This article deals with fixed points of mappings in a metric space . The main result is concerned with points of coincidence for a pair of self-mappings and . It is assumed that: (a) there exists a sequence in such that ((E.A) property); (b) for all , the inequality
where is a semi-continuous function with the following properties: is non-increasing in the variable and , there exists such that, for every , the relations and imply , and for all (the authors call such functions “implicit functions of Popa”); and (c) is a complete subspace of . Under these assumptions, the pair has a point of coincidence. Moreover, under the additional assumption that is weakly compatible, the pair has a common fixed point.
As application, the problem of the existence of common fixed points for two finite families of mappings is considered. The article also presents some illustrative examples.