On the Properties of Iterations Generated with Composition Maps of Cyclic Contractive Self-Mappings and Strict Contractions in Metric Spaces

Manuel De la Sen ()
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Manuel De la Sen: Automatic Control Group–ACG and Institute of Research and Development of Processes, Faculty of Science and Technology, Department of Electricity and Electronics, University of the Basque Country (UPV/EHU), 48940 Leioa, Spain

Mathematics, 2025, vol. 13, issue 14, 1-31

Abstract: This paper studies the convergence of distances between sequences of points and that of sequences of points in metric spaces. This investigation is focused on the iterative processes built with composed self-mappings of a cyclic contraction, which can involve more than two nonempty closed subsets in a metric space, which are combined with compositions of a strict contraction with itself, which operates in each of the individual subsets, in any order and any number of mutual compositions. It is admitted, in the most general case, the involvement of any number of repeated compositions of both self-maps with themselves. It is basically seen that, if one of the best-proximity points in the cyclic disposal is unique in a boundedly compact subset of the metric space is sufficient to achieve unique asymptotic cycles formed by a best-proximity point per each adjacent subset. The same property is achievable if such a subset is strictly convex and the metric space is a uniformly convex Banach space. Furthermore, all the sequences with arbitrary initial points in the union of all the subsets of the cyclic disposal converge to such a limit cycle.

Keywords: impulsive actions; discontinuities of the first kind; dynamic systems; impulsive dynamic systems; global stability; global asymptotic stability (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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