On Some Classes of Enriched Cyclic Contractive Self-Mappings and Their Boundedness and Convergence Properties

This paper focuses on dealing with several types of enriched cyclic contractions defined in the union of a set of non-empty closed subsets of normed or metric spaces. In general, any finite number p ≥ 2 of subsets is permitted in the cyclic arrangement. The types of examined single-valued enriched cyclic contractions are, in general, less stringent from the point of view of constraints on the self-mappings compared to p-cyclic contractions while the essential properties of these last ones are kept. The convergence of distances is investigated as well as that of sequences generated by the considered enriched cyclic mappings. It is proved that, both in normed spaces and in simple metric spaces, the distances of sequences of points in adjacent subsets converge to the distance between such subsets under weak extra conditions compared to the cyclic contractive case, which is simply that the contractive constant be less than one. It is also proved that if the metric space is a uniformly convex Banach space and one of the involved subsets is convex then all the sequences between adjacent subsets converge to a unique set of best proximity points, one of them per subset which conform a limit cycle, although the sets of best proximity points are not all necessarily singletons in all the subsets.