×

zbMATH — the first resource for mathematics

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 \(X\).

MSC:
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)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bourgin, D. G., Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J., 16, 385-397, (1949) · Zbl 0033.37702
[2] Brzdȩk, J.; Chudziak, J.; Páles, Zs., A fixed point approach to stability of functional equations, Nonlinear Anal., 74, 6728-6732, (2011) · Zbl 1236.39022
[3] L. Cǎdariu and V. Radu, Fixed point and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math., 2 (2003), Art. ID 4. · Zbl 1219.39016
[4] Hyers, D. H., On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A., 27, 222-224, (1941) · Zbl 0061.26403
[5] Jung, S.-M., Hyers-Ulam-Rassias stability of jensen’s equation, Proc. Amer. Math. Soc., 126, 3137-3143, (1998) · Zbl 0909.39014
[6] Jung, S.-M.; Moslehian, M. S.; Sahoo, P. K., Stability of a generalized Jensen equation on restricted domains, J. Math. Ineq., 4, 191-206, (2010) · Zbl 1219.39016
[7] Kominek, Z., On a local stability of the Jensen functional equation, Demonstratio Math., 22, 499-507, (1989) · Zbl 0702.39007
[8] Lee, Y.-H.; Jun, K.-W., A generalization of the Hyers-Ulam-Rassias stability of jensens equation, J. Math. Anal. Appl., 238, 305-315, (1999) · Zbl 0933.39053
[9] Maksa, Gy.; Páles, Zs., Hyperstability of a class of linear functional equations, Acta Math. Acad. Paedag. Nyíregyháziensis, 17, 107-112, (2001) · Zbl 1004.39022
[10] Rassias, J. M., On the Ulam stability of Jensen and Jensen type mappings on restricted domains, J. Math. Anal. Appl., 281, 516-524, (2003) · Zbl 1028.39011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.