Notes on stability of the generalized gamma functional equation
type: Item
instance of: scholarly article
title: Notes on stability of the generalized gamma functional equation
zbMATH Open document ID: 1013.39015
DOI: 10.1155/S0161171202112129
author: Gwang Hui Kim
author: Bing Xu
author: Weinian Zhang
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2002-10-29T00:00:00Z
full work available at URL: https://eudml.org/doc/50063
review text: The authors consider the generalized gamma functional equation (f.e.) \(g(x+ p)= a(x) g(x)\) and prove the following theorem. Let \(f: (0,\infty)\to \mathbb{R}\) be an approximate solution which satisfies \(|f(x+ p)- a(x)f(x)|\leq\delta\), \(\forall x> n_0\). If the function \(a(x)\) satisfies \[ \liminf_{k\to\infty} a(x+ pk)> 1,\quad \forall x> n_0 \] then the mentioned f.e. has the Hyers-Ulam stability. The authors consider also the generalized Hyers-Ulam-Rassias stability of the same equation. Let \(f: (0,\infty)\to \mathbb{R}\) be an approximate solution of the considered f.e. such that \(|f(x+ p)- a(x)f(x)|\leq \psi(x)\), \(\forall x> n_0\), \(\psi: (0,\infty)\to (0,\infty)\) fixed. If \[ \liminf_{k\to\infty} {\psi(x+ p(k-1))\over \psi(x+ pk)} a(x+ pk)< 1,\;\forall x> n_0 \] then the considered f.e. has the generalized Hyers-Ulam-Rassias stability, i.e. there exists a solution \(g\) of the f.e. such that \(|f(x)- g(x)|\leq\phi(x)\), for some fixed function \(\phi(x)\) depending only on \(\psi(x)\).
reviewed by: Borislav Crstici
Mathematics Subject Classification ID: 39B82
Mathematics Subject Classification ID: 39B22
zbMATH DE Number: 1821312
zbMATH Keywords: generalized gamma functional equation
zbMATH Keywords: generalized Hyers-Ulam-Rassias stability
MaRDI profile type: Publication
OpenAlex ID: W2136858878
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description: scientific article; zbMATH DE number 1821312