Minimization Theorems in $ (q_{1},q_{2}) $-Quasimetric Spaces with Applications

In this paper, we first present some minimization theorems for a new class of nonnegative functionals defined on $ (q_{1},q_{2}) $ -quasimetric spaces. We then apply these results to derive some new coincidence point and fixed point results for mappings in $ (q_{1},q_{2}) $ -quasimetric spaces. In particular, we study sufficient conditions for the existence of coincidence points of a filling mapping and a Lipschitz mapping. Our results improve and generalize several results in the literature including those in Feng and Liu [J. Math. Anal. Appl., vol. 317, no. 1, 103–112 (2006)], Arutyunov [Dokl. Math., vol. 76, no. 2, 665–668 (2007)], Arutyunov and Gel’man [Comput. Math. Math. Phys., vol. 49, no. 7, 1111–1118 (2009)], Fomenko [Topol. Appl., vol. 157, no. 4, 760–773 (2010); Moscow Univ. Math. Bull., vol. 74, no. 6, 227–234 (2019)], and Nguyen and Pasynkov [Topol. Appl., vol. 201, 57–77 (2016)]. Examples are also given to illustrate our results.