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Fixed Points and Completeness in Metric and Generalized Metric Spaces

Abstract

The famous Banach contraction principle holds in complete metric spaces, but completeness is not a necessary condition: there are incomplete metric spaces on which every contraction has a fixed point. The aim of this paper is to present various circumstances in which fixed point results imply completeness. For metric spaces, this is the case of Ekeland’s variational principle and of its equivalent, Caristi’s fixed point theorem. Other fixed point results having this property will also be presented in metric spaces, in quasi-metric spaces, and in partial metric spaces. A discussion on topology and order and on fixed points in ordered structures and their completeness properties is included as well.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 22, No. 1, pp. 127–215, 2018.

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Cobzaş, S. Fixed Points and Completeness in Metric and Generalized Metric Spaces. J Math Sci 250, 475–535 (2020). https://doi.org/10.1007/s10958-020-05027-1

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