Leader type contractions, periodic and fixed points and new completivity in quasi-gauge spaces with generalized quasi-pseudodistances

Abstract

Leaderʼs fixed point theorem – being more general as some Banach, Boyd and Wong, Browder, Burton, Caccioppoli, Dugundji and Granas, Geraghty, Krasnoselʼskiĭ et al., Matkowski, Meir and Keeler, Mukherjea, Rakotch, Tasković, Walter and othersʼ results – have played a great role in metric fixed point theory; in the literature the investigations of periodic points of contractions of Leader or Leader type are not known. We want to show how the introduced here generalized quasi-pseudodistances in quasi-gauge spaces can be used, in a natural way, to define contractions of Leader type and to obtain, for these contractions, the periodic and fixed point theorems without Hausdorff and sequentially complete assumptions about these spaces and without complete graph assumptions about these contractions, which was not done in the previous publications on this subject. The definitions, results and methods presented here are new for maps in quasi-gauge, topological, quasi-pseudometric and quasi-metric spaces. Examples are provided.