Analytic Solutions of a Polynomial-Like Iterative Functional Equation near Resonance
Received:September 26, 2007  Revised:April 16, 2008
Key Words: iterative functional equation   analytic solutions   diophantine condition   Brjuno condition   resonance.  
Fund Project:the Natural Science Foundation of Shandong Province (No.2006ZRB01066).
Author NameAffiliation
LIU Ling Xia Department of Mathematics, Weifang University, Shandong 261041, China 
SI Jian Guo Department of Mathematics, Shandong University, Shandong 250100, China 
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Abstract:
      In this paper existence of local analytic solutions of a polynomial-like iterative functional equation is studied. As well as in previous work, we reduce this problem with the Schr\"oder transformation to finding analytic solutions of a functional equation without iteration of the unknown function $f$. For technical reasons, in previous work the constant $\alpha$ given in the Schr\"oder transformation, i.e., the eigenvalue of the linearized $f$ at its fixed point $O,$ is required to fulfill that $\alpha$ is off the unit circle $S^1$ or lies on the circle with the Diophantine condition. In this paper, we obtain results of analytic solutions in the case of $\alpha$ at resonance, i.e., at a root of the unity and the case of $\alpha$ near resonance under the Brjuno condition.
Citation:
DOI:10.3770/j.issn:1000-341X.2009.04.021
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