Around metric coincidence point theory

Abstract

Let $(X,d)$ be a complete metric space, $(Y,\rho)$ be a metric space and $f,g:X\to Y$ be two mappings. The problem is to give metric conditions which imply that, $C(f,g):=\{x\in X\ |\ f(x)=g(x)\}\not=\emptyset$. In this paper we give an abstract coincidence point result with respect to which some results such as of Peetre-Rus (I.A. Rus, \emph{Teoria punctului fix \^in analiza func\c tional\u a}, Babe\c s-Bolyai Univ., Cluj-Napoca, 1973), A. Buic\u a (A. Buic\u a, \emph{Principii de coinciden\c t\u a \c si aplica\c tii}, Presa Univ. Clujean\u a, Cluj-Napoca, 2001) and A.V. Arutyunov (A.V. Arutyunov, \emph{Co\-vering mappings in metric spaces and fixed points}, Dokl. Math., 76(2007), no.2, 665-668) appear as corollaries. In the case of multivalued mappings our result generalizes some results given by A.V. Arutyunov and by A. Petru\c sel (A. Petru\c sel, \emph{A generalization of Peetre-Rus theorem}, Studia Univ. Babe\c s-Bolyai Math., 35(1990), 81-85). The impact on metric fixed point theory is also studied.