On fixed points of contraction maps acting in (q 1 ; q 2 )-quasimetric spaces and geometric properties of these spaces

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. © 2017, Eurasian Mathematical Journal.