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Zbl 1053.39042
Agarwal, Ravi P.; Xu, Bing; Zhang, Weinian
Stability of functional equations in single variable. (English)
[J] J. Math. Anal. Appl. 288, No. 2, 852-869 (2003). ISSN 0022-247X

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 $G\bigl(\varphi(x),\varphi^2(x),\dots,\varphi^n(x)\bigr)=F(x)$. \par 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 $\varphi\bigl(f(x)\bigr)=\varphi(x)^p$ ($p\ne 1$) and of the iterative equation $G\bigl(x,\varphi(x),\varphi^2(x),\dots,\varphi^n(x)\bigr)=F(x)$ is established.
[Szymon Wasowicz (Bielsko-Biała)]

MSC 2000:: *39B82 Stability, separation, extension, and related topics
26A18 Iteration of functions of one real variable
39B12 Iteraterative functional equations
39B22 Functional equations for real functions

Keywords: functional equations; iteration; Hyers-Ulam stability; iterative stability; Böttcher's equation

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Zentralblatt MATH Berlin [Germany]