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We revisit Perov's fixed point theorem for selfmaps of a set endowed with a vector metric taking values in the Euclidean space Rm. In particular, we show that this result is subsumed by the classical Banach contraction principle. We also obtain a generalization of Perov's theorem by con- sidering mappings on K-metric spaces satisfying a nonlinear Lipschitz condition. Two applications are presented and some characterizations of convergence in K-metric spaces are given. [ABSTRACT FROM AUTHOR]

Abstract: 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 Harjani and Sadarangani [J. Harjani, K. Sadarangani, Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Anal. 72 (2010) 1188–1197], and Amini-Harandi and Emami [A. Amini-Harandi, H. Emami A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations, Nonlinear Anal. 72 (2010) 2238–2242] do follow from an earlier result of O’Regan and Petruşel [D. O’Regan and A. Petruşel, Fixed point theorems for generalized contractions in ordered metric spaces, J. Math. Anal. Appl. 341 (2008) 1241–1252]. [Copyright &y& Elsevier]

Abstract: Recently, Ćirić [Lj.B. Ćirić, Solving the Banach fixed point principle for nonlinear contractions in probabilistic metric spaces, Nonlinear Anal. 72 (2010) 2009–2018] obtained a fixed point theorem with the intention to get a probabilistic version of the Boyd–Wong theorem [D.W. Boyd, J.S.W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969) 458–464]. We give a counterexample to the key lemma of Ćirić, and we establish a corrected version of his main theorem. We also discuss the relations between our result and fixed point theorems of Browder [F.E. Browder, On the convergence of successive approximations for nonlinear functional equations, Nederl. Akad. Wetensch. Proc. Ser. A 71=Indag. Math. 30 (1968) 27–35] and Matkowski [J. Matkowski, Integrable solutions of functional equations, Dissertationes Math. (Rozprawy Mat.) 127 (1975), 68 pp.]. [Copyright &y& Elsevier]

Abstract: We show that most contractive conditions of integral type given recently by many authors coincide with classical ones. This is done with the help of a geometric lemma on subsets of the quadrant . In particular, we extend a recent result of Suzuki who observed that an integral version of the Banach contraction principle given by Branciari could be derived from a classical fixed point theorem of Meir and Keeler. Nevertheless, we also give a new contractive condition of integral type which is independent of classical ones since it does not force the nonexpansiveness of a mapping. [Copyright &y& Elsevier]

Abstract: Recently Kirk introduced the notion of asymptotic contractions on a metric space and using ultrapower techniques he obtained an asymptotic version of the Boyd–Wong fixed point theorem. In this paper we extend this result and moreover, we give a constructive proof of it. Furthermore, we obtain a complete characterization of asymptotic contractions on a compact metric space. As a by-product we establish a separation theorem for upper semicontinuous functions satisfying some limit condition. [Copyright &y& Elsevier]

Abstract: We give a stationary point theorem for some set-valued mappings on a metric space. The existence of fixed points of such mappings characterizes the metric completeness. Our result easily yields the order-theoretic Cantor theorem of Granas and Horvath, and the famous Ekeland variational principle. [ABSTRACT FROM AUTHOR]

Abstract: W.A. Kirk [W.A. Kirk, Fixed points of asymptotic contractions, J. Math. Anal. Appl. 277 (2003) 645–650] defined the notion of an asymptotic contraction on a metric space and using ultrapower techniques he gave a nonconstructive proof of an asymptotic version of the Boyd–Wong fixed point theorem. Subsequently, I.D. Arand̄elović [I.D. Arand̄elović, On a fixed point theorem of Kirk, J. Math. Anal. Appl. 301 (2005) 384–385] established somewhat more general version of Kirk''s result and he gave an elementary proof of it. However, our purpose is to show that there is an error in this proof and, moreover, Arand̄elović''s theorem is false. We also explain how to correct this result. [Copyright &y& Elsevier]