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Fixed-Point Theory for Generalized Metric Spaces
Published in Pascal Hitzler, Anthony Seda, Mathematical Aspects of Logic Programming Semantics, 2016
It should be noted that Theorem 4.2.5 is not a true converse of the Banach theorem in that we do not start out with a metrizable space and attempt to obtain a metric for it relative to which f is a contraction. Thus, Theorem 4.2.5 is quite different from those discussed, for example, in Section 3.6 of the text [Istrǎțescu, 1981], in which a number of converses of the Banach theorem are considered. Even the result of Bessaga discussed there, which applies to an abstract set, is very different from ours in that we do not require all iterations of f to have a unique fixed point, but we do require topological convergence of the iterates of any point. Indeed, we can only make the following observations on the relationship between the original topology and the one created by the metric constructed in the proof of Theorem 4.2.5.
b-Metric Spaces
Published in Dhananjay Gopal, Praveen Agarwal, Poom Kumam, Metric Structures and Fixed Point Theory, 2021
Nguyen Van Dung, Wutiphol Sintunavarat
Since every b-metric space is a metrizable space, we find that all topological properties of b-metric spaces and metric spaces are coincident. However, some authors did not know this fact and did some redundant work. For example, in [10], An et al. showed a weaker result that every b-metric space with the topology induced by its convergence is a semi-metrizable space and thus many properties of b-metric spaces used in the literature are obvious. Then, the authors proved the Stone-type theorem on b-metric spaces and found a sufficient condition for a b-metric space to be metrizable.
Densely relaxed pseudomonotone and quasimonotone generalized variational-like inequalities
Published in Optimization, 2021
Bijaya K. Sahu, George Nguyen, Gayatri Pany, Ouayl Chadli
Since every metrizable space is paracompact, then from the Michael's selection theorem (Lemma 2.17) we deduce that there exists a continuous mapping such that for any . First, we verify that F is generalized densely relaxed pseudomonotone. Indeed, since Φ is generalized densely relaxed pseudomonotone, it follows that there exists a segment-dense set such that Φ is generalized relaxed pseudomonotone on . Let and such that . By taking account of the fact that and Φ is generalized relaxed pseudomonotone at , we obtain that Particularly, if we take in the previous inequality, we get Therefore, F is generalized densely pseudomonotone. Hence, from Theorem 3.1 we deduce that there exists such that Thus, there exists such that Which completes the proof of the theorem.