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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The research was supported, by the Czech Science Foundation, project 201/10/0887.
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Reich, L., Smítal, J. & Štefánková, M. Singular Solutions of the Generalized Dhombres Functional Equation. Results. Math. 65, 251–261 (2014). https://doi.org/10.1007/s00025-013-0345-3
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DOI: https://doi.org/10.1007/s00025-013-0345-3