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Conjugacy of P-configurations and nonlinear solutions to a certain conditional Cauchy equation. (English) Zbl 1157.39013

A homogeneous Cauchy type functional equation is an equation of the form

$f\left(t\right)=f\left({\delta }_{1}\left(t\right)\right)+f\left({\delta }_{2}\left(t\right)\right),$

where $t\in \left[-1,1\right]$ and $f$ is an unknown function and ${\delta }_{1},{\delta }_{2}$ are two increasing maps on $\left[-1,1\right]$ which satisfy ${\delta }_{1}\left(t\right)+{\delta }_{2}\left(t\right)=t$ and certain additional conditions. Such functions ${\delta }_{1},{\delta }_{2}$ are said to form a P-configuration in $\left[-1,1\right]$.

B. Paneah [Discrete Contin. Dyn. Syst. 10, No. 1–2, 497–505 (2004); erratum ibid. 11, No. 2–3, 744 (2004; Zbl 1057.39022)] showed that every continuously differential solution of the equation above is linear. In this paper the author by an analysis of P-configuration dynamical systems shows that the equation above and, in particular, the functional equation

$f\left(t\right)=f\left(\frac{t+1}{2}\right)+f\left(\frac{t-1}{2}\right)$

have a continuous nonlinear solution.

##### MSC:
 39B22 Functional equations for real functions 39B55 Orthogonal additivity and other conditional functional equations