Nonlinear contractive conditions: A comparison and related problems

• Volume: 77, Issue: 1, page 123-146
• ISSN: 0137-6934

top

Abstract

top
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.

How to cite

top

Jacek Jachymski, and Izabela Jóźwik. "Nonlinear contractive conditions: A comparison and related problems." Banach Center Publications 77.1 (2007): 123-146. <http://eudml.org/doc/281677>.

@article{JacekJachymski2007,
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.},
author = {Jacek Jachymski, Izabela Jóźwik},
journal = {Banach Center Publications},
keywords = {nonlinear contractive conditions; -contractive map; contractive fixed point; approximate fixed point; domain invariance theorem; separation theorem; semi-continuous function; monotonic function; concave function; subadditive function},
language = {eng},
number = {1},
pages = {123-146},
title = {Nonlinear contractive conditions: A comparison and related problems},
url = {http://eudml.org/doc/281677},
volume = {77},
year = {2007},
}

TY - JOUR
AU - Jacek Jachymski
AU - Izabela Jóźwik
TI - Nonlinear contractive conditions: A comparison and related problems
JO - Banach Center Publications
PY - 2007
VL - 77
IS - 1
SP - 123
EP - 146
AB - 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.
LA - eng
KW - nonlinear contractive conditions; -contractive map; contractive fixed point; approximate fixed point; domain invariance theorem; separation theorem; semi-continuous function; monotonic function; concave function; subadditive function
UR - http://eudml.org/doc/281677
ER -

NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.