The hyperstability of general linear equation via that of Cauchy equation.

*(English)*Zbl 1417.39085The 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.

Reviewer: Mohammad Sal Moslehian (Mashhad)

##### MSC:

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) |

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\textit{T. Phochai} and \textit{S. Saejung}, Aequationes Math. 93, No. 4, 781--789 (2019; Zbl 1417.39085)

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##### References:

[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 |

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