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2007 | 77 | 1 | 123-146
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Nonlinear contractive conditions: A comparison and related problems

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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.
Słowa kluczowe
  • Institute of Mathematics, Technical University of Łódź, Wólczańska 215, 93-005 Łódź, Poland
  • Institute of Mathematics and Centre of Mathematics and Physics, Technical University of Łódź, al. Politechniki 11, 90-924 Łódź, Poland
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