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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 $$f(t)=f(\delta_1(t))+f(\delta_2(t)),$$ where $t\in [-1,1]$ and $f$ is an unknown function and $\delta_1, \delta_2$ are two increasing maps on $[-1,1]$ which satisfy $\delta_1(t)+\delta_2(t)=t$ and certain additional conditions. Such functions $\delta_1, \delta_2$ are said to form a P-configuration in $[-1,1]$. {\it 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 $$f(t)=f(\frac{t+1}{2})+f(\frac{t-1}{2})$$ have a continuous nonlinear solution.
39B22Functional equations for real functions
39B55Orthogonal additivity and other conditional functional equations
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