Generalizations of Caristi Kirk's Theorem on Partial Metric Spaces

Erdal Karapinar

doi:10.1186/1687-1812-2011-4

Abstract

AbstractIn this article, lower semi-continuous maps are used to generalize Cristi-Kirk's fixed point theorem on partial metric spaces. First, we prove such a type of fixed point theorem in compact partial metric spaces, and then generalize to complete partial metric spaces. Some more general results are also obtained in partial metric spaces.2000 Mathematics Subject Classification 47H10,54H25

Highlights

Introduction and preliminariesIn 1992, Matthews [1,2] introduced the notion of a partial metric space which is a generalization of usual metric spaces in which d(x, x) are no longer necessarily zero
Let T : X ® X be an arbitrary self-mapping on X such that p(x, Tx) ≤ φ(x) − φ(Tx) for all x ∈ X
The following lemma will be used in the proof of the main theorem

Summary

Introduction

Introduction and preliminariesIn 1992, Matthews [1,2] introduced the notion of a partial metric space which is a generalization of usual metric spaces in which d(x, x) are no longer necessarily zero. 2. Main Results Let (X, p) be a PMS, c ⊂ X and : C ® R+ a function on C. The function is called a lower semi-continuous (l.s.c) on C whenever lim n→∞ Let T : X ® X be an arbitrary self-mapping on X such that p(x, Tx) ≤ φ(x) − φ(Tx) for all x ∈ X.

Results

Conclusion