We develop an axiomatic approach to the theory of Sobolev spaces on metric measure spaces and we show that this axiomatic construction covers the main known examples (Hajtasz Sobolev spaces, weighted Sobolev spaces, Upper-gradients, etc). We then introduce the notion of variational p-capacity and discuss its relation with the geometric properties of the metric space. The notions of p-parabolic ...

Diophantine approximation is traditionally the study of how well real numbers are approximated by rationals. We propose a model for studying in an arbitrary totally bounded metric space where rationals replaced with countable hierarchy “well-spread” points, which we refer to as abstract prove various Jarník–Besicovitch type dimension bounds and investigate their sharpness.

The xed point result in Mustafa-Sims metrical structures obtained by Karapinar and Agarwal[Fixed Point Th. Appl., 2013, 2013:154] is deductible from a corresponding one stated in terms ofanticipative contractions over the associated (standard) metric space.

The purpose of this paper is to establish fixed point results for a single mapping in a partially ordered modular metric space, and to prove a common fixed point theorem for two self-maps satisfying some weak contractive inequalities.