On a factorization of mappings with a prescribed behaviour of the Cauchy difference

Abstract

We deal with functional congruences of the form /(* + y) - /(*) - f(9) € U + V where U and V are given sets subjected to satisfy some "separability" conditions essentially weaker than that occurring in [4] which proved to be pretty useful especially while investigating various types of Hyers-Ulam stability problems. The goal is to factorize / into a sum of two functions whose Cauchy differences remain in U and V, respectively, or, at least to obtain an approximation of / by such a sum. An application of the newly established result in that spirit is given. Moreover, a stability result for the celebrated cocycle equation is presented and, finally, the behaviour of mappings whose Cauchy differences fall into a given Haniel basis is described.