Notes on stability of the generalized gamma functional equation
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 \(\psi(x)\).
generalized Hyers-Ulam-Rassias stability
scientific article; zbMATH DE number 1821312