Hu’s characterization of metric completeness revisited – Advances in the theory of nonlinear analysis and its applications

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Original research article

Hu's characterization of metric completeness revisited

Author(s):

Salvador Romaguera

Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, 46022 Valencia, Spain.

Advances in the Theory of Nonlinear Analysis and its Applications 6(4), 476-480.

Received: March 18, 2022

Accepted: July 3, 2022

Published: July 7, 2022

Abstract

In this note we show the somewhat surprising fact that the proof of the `if part’ of the distinguished characterizations of metric completeness due to Kirk, and Suzuki and Takahashi, respectively, can be deduced in a straightforward manner from Hu’s theorem that a metric space is complete if and only if any Banach contraction on bounded and closed subsets thereof has a fixed point. We also take advantage of this approach to easily deduce a characterization of metric completeness via fixed point theorems for α−ψ-contractive mappings.

Keywords: Fixed point, Complete metric space, Hu, Caristi–Kirk, Suzuki–Takahashi.

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APA Style

Bonilla, S. R. Hu’s characterization of metric completeness revisited. Advances in the Theory of Nonlinear Analysis and its Application, 6(4), 476-480.