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Zbl 1201.54034
Jachymski, Jacek
Equivalent conditions for generalized contractions on (ordered) metric spaces.
(English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 3, 768-774 (2011). ISSN 0362-546X

Summary: We establish a geometric lemma giving a list of equivalent conditions for some subsets of the plane. As its application, we get that various contractive conditions using the so-called altering distance functions coincide with classical ones. We consider several classes of mappings both on metric spaces and ordered metric spaces. In particular, we show that unexpectedly, some very recent fixed point theorems for generalized contractions on ordered metric spaces obtained by {\it J.\,Harjani} and {\it K.\,Sadarangani} [Nonlinear Anal., Theory Methods Appl., Ser.\,A, Theory Methods 72, No.\, 3--4, A, 1188--1197 (2010; Zbl 1220.54025)] and {\it A.\,Amini-Harandi} and {\it H.\,Emami} [ibid.\ 72, No.\,5, A, 2238--2242 (2010; Zbl 1197.54054)] do follow from an earlier result of {\it D.\,O'Regan} and {\it A.\,Petruşel} [J.~Math.\ Anal.\ Appl.\ 341, No.\,2, 1241--1252 (2008; Zbl 1142.47033)].
MSC 2000:
*54H25 Fixed-point theorems in topological spaces
54F05 Ordered topological spaces
47H09 Mappings defined by "shrinking" properties

Keywords: fixed point; generalized contraction; altering distance function; metric space; ordered metric space

Citations: Zbl 1220.54025; Zbl 1197.54054; Zbl 1142.47033

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