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C ∞-solutions for the second type of generalized Feigenbaum’s functional equations

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Abstract

This work focuses on the second type of generalized Feigenbaum’s equation

$\left\{ \begin{gathered} \phi (f(x)) = f(f(\phi (x))), \hfill \\ f(0) = 1,0 \leqslant f(x) \leqslant 1,x \in [0,1], \hfill \\ \end{gathered} \right. $

where φ(x) is C ∞-increasing function on [0, 1] and satisfies that φ(0) = 0,0 < φ′(x) < 1 (x ∈ [0, 1]). Using constructive method, we discuss the existence of C ∞-single-valley solutions whose derivatives are not equal to 0 on origin of the above equation.

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Authors and Affiliations

  1. College of Science, China University of Petroleum, Qingdao, 266555, P. R. China

    Min Zhang

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  1. Min Zhang
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Correspondence to Min Zhang.

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Supported by National Natural Science Foundation of China, Tian Yuan Foundation (Grant No. 11326129) and the Fundamental Research Funds for the Central Universities (Grant No. 14CX02152A)

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Zhang, M. C ∞-solutions for the second type of generalized Feigenbaum’s functional equations. Acta. Math. Sin.-English Ser. 30, 1785–1794 (2014). https://doi.org/10.1007/s10114-014-2289-2

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  • DOI: https://doi.org/10.1007/s10114-014-2289-2

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