Recent results on functional equations in a single variable, perspectives and open problems.

*(English)*Zbl 0972.39011In 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:

\(\sum_{j=0}^N a_j f^j(x)=F(x)\) \((f^j\) 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;

linear equations \(\sum_{j=0}^N A_j(x)f[g_j (x)]=F(x);\)

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 \(f(x)=\int_S f(x+y)\sigma(dy) (S\) is a Borel set, \(\sigma\) a Borel measure on it);

the generalized dilatation equation \(f(x)=\sum_{j=0}^N c_j f(a_j x+b_j);\)

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) \(\sum_{j=0}^{n-1}f[(x+j)/n]=\sum_{k=1}^\infty g_n(k)f(kx)\) \((n=1,2, \dots)\) (some “pathological functions” are characterized by similar equations);

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.

\(\sum_{j=0}^N a_j f^j(x)=F(x)\) \((f^j\) 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;

linear equations \(\sum_{j=0}^N A_j(x)f[g_j (x)]=F(x);\)

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 \(f(x)=\int_S f(x+y)\sigma(dy) (S\) is a Borel set, \(\sigma\) a Borel measure on it);

the generalized dilatation equation \(f(x)=\sum_{j=0}^N c_j f(a_j x+b_j);\)

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) \(\sum_{j=0}^{n-1}f[(x+j)/n]=\sum_{k=1}^\infty g_n(k)f(kx)\) \((n=1,2, \dots)\) (some “pathological functions” are characterized by similar equations);

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.

Reviewer: János Aczél (Waterloo/Ontario)

##### MSC:

39B12 | Iteration theory, iterative and composite equations |

39-02 | Research exposition (monographs, survey articles) pertaining to difference and functional equations |

26A18 | Iteration of real functions in one variable |

26A27 | Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives |

26E25 | Set-valued functions |

28C20 | Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) |

39B22 | Functional equations for real functions |

39B32 | Functional equations for complex functions |

39B62 | Functional inequalities, including subadditivity, convexity, etc. |