Geodesic
On fixed points of contraction maps acting in $(q_1, q_2)$-quasimetric spaces and geometric properties of these spaces
Eurasian mathematical journal, Tome 8 (2017) no. 3, pp. 70-76.

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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. 
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R. Sengupta. On fixed points of contraction maps acting in $(q_1, q_2)$-quasimetric spaces and geometric properties of these spaces. Eurasian mathematical journal, Tome 8 (2017) no. 3, pp. 70-76. http://geodesic.mathdoc.fr/item/EMJ_2017_8_3_a7/

[1] A. V. Arutyunov, A. V. Greshnov, “Theory of $(q_1, q_2)$-quasimetric spaces and coincidence points”, Doklady Mathematics, 94:1 (2016), 434–437 | DOI | MR | Zbl