The stability of the functional equation \[ f\bigl(\varphi(X)\bigr)=\phi(X)f(X)\tag{1} \] in the sense of Hyers, Ulam, Rassias and Găvruta is dealt with. The main result of the paper is the following
{Theorem.} Let \(S\neq\emptyset\) and \(B\) be a Banach space over a field \({\mathbb K}\in\{{\mathbb R},{\mathbb C}\}\). Let \(\varphi:S^n\to S^n\), \(\phi:S^n\to{\mathbb K}\setminus\{0\}\) and \(\varepsilon:S^n\to[0,\infty)\). Assume that \(\varphi,\phi,\varepsilon\) satisfy the condition \[ \omega(X):=\sum_{k=0}^{\infty}\frac{\varepsilon\bigl(\varphi_k(X)\bigr)}{\prod_{j=0}^k\Bigl| \phi\bigl(\varphi_j(X)\bigr)\Bigr| }<\infty,\quad X\in S^n \] (\(\varphi_k\) stands for the \(k\)-th iterate of \(\varphi\)). If \(f:S^n\to B\) satisfies the inequality \[ \Bigl\| f\bigl(\varphi(X)\bigr)-\phi(X)f(X)\Bigr\| \leq\varepsilon(X),\quad X\in S^n, \] then there exists the unique solution \(g:S^n\to B\) of the equation (1) with \[ \bigl\| g(X)-f(X)\bigr\| \leq\omega(X),\quad X\in S^n. \] The particular case \(\varphi(X)=X+P\) is also considered. Numerous applications to beta-type functional equations, gamma, \(G\), Schröder, iterative and other functional equations are given.
For the single variable and two-variable cases apart from the papers quoted in References see also K. W. Jun and H. M. Kim [J. Comput. Anal. Appl. 7, No. 4, 397–407 (2005; Zbl 1087.39028)].