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Zbl 0972.39011
Baron, Karol; Jarczyk, Witold
Recent results on functional equations in a single variable, perspectives and open problems. (English)
[J] Aequationes Math. 61, No.1-2, 1-48 (2001). ISSN 0001-9054; ISSN 1420-8903/e

In 1990 {\it M. Kuczma, B. Choczewski}, and {\it R. Ger} published their book ``Iterative functional equations'' in the Encyclopedia of Mathematics and Its Applications series of Cambridge University Press (1990; Zbl 0703.39005). The present very thorough survey (with 281 references!) picks up the thread of works on iterative functional equations (``functional equations in a single variable'': not (only) the unknown functions are of a single variable; also equation contains just one variable) where the 1990 book left and runs with it (and with some prior results). The detailed discussion is, of necessity, more selective. The following very incomplete sampling of some (families of) functional equations discussed (in general for real valued solutions $f,$ for real variable(s)) shows, however, the richness of its contents: \par $\sum_{j=0}^N a_j f^j(x)=F(x)$ $(f^j$ is the $j$th iterate of $f$); its particular case $a_0=\dots= a_{N-1}=0, a_N=1$, determining $N$th iterative roots; \par linear equations $\sum_{j=0}^N A_j(x)f[g_j (x)]=F(x);$ \par the composite equations $f(F[x,f(x)])=G[x,f(x)]$ of invariant curves; \par Feigenbaum's equation $f[f(cx)]+cf(x)=0;$ \par the integrated Cauchy equation $f(x)=\int_S f(x+y)\sigma(dy) (S$ is a Borel set, $\sigma$ a Borel measure on it); \par the generalized dilatation equation $f(x)=\sum_{j=0}^N c_j f(a_j x+b_j);$ \par Schilling's equation $4qf(qx)=f(x-1)+f(x+1)+2f(x)$ (with solutions of remarkably different nature for different $q$); \par Daróczy's equation $f(x)=f(x+1)+f[x(x+1)];$ \par extended (systems of) replicative equation(s) $\sum_{j=0}^{n-1}f[(x+j)/n]=\sum_{k=1}^\infty g_n(k)f(kx)$ $(n=1,2, \dots)$ (some ``pathological functions" are characterized by similar equations); \par Abel's equation $f[g(x)]=f(x)+1;$ \par Schröder's equation $f[g(x)]=cf(x);$ \par and (this one on complex functions) $P[z,f(z),f(qz)]=0,$ where $P$ is a polynomial. \par Some functional inequalities and equations for set valued functions are also discussed.
[János Aczél (Waterloo/Ontario)]

MSC 2000:: *39B12 Iteraterative functional equations
39-02 Research monographs (functional equations)
26A18 Iteration of functions of one real variable
26A27 Nondifferentiability of functions of one real variable
26E25 Set-valued real functions
28C20 Set functions and measures and integrals in infinite-dim. spaces
39B22 Functional equations for real functions
39B32 Functional equations for complex functions
39B62 Systems of functional equations

Keywords: iterative functional equations; iterative roots; Feigenbaum functional equations; linear dilatation; simultaneous and replicative equations; integrated Cauchy equation. inequalities; real functions; functional inequalities; research survey; Daroczy functional equations; Schilling functional equations; Schröder functional equations; Abel functional equations; complex functions; set-valued functions

Citations: Zbl 0703.39005

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