In this paper we prove a fixed‐point theorem for a class of operators with suitable properties, in very general conditions. Also, we show that some recent fixed‐points results in Brzdęk et al., (2011) and Brzdęk and Ciepliński (2011) can be obtained directly from our theorem. Moreover, an affirmative answer to the open problem of Brzdęk and Ciepliński (2011) is given. Several corollaries, obtained directly from our main result, show that this is a useful tool for proving properties of generalized Hyers‐Ulam stability for some functional equations in a single variable.