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Characterizations of quasi-metric and G-metric completeness involving w-distances and fixed points

Karapınar Erdal, Romaguera Salvador, Tirado Pedro

Demonstratio Mathematica

2022

Vol. 55, nr 1

939--951

Involving w-distances we prove a fixed point theorem of Caristi-type in the realm of (non-necessarily T1) quasi-metric spaces. With the help of this result, a characterization of quasi-metric completeness is obtained. Our approach allows us to retrieve several key examples occurring in various fields of mathematics and computer science and that are modeled as non- T1 quasi-metric spaces. As an application, we deduce a characterization of complete G -metric spaces in terms of a weak version of Caristi’s theorem that involves a G-metric version of w-distances.

Some extensions of banach contraction principle in G-metric spaces

Gaba Y. U.

Fasciculi Mathematici

2018

Nr 61

55--70

We present different extensions of the Banach contraction principle in the G-metric space setting. More precisely, we consider mappings for which the contractive condition is satisfied by a power of the mapping and for which the power depends on the specified point in the space. We first state the result in the continuous case and later, show that the continuity is indeed not necessary. Imitating some techniques obtained in the metric case, we prove that under certain conditions, it is enough for the contractive condition to be verified on a proper subset of the space under consideration. These results generalize well known comparable results.