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Iteration, translation, commuting, and differential equations. (English) Zbl 0651.39005

Ber. Math.-Stat. Sekt. Forschungsges. Joanneum-Graz 285-296, 285/1-285/6 (1988).
The translation equation (T.E.) \(F[F(x,s),t]=F(x,s+t)\) describes autonomous (semi-)dynamical systems. If F is suitably differentiable, three functional-differential equations can be derived from the T.E., as noticed by J. Aczél in 1950 and, in a different context, by E. Jabotinsky in 1955. These we call Aczél-Jabotinsky (A-J) equations; the authors speak of Jabotinsky equations, following earlier usage by the present reviewer, who later became aware of Aczél’s priority.
The question is discussed under what additional conditions some or all of the A-J equations imply the T.E.; in other words: under what circumstances are the “weak dynamical systems” defined by the A-J equations true dynamical systems. The answers turn out to be interesting and highly non-trivial. A concise discussion of the arising possibilities is given.
Reviewer: Gy.Targonski

MSC:

39B52 Functional equations for functions with more general domains and/or ranges
39B99 Functional equations and inequalities
34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
37-XX Dynamical systems and ergodic theory