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Zbl 1157.39013
Shalit, Orr Moshe
Conjugacy of P-configurations and nonlinear solutions to a certain conditional Cauchy equation.
(English)
[J] Banach J. Math. Anal. 3, No. 1, 28-35, electronic only (2009). ISSN 1735-8787/e

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.
MSC 2000:
*39B22 Functional equations for real functions
39B55 Orthogonal additivity and other conditional equations

Keywords: conditional functional equation; Cauchy type functional equation; $P$-configuration; guided dynamical system; continuous nonlinear solution

Citations: Zbl 1057.39022

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