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, Teoria punctului fix în analiza funcțională, Babeș-Bolyai Univ., Cluj-Napoca, 1973), A. Buică (A. Buică, Principii de coincidență și aplicații, Presa Univ. Clujeană, Cluj-Napoca, 2001) and A.V. Arutyunov (A.V. Arutyunov, Covering 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. Petrușel (A. Petrușel, A generalization of Peetre-Rus theorem, Studia Univ. Babeș-Bolyai Math., 35(1990), 81-85). The impact on metric fixed point theory is also studied.