On extended versions of Dancs-Hegedus-Medvegyev's fixed-point theorem

In this article, we establish some fixed-point (known also as critical point, invariant point) theorems in quasi-metric spaces. Our results unify and further extend in some regards the fixed-point theorem proposed by Dancs, S.; Hegedus, M.; Medvegyev, P. (A general ordering and fixed-point principle in complete metric space. Acta Sci. Math. 1983;46:381-388), the results given by Khanh, P.Q., Quy D.N. (A generalized distance and enhanced Ekeland?s variational principle for vector functions. Nonlinear Anal. 2010;73:2245-2259), the preorder principles established by Qiu, J.H. (A pre-order principle and set-valued Ekeland variational principle. J. Math. Anal. Appl. 2014;419:904-937) and the results obtained by Bao, T.Q., Mordukhovich, B.S., Soubeyran, A. (Fixed points and variational principles with applications to capability theory of wellbeing via variational rationality. Set-Valued Var. Anal. 2015;23:375-398). In addition, we provide examples to illustrate that the improvements of our results are significant.