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Stability of functional equations in single variable. (English) Zbl 1053.39042

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.

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
26A18 Iteration of real functions in one variable
39B12 Iteration theory, iterative and composite equations
39B22 Functional equations for real functions
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