A Nemytskii-Edelstein type fixed point theorem for partial metric spaces

Naseer Shahzad,Oscar Valero

doi:10.1186/s13663-015-0266-9

Abstract

AbstractIn 1994, Matthews obtained an extension of the celebrated Banach fixed point theorem to the partial metric framework (Ann. N.Y. Acad. Sci. 728:183-197, 1994). Motivated by the Matthews extension of the Banach theorem, we present a Nemytskii-Edelstein type fixed point theorem for self-mappings in partial metric spaces in such a way that the classical one can be retrieved as a particular case of our new result. We give examples which show that the assumed hypothesis in our new result cannot be weakened. Moreover, we show that our new fixed point theorem allows one to find fixed points of mappings in some cases in which the Matthews result and the classical Nemytskii-Edelstein one cannot be applied. Furthermore, we provide a negative answer to the question about whether our new result can be retrieved as a particular case of the classical Nemytskii-Edelstein one whenever the metrization technique, developed by Hitzler and Seda (Mathematical Aspects of Logic Programming Semantics, 2011), is applied to partial metric spaces.

Highlights

1 Introduction In, S Banach proved in the context of metric spaces his celebrated fixed point result
Motivated by this intense research activity in fixed point theory in partial metric spaces, we present a Nemytskii-Edelstein type fixed point theorem for self-mappings in partial metric spaces in such a way that the classical one can be retrieved as a particular case of our new result
In spite of this handicap, we prove that an additional assumption, which is not too much restrictive, on the self-mapping is enough to provide Nemytskii-Edelstein type fixed point theorems in the spirit of the above conjectures

Summary

Introduction

In , S Banach proved in the context of metric spaces his celebrated fixed point result. Let us recall that a function f from a topological space (X, T ) into (R+, T (| · |)) is lower semicontinuous on (X, T ) if and only if f is continuous from (X, T ) to (R+, T (du– )) (see [ ]), where du is the upper quasi-metric introduced in Example .

Results

Conclusion