zbMATH — the first resource for mathematics

The hyperstability of general linear equation via that of Cauchy equation. (English) Zbl 1417.39085
The task of a hyperstability problem is to understand when a function which approximately satisfies a functional equation is also a solution of it. M. Piszczek [Aequationes Math. 88, No. 1–2, 163–168 (2014; Zbl 1304.39033)] proved a hyperstability result for general linear equation \(f(ax + by) = Af(x) + Bf(y)\). J. Brzdęk [Acta Math. Hung. 141, No. 1–2, 58–67 (2013; Zbl 1313.39037)] and proved the hyperstability of the Cauchy equation \(f(x+y)=f(x)+f(y)\). In the paper under review, the authors show that the result of Piszczek can be deduced from that of Brzdęk.

39B82 Stability, separation, extension, and related topics for functional equations
39B62 Functional inequalities, including subadditivity, convexity, etc.
47H14 Perturbations of nonlinear operators
47J20 Variational and other types of inequalities involving nonlinear operators (general)
PDF BibTeX Cite
Full Text: DOI
[1] Bahyrycz, A.; Olko, J., Hyperstability of general linear functional equation, Aequationes Math., 90, 527-540, (2016) · Zbl 1341.39013
[2] Bahyrycz, A., On stability and hyperstability of an equation characterizing multi-additive mappings, Fixed Point Theory, 18, 445-456, (2017) · Zbl 1387.39017
[3] Bahyrycz, A.; Olko, J., On stability and hyperstability of an equation characterizing multi-Cauchy-Jensen mappings, Results Math., 73, 55, (2018) · Zbl 1404.39026
[4] Brzdȩk, J.; Chudziak, J.; Páles, Z., A fixed point approach to stability of functional equations, Nonlinear Anal., 74, 6728-6732, (2011) · Zbl 1236.39022
[5] Brzdȩk, J., Hyperstability of the Cauchy equation on restricted domains, Acta Math. Hung., 141, 58-67, (2013) · Zbl 1313.39037
[6] Brzdȩk, J., Remarks on stability of some inhomogeneous functional equations, Aequationes Math., 89, 83-96, (2015) · Zbl 1316.39011
[7] Ng, CT, Jensen’s functional equation on groups, Aequationes Math., 39, 85-99, (1990) · Zbl 0688.39007
[8] Parnami, JC; Vasudeva, HL, On Jensen’s functional equation, Aequationes Math., 43, 211-218, (1992) · Zbl 0755.39008
[9] Piszczek, M., Remark on hyperstability of the general linear equation, Aequationes Math., 88, 163-168, (2014) · Zbl 1304.39033
[10] Piszczek, M., Hyperstability of the general linear functional equation, Bull. Korean Math. Soc., 52, 1827-1838, (2015) · Zbl 1334.39062
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.