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A minimum condition and some realted fixed-point theorems

Part of: Nonlinear operators and their properties Connections with other structures, applications

Published online by Cambridge University Press:  09 April 2009

Jacek R. Jachymski  and
James D. Stein Jr
Jacek R. Jachymski
Affiliation:
Institute of Mathematics, Technical University of Łódź, Żwirki 36, 90-924 Łódź, Poland e-mail: jachym@ck-sg.p.lodz.pl
James D. Stein Jr
Affiliation:
Department of Mathematics, California State University, Long Beach, 1250 Bellflower Blvd., Long Beach, California 90840, USA e-mail: jimstein@csulb.edu
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Abstract

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The classic Banach Contraction Principle assumes that the self-map is a contraction. Rather than requiring that a single operator be a contraction, we weaken this hypothesis by considering a minimum involving a set of iterates of that operator. This idea is a central motif for many of the results of this paper, in which we also study how this weakended hypothesis may be applied in Caristi's theorem, and how combinatorial arguments may be used in proving fixed-point theorems.

Keywords

metric spaceasymptotically regular mapcontractionuniformly continuousmapsemi-continuous functionfixed pointperiodic point

MSC classification

Secondary: 47H10: Fixed-point theorems 54H25: Fixed-point and coincidence theorems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 66 , Issue 2 , April 1999 , pp. 224 - 243
Copyright
Copyright © Australian Mathematical Society 1999

References

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