数学物理学报(英文版) ›› 2013, Vol. 33 ›› Issue (1): 207-217.doi: 10.1016/S0252-9602(12)60205-7

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LOCAL ANALYTIC SOLUTIONS OF A MORE GENERALIZED DHOMBRES EQUATION

张倩   

  1. School of Science, Southwest University of Science and Technology, Mianyang 621010, China
  • 收稿日期:2011-09-01 出版日期:2013-01-20 发布日期:2013-01-20
  • 基金资助:

    This work is supported by National Natural Science Foundation of China (11101295).

LOCAL ANALYTIC SOLUTIONS OF A MORE GENERALIZED DHOMBRES EQUATION

 ZHANG Qian   

  1. School of Science, Southwest University of Science and Technology, Mianyang 621010, China
  • Received:2011-09-01 Online:2013-01-20 Published:2013-01-20
  • Supported by:

    This work is supported by National Natural Science Foundation of China (11101295).

摘要:

We study the local analytic solutions f of the functional equation fΨ(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 < |0| < 1 and the case of |0| = 1 with the Brjuno condition.

关键词: local analytic solutions, the majorant series, functional equation, Brjuno condition

Abstract:

We study the local analytic solutions f of the functional equation fΨ(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 < |0| < 1 and the case of |0| = 1 with the Brjuno condition

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

中图分类号: 

  • 39B12