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Fixed points and generalized Hyers-Ulam stability. (English) Zbl 1252.39030
Summary: 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 [J. Brzdȩk, J. Chudziak and Z. Páles, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 17, 6728–6732 (2011; Zbl 1236.39022); J. Brzdȩk and K. Ciepliński, ibid., 74, No. 18, 6861-6867 (2011; Zbl 1237.39022)] can be obtained directly from our theorem. Moreover, an affirmative answer to the open problem of Brzdȩk and Ciepliński [loc. cit.] 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.

MSC:
39B82 Stability, separation, extension, and related topics for functional equations
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