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Hyers-Ulam-Rassias stability of Jensen’s equation and its application. (English) Zbl 0909.39014

The author examines the Hyers-Ulam-Rassias stability [see D. H. Hyers, G. Isac and Th. M. Rassias, Stability of functional equations in several variables, Birkhäuser, Boston (1998)] of the Jensen functional equation and proves that a mapping \(f\) from a real normed space \(X\) into a real Banach space \(Y\) satisfying \(f(0)=0\) is additive if and only if \(\| 2f((x+y)/2) -f(x)- f(y)\|\to 0\) as \(\| x\| +\| y\|\to \infty\).

MSC:

39B72 Systems of functional equations and inequalities
39B52 Functional equations for functions with more general domains and/or ranges
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