Singular Solutions of the Generalized Dhombres Functional Equation |
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Authors: | L Reich J Smítal M Štefánková |
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Institution: | 1. Institut für Mathematik Karl-Franzens Universit?t Graz, 8010, Graz, Austria 2. Mathematical Institute Silesian University, 746 01, Opava, Czech Republic
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Abstract: | We consider singular solutions of the functional equation ${f(xf(x)) = \varphi (f(x))}$ where ${\varphi}$ is a given and f an unknown continuous map ${\mathbb R_{+} \rightarrow \mathbb R_{+}}$ . A solution f is regular if the sets ${R_f \cap (0, 1]}$ and ${R_f \cap 1, \infty)}$ , where R f is the range of f, are ${\varphi}$ -invariant; otherwise f is singular. We show that for singular solutions the associated dynamical system ${({R_f}, \varphi|_{R_f})}$ can have strange properties unknown for the regular solutions. In particular, we show that ${\varphi |_{R_f}}$ can have a periodic point of period 3 and hence can be chaotic in a strong sense. We also provide an effective method of construction of singular solutions. |
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