Analysis of spaces equipped with functions of the type of distance, different ways of introducing topology, metizability, applications of remetrization techniques, fixed point theorems, iterated function systems on semimetric spaces.
Solving the problem of Dung and Hang set in 2017 in J. Fixed Point Theory Appl. - a full characterization of regular semimetric spaces was obtained. A general method was given for constructing semimetrics satisfying the so-called generalized polygonal inequality and having the property that no open ball is an open set, and at the same time no closed ball is a closed set, which answers the question put by Khamsi and Hussain in 2010 in Nonlinear Anal. The optimal constant occurring in the inequality stating on Lipschitz's equivalence of quasimetric and metric was obtained, which strengthens Schroeder's 2006 result published in Conform. Geom. Dyn. Cantor's theorem on the intersection of a sequence of subsets (not necessarily closed) of a quasimetric space has been generalized, which allowed to provide a new proof of the quasimetric version of the Banach fixed point principle obtained by Bakhtin in 1989. PhD thesis (K. Chrząszcz) - completed. Another PhD thesis (F. Turoboś) in preparation.
K. Chrząszcz, J. Jachymski, F. Turoboś, Two refinements of Frink's metrization theorem and fixed point results for Lipschitzian mappings on quasimetric spaces, Aequationes Mathematicae (25 pkt.), 93 (2019), 277-297, JCR
J. Jachymski, An extension of Cantor's intersection theorem and well posedness of the fixed point problem for discontinuous mappings, Journal of Nonlinear and Convex Analysis (25 pkt.), Volume 19 (Nr 6) (2018), 995-1004, JCR
K. Chrząszcz, J. Jachymski, F. Turoboś, On characterizations and topology of regular semimetric spaces, Publicationes Mathematicae-Debrecen (20 pkt.), 93 (2018), 87-105, JCR