Analytic solutions of the functional equation (xy)x=x(yx)

1991 ◽  
Vol 50 (4) ◽  
pp. 1073-1078 ◽  
Author(s):  
A. M. Shelekhov
Keyword(s):  
Functional Equation ◽  
Analytic Solutions

XXIV.—The Solution of a Functional Equation

1952 ◽  
Vol 63 (4) ◽  
pp. 336-345
Author(s):  
A. H. Read
Keyword(s):  
Functional Equation ◽  
Recurrence Relation ◽  
Unknown Function ◽  
Analytic Functions ◽  
Analytic Solutions ◽  
Sequence Of Functions

SynopsisAnalytic solutions of the functional equation f[z, φ{g(z)}] = φ(z), in which f(z, w) and g(z) are given analytic functions and φ(z) is the unknown function, are investigated in the neighbourhood of points ζ such that g(ζ) = ζ. Conditions are established under which each solution φ(z) may be given as the limit of a sequence of functions φn(z), defined by the recurrence relation φn+1(Z) = ƒ[z, φn{g(z)}], the function φn(z) being to a large extent arbitrary.

scholarly journals The existence of Siegel disks for the Cremona map

Filomat ◽  
2017 ◽  
Vol 31 (9) ◽  
pp. 2705-2712
Author(s):  
Yong-Guo Shi ◽  
Qian Zhang
Keyword(s):  
Functional Equation ◽  
Analytic Solutions ◽  
Small Divisors ◽  
Brjuno Condition

This paper is concerned with the existence of Siegel disks of the Cremona map F?(x,y)=(x cos?-(y-x2) sin?, x sin?+(y-x2) cos?) with the parameter ? ? [0, 2?). This problem is reduced to the existence of local invertible analytic solutions to a functional equation with small divisors ?n + ?-n - ? - ?-1. The main aim of this paper is to investigate whether this equation with |?|=1 has such a solution under the Brjuno condition.

scholarly journals Series Solution of a Functional Equation

1965 ◽  
Vol 5 (1) ◽  
pp. 48-55
Author(s):  
W. Pranger
Keyword(s):  
Functional Equation ◽  
Linear Operator ◽  
Operator Theory ◽  
Inversion Formula ◽  
Linear Functional ◽  
Series Solution ◽  
Analytic Solutions ◽  
Linear Functional Equation ◽  
General Operator ◽  
Image Position

In this we will study analytic solutions to the linear functional equation where f and h are given functions, x is a given complex number and the function g is to be found. This is a generalization of Schröder's functional equation. The results obtained are global in nature and the solutions holomorphic. The equation will be viewed from the standpoint of linear operator theory. When studied in this manner one arrives at a general operator inversion formula.

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