Nonlinear contractive conditions: A comparison and related problems

• Autorzy: Jacek Jachymski , Izabela Jóźwik
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 26A48: Monotonic functions, generalizations
• 54C10: Special maps on topological spaces (open, closed, perfect, etc.)
• 26A18: Iteration
• 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.)
• 26A51: Convexity, generalizations
• 47H10: Fixed-point theorems
• 54H25: Fixed-point and coincidence theorems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2007
• Tom: 77
• Numer: 1
• Strony: 123-146

Daty

wydano

2007

Twórcy

autor

Jacek Jachymski

• Institute of Mathematics, Technical University of Łódź, Wólczańska 215, 93-005 Łódź, Poland

autor

Izabela Jóźwik

• Institute of Mathematics and Centre of Mathematics and Physics, Technical University of Łódź, al. Politechniki 11, 90-924 Łódź, Poland

Identyfikatory

DOI

10.4064/bc77-0-10