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On a local stability of the Jensen functional equation. (English) Zbl 0702.39007
A theorem of F. Skof [Rend. Semin. Mat. Fis. Milano 53, 113-129 (1983; Zbl 0599.39007)] asserts that Cauchy’s functional equation on a restricted domain in is stable. In this paper, the above result is extended to higher dimensional N ; and similar results are observed for the Jensen functional equation. In particular, it is shown that if X is a real Banach space and g: (-a,a) N X is ϵ-Jensen in the sense that 2g((x+y)/2)-g(x)-g(y)ϵ for all x, y in (-a,a) N , then there exists a Jensen function G: N X such that G(x)-g(x)(25N-4)ϵ.
Reviewer: C.T.Ng

MSC:
39B72Systems of functional equations and inequalities
39B52Functional equations for functions with more general domains and/or ranges
26B25Convexity and generalizations (several real variables)