Acta mathematica scientia, Series B >
LOCAL ANALYTIC SOLUTIONS OF A MORE GENERALIZED DHOMBRES EQUATION
Received date: 2011-09-01
Online published: 2013-01-20
Supported by
This work is supported by National Natural Science Foundation of China (11101295).
We study the local analytic solutions f of the functional equation f( Ψ(zf(z))) =φ(f(z)) for z in some neighborhood of the origin. Whether the solution f vanishes at z = 0 or not plays a critical role for local analytic solutions of this equation. In this paper, we obtain results of analytic solutions not only in the case f(0) = 0 but also for f(0) 6= 0. When assuming f(0) = 0, for technical reasons, we just get the result for f′(0) ≠ 0. Then when assuming f(0) =ω0 ≠ 0, Ψ′(0) = s ≠ 0, (z) is analytic at z = 0 and φ'(z) is analytic at z =ω0, we give the existence of local analytic solutions f in the case of 0 < |sω0| < 1 and the case of |sω0| = 1 with the Brjuno condition
ZHANG Qian . LOCAL ANALYTIC SOLUTIONS OF A MORE GENERALIZED DHOMBRES EQUATION[J]. Acta mathematica scientia, Series B, 2013 , 33(1) : 207 -217 . DOI: 10.1016/S0252-9602(12)60205-7
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