Quasicone Metric Spaces and Generalizations of Caristi Kirk's Theorem

Thabet Abdeljawad,Erdal Karapinar

doi:10.1155/2009/574387

Abstract

Cone-valued lower semicontinuous maps are used to generalize Cristi-Kirik's fixed point theorem to Cone metric spaces. The cone under consideration is assumed to be strongly minihedral and normal. First we prove such a type of fixed point theorem in compact cone metric spaces and then generalize to complete cone metric spaces. Some more general results are also obtained in quasicone metric spaces.

Highlights

Introduction and PreliminariesIn 2007, Huang and Zhang 1 introduced the notion of cone metric spaces CMSs by replacing real numbers with an ordering Banach space
The authors there gave an example of a function which is contraction in the category of cone metric spaces but not contraction if considered over metric spaces and by proving a fixed point theorem in cone metric spaces, ensured that this map must have a unique fixed point
After that series of articles about cone metric spaces started to appear. Some of those articles dealt with the extension of certain fixed point theorems to cone metric spaces see, e.g., 2–5, and some other with the structure of the spaces themselves see, e.g., 3, 6

Summary

Introduction

Introduction and PreliminariesIn 2007, Huang and Zhang 1 introduced the notion of cone metric spaces CMSs by replacing real numbers with an ordering Banach space. Some results about fixed points of multifunctions on cone metric spaces with normal cones have been obtained as well 8 . For the use of lower semicontinuous functions in obtaining fixed point theorems in cone metric spaces we refer to 9 .

Results

Conclusion