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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 $ℝ$ is stable. In this paper, the above result is extended to higher dimensional ${ℝ}^{N}$; 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)${}^{N}\to X$ is $ϵ$-Jensen in the sense that $\parallel 2g\left(\left(x+y\right)/2\right)-g\left(x\right)-g\left(y\right)\parallel \le ϵ$ for all x, y in ${\left(-a,a\right)}^{N}$, then there exists a Jensen function G: ${ℝ}^{N}\to X$ such that $\parallel G\left(x\right)-g\left(x\right)\parallel \le \left(25N-4\right)ϵ$.
Reviewer: C.T.Ng

##### MSC:
 39B72 Systems of functional equations and inequalities 39B52 Functional equations for functions with more general domains and/or ranges 26B25 Convexity and generalizations (several real variables)