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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.

Dedicated to Professor Th. M. Rassias, on the occasion of his 60th birthdayWe prove the stability of some equations of a single variable, including a nonlinear functional equation, a linear functional equation as well as a Volterra integral equation, by using the weighted space method. Our results generalize and extend some recent theorems given in this field, with simplified proofs. Several direct applications of these results are also obtained.

In this paper, we obtain the general solution and we prove the generalized Hyers-Ulam stability for an affine functional equation.