Stability of functional equations in single variable. (Q1419754)

Some functional equations in a single variable are considered: the linear equation \(\varphi\bigl(f(x)\bigr)=g(x)\varphi(x)+h(x)\) with given functions \(f,g,h\) and an unknown function \(\varphi\), the linear equation \(\varphi(x)=g(x)\varphi\bigl(f(x)\bigr)+h(x)\), the nonlinear equation \(\varphi(x)=F\bigl(x,\varphi\bigl(f(x)\bigr)\bigr)\) and the iterative equation \(G\bigl(\varphi(x),\varphi^2(x),\dots,\varphi^n(x)\bigr)=F(x)\). The known results concerning Hyers-Ulam stability and the iterative stability of these equations and of their special cases are surveyed. The authors give also some new results. Namely, the Hyers-Ulam stability of Böttcher's equation \(\varphi\bigl(f(x)\bigr)=\varphi(x)^p\) (\(p\neq 1\)) and of the iterative equation \(G\bigl(x,\varphi(x),\varphi^2(x),\dots,\varphi^n(x)\bigr)=F(x)\) is established.