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Hyperstability of the Jensen functional equation. (English) Zbl 1299.39022
S.-M. Jung, M. S. Moslehian and P. K. Sahoo [J. Math. Inequal. 4, No. 2, 191–206 (2010; Zbl 1219.39016)] investigated the conditional stability of the generalized Jensen functional equation \(f(ax+by)=af(x)+bf(y)\). Based on a fixed point method, the authors of the present paper consider the hyperstability problem of the classical Jensen equation \(f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2}\), where \(f\) is a mapping from a normed space \(X\) into a Banach space \(Y\) such that \(x, y, (x+y)/2\) are in a nonempty subset \(U\) of \(X\).

39B82 Stability, separation, extension, and related topics for functional equations
39B62 Functional inequalities, including subadditivity, convexity, etc.
47H14 Perturbations of nonlinear operators
47H10 Fixed-point theorems
47J20 Variational and other types of inequalities involving nonlinear operators (general)
Full Text: DOI
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