A fixed point theorem for contractions in modular metric spaces

The notion of a (metric) modular on an arbitrary set and the corresponding modular space, more general than a metric space, were introduced and studied recently by the author [V. V. Chistyakov, Metric modulars and their application, Dokl. Math. 73(1) (2006) 32-35, and Modular metric spaces, I: Basic concepts, Nonlinear Anal. 72(1) (2010) 1-14]. In this paper we establish a fixed point theorem for contractive maps in modular spaces. It is related to contracting rather ``generalized average velocities'' than metric distances, and the successive approximations of fixed points converge to the fixed points in a weaker sense as compared to the metric convergence.Comment: 31 pages, LaTeX, uses elsarticle.st

Regularity of maximal operators: recent progress and some open problems

This is an expository paper on the regularity theory of maximal operators, when these act on Sobolev and BV functions, with a special focus on some of the current open problems in the topic. Overall, a list of fifteen research problems is presented. It summarizes the contents of a talk delivered by the author at the CIMPA 2017 Research School - Harmonic Analysis, Geometric Measure Theory and Applications, in Buenos Aires, Argentina.Comment: 19 pages. Expository paper with the contents of a lecture given at the in the CIMPA 2017 Research School - Harmonic Analysis, Geometric Measure Theory and Applications, in Buenos Aires, Argentin