LOCAL ANALYTIC SOLUTIONS OF A MORE GENERALIZED DHOMBRES EQUATION

doi:10.1016/S0252-9602(12)60205-7

Abstract

Abstract:

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

Key words: local analytic solutions, the majorant series, functional equation, Brjuno condition

CLC Number:

39B12

ZHANG Qian. LOCAL ANALYTIC SOLUTIONS OF A MORE GENERALIZED DHOMBRES EQUATION[J].Acta mathematica scientia,Series B, 2013, 33(1): 207-217.

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