Hu's characterization of metric completeness revisited

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 $\alpha -\psi $-contractive mappings.

Keywords:

Fixed point, , complete metric space, Hu, Caristi-Kirk Suzuki-Takahashi,

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