x , y), d (x , g(x)), d ( y, g( y))} For each x , y ∈ X (x ≠ y) for which the right hand side of above inequality is not zero. U ∈ X is a periodic point of g if and only if u is the unique fixed point of g

To illustrate the above theorem we have the following Example

Which is a contradiction