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Construction of convex solutions for the second type of Feigenbaum’s functional equations

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Abstract

In this paper, convex solutions for the second type of Feigenbaum’s equation

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

are investigated. Using constructive methods, we discuss the existence and uniqueness of continuous convex solutions, C 1-convex solutions and C 2-convex solutions of the above equation.

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Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, 250100, China

    JianGuo Si & Min Zhang

Authors
  1. JianGuo Si
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  2. Min Zhang
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Correspondence to JianGuo Si.

Additional information

This work was supported by National Natural Science Foundation of China (Grant No. 10871117) and Natural Science Foundation of Shandong Province (Grant No. Y2006A07)

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Si, J., Zhang, M. Construction of convex solutions for the second type of Feigenbaum’s functional equations. Sci. China Ser. A-Math. 52, 1617–1638 (2009). https://doi.org/10.1007/s11425-009-0004-z

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  • DOI: https://doi.org/10.1007/s11425-009-0004-z

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