The Existence of Analytic Solutions of a Functional Equation for Invariant Curves

2012 ◽  
pp. 453-461
Author(s):  
Lingxia Liu
Keyword(s):  
Functional Equation ◽  
Analytic Solutions ◽  
Invariant Curves

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.

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

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.

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