zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.

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)