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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(δ 1 (t))+f(δ 2 (t)),

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

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(t+1 2)+f(t-1 2)

have a continuous nonlinear solution.

MSC:
39B22Functional equations for real functions
39B55Orthogonal additivity and other conditional functional equations