On Fixed Points of Contraction Maps Acting in \((q_1, q_2)\)-Quasimetric Spaces and Geometric Properties of These Spaces

Abstract

We study geometric properties of \((q_1, q_2)\)-quasimetric spaces and fixed point theorems in these spaces. In paper [1], a fixed point theorem was obtained for a contraction map acting in a complete \((q_1, q_2)\)-quasimetric space. The graph of the map was assumed to be closed. In this paper, we show that this assumption is essential, i.e. we provide an example of a complete quasimetric space and a contraction map acting in it whose graph is not closed and which is fixed-point-free. We also describe some geometric properties of such spaces.