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Equivalent conditions for generalized contractions on (ordered) metric spaces. (English) Zbl 1201.54034
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 J. Harjani and K. Sadarangani [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 3–4, A, 1188–1197 (2010; Zbl 1220.54025)] and A. Amini-Harandi and H. Emami [ibid. 72, No. 5, A, 2238–2242 (2010; Zbl 1197.54054)] do follow from an earlier result of D. O’Regan and A. Petruşel [J. Math. Anal. Appl. 341, No. 2, 1241–1252 (2008; Zbl 1142.47033)].

MSC:
54H25 Fixed-point and coincidence theorems (topological aspects)
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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