On a factorization of mappings with a prescribed behaviour of the Cauchy difference
Abstract
We deal with functional congruences of the form
f(x+y) - f(x) - f(y) ∈ 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 f into a sum of two functions whose Cauchy differences remain in U and V, respectively, or, at least to obtain an approximation of f 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 Hamel basis is described.
References
2. G.L. Forti, On an alternative functional equation related to the Cauchy equation, Aequationes Math. 24 (1982), 195-206.
3. R. Ger, The singular case in the stability behaviour of linear mappings, Grazer Mathematische Berichte 316 (1992), 59-70.
4. R. Ger, P. Šemrl, The stability of the exponential equation, Proc. Amer. Math. Soc. (to appear).
5. M. Hosszú, On the functional equation F(x+y, z) + F(x, y) = F(x, y+z) + F(y, z), Period. Math. Hungar. 1 (1971), 213-216.
6. M. Kuczma, An introduction to the theory of functional equations and inequalities, Polish Scientific Publishers & Silesian University, Warszawa-Kraków-Katowice, 1985.
7. J. Rätz, On approximately additive mappings, General Inequalities 2, Internat. Ser. Numer. Math., vol. 47, Birkhäuser Verlag, Basel, 1980, pp. 233-251.
8. L. Székelyhidi, Stability properties of functional equations in several variables, Technical Report # 91/02, Department of Mathematics, Lajos Kossuth University (1991), 1-6.
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
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