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Generalized stability of the Cauchy and the Jensen functional equations on spheres. (English) Zbl 1157.39019

The author investigates the conditional stability problems for the Cauchy and the Jensen functional equation. One of the main theorems concerning the Cauchy equation states: Let \((X, +)\) be an abelian group uniquely divisible by \(2\), \((Y,\|\cdot\|)\) be a Banach space and \(Z\) be a nonempty set. Assume that \(\gamma: X\to Z\) and \(\varphi : X\times X\to [0, \infty)\) satisfy some given conditions. For any function \(f : X\to Y\) with the property
\[ \gamma(x) = \gamma(y)\text{ implies } \| f (x + y) - f (x) - f (y)\|\leq\varphi(x, y), \]
there exists a unique function \(F : X\to Y\) with the properties
\[ \gamma(x) = \gamma(y)\text{ implies }F (x + y) = F (x) + F (y) \]
and
\[ \|f (x) - F (x)\|\leq\Phi(x) \]
for all \(x\in X\) (see Theorem 2.1 of this paper for the definition of \(\Phi\)).

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B55 Orthogonal additivity and other conditional functional equations
39B52 Functional equations for functions with more general domains and/or ranges
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