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On a local stability of the Jensen functional equation. (English) Zbl 0702.39007
A theorem of {\it 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 ${\bbfR}$ is stable. In this paper, the above result is extended to higher dimensional ${\bbfR}\sp N$; 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)${}\sp N\to X$ is $\epsilon$-Jensen in the sense that $\Vert 2g((x+y)/2)-g(x)-g(y)\Vert \le \epsilon$ for all x, y in $(-a,a)\sp N$, then there exists a Jensen function G: ${\bbfR}\sp N\to X$ such that $\Vert G(x)-g(x)\Vert \le (25N- 4)\epsilon$.
Reviewer: C.T.Ng

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