Asymmetric structures, discontinuous contractions and iterative approximation of fixed and periodic points

In quasi-pseudometric spaces (X, p) (not necessarily Hausdorff), the concepts of the left quasi-closed maps (generalizing continuous maps) and generalized quasi-pseudodistances J : X x X -> [0,infinity) (generalizing in metric spaces: metrics, Tataru distances, w-distances of Kada et al., tau-distances of Suzuki and tau-functions of Lin and Du) are introduced, the asymmetric structures on X determined by J (generalizing the asymmetric structure on X determined by quasi-pseudometric p) are described and the contractions T : X -> X with respect to J (generalizing Banach and Rus contractions) are defined. Moreover, if (X, p) are left sequentially complete (in the sense of Reilly, Subrahmanyam and Vamanamurthy), then, for these contractions T : X -> X such that T-[q] is left quasi-closed for some q is an element of N, the global minimum of the map x -> J(x, T-[q](x)) is studied and theorems concerning the existence of global optimal approximate solutions of the equation T-[q](x) = x are established. The results are new in quasi-pseudometric and quasi-metric spaces and even in metric spaces. Examples showing the difference between our results and the well-known ones are provided. In the literature the fixed and periodic points in not Hausdorff spaces were not studied.