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In quasi-gauge spaces (X,P) (in the sense of Dugundji and Reilly), we introduce the concept of the left (right) J-family of generalized quasi-pseudodistances, and we use this J-family to define the new kind of left (right) J-sequential completeness, which extends (among others) the usual P-sequential completeness. We use this J-family to construct more general contractions than those of Banach and Rus, and for such contractions (which are not necessarily continuous), we establish the conditions guaranteeing the existence of periodic points (when (X,P) is not Hausdorff), fixed points (when (X,P) is Hausdorff), and iterative approximation of these points. The results are new in quasi-gauge, topological and quasi-uniform spaces and, in particular, generalize the well-known theorems of Banach and Rus types in the matter of fixed points. Various examples illustrating ideas, methods of investigations, definitions and results, and fundamental differences between our results and the well-known ones are given.

New completeness and periodic points of discontinuous contractions of Banach-type in quasi-gauge spaces without Hausdorff property