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Recent results on functional equations in a single variable, perspectives and open problems. (English) Zbl 0972.39011

In 1990 M. Kuczma, B. Choczewski, and 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:

j=0 N a j f j (x)=F(x) (f j is the jth iterate of f); its particular case a 0 ==a N-1 =0,a N =1, determining Nth iterative roots;

linear equations j=0 N A j (x)f[g j (x)]=F(x);

the composite equations f(F[x,f(x)])=G[x,f(x)] of invariant curves;

Feigenbaum’s equation f[f(cx)]+cf(x)=0;

the integrated Cauchy equation f(x)= S f(x+y)σ(dy)(S is a Borel set, σ a Borel measure on it);

the generalized dilatation equation f(x)= j=0 N c j f(a j x+b j );

Schilling’s equation 4qf(qx)=f(x-1)+f(x+1)+2f(x) (with solutions of remarkably different nature for different q);

Daróczy’s equation f(x)=f(x+1)+f[x(x+1)];

extended (systems of) replicative equation(s) j=0 n-1 f[(x+j)/n]= k=1 g n (k)f(kx) (n=1,2,) (some “pathological functions” are characterized by similar equations);

Abel’s equation f[g(x)]=f(x)+1;

Schröder’s equation f[g(x)]=cf(x);

and (this one on complex functions) P[z,f(z),f(qz)]=0, where P is a polynomial.

Some functional inequalities and equations for set valued functions are also discussed.

MSC:
39B12Iterative and composite functional equations
39-02Research monographs (functional equations)
26A18Iteration of functions of one real variable
26A27Nondifferentiability of functions of one real variable; discontinuous derivatives
26E25Set-valued real functions
28C20Set functions and measures and integrals in infinite-dimensional spaces
39B22Functional equations for real functions
39B32Functional equations for complex functions
39B62Functional inequalities, including subadditivity, convexity, etc. (functional equations)