A+ CATEGORY SCIENTIFIC UNIT

Nonlinear contractive conditions: A comparison and related problems

Volume 77 / 2007

Jacek Jachymski, Izabela Jóźwik Banach Center Publications 77 (2007), 123-146 MSC: Primary: 47H10, 54H25; Secondary: 26A15, 26A18, 26A48, 26A51, 54C10. DOI: 10.4064/bc77-0-10

Abstract

We establish five theorems giving lists of nonlinear contractive conditions which turn out to be mutually equivalent. We derive them from some general lemmas concerning subsets of the plane which may be applied both in the single- or set-valued case as well as for a family of mappings. A separation theorem for concave functions is proved as an auxiliary result. Also, we discuss briefly the following problems for several classes of contractions: stability of procedure of successive approximations, existence of approximate fixed points, continuous dependence of fixed points on parameters, existence of invariant sets for iterated function systems. Moreover, James Dugundji's contribution to the metric fixed point theory is presented. Using his notion of contractions, we also establish an extension of a domain invariance theorem for contractive fields.

Authors

  • Jacek JachymskiInstitute of Mathematics
    Technical University of Łódź
    Wólczańska 215
    93-005 Łódź, Poland
    e-mail
  • Izabela JóźwikInstitute of Mathematics and Centre of Mathematics and Physics
    Technical University of /L/od/x
    al. Politechniki 11
    90-924 Łódź, Poland
    e-mail

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