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Some differential equations related to iteration theory. (English) Zbl 0648.35015

The three Jabotinsky equations \[ (1)\quad (\partial /\partial t)F(x,t)=(\partial /\partial x)F(x,t)\cdot G(x),\quad (2)\quad \partial F/\partial t(x,t)=G[F(x,t)], \]
\[ (3)\quad (\partial /\partial x)F(x,t)\cdot G(x)=G[F(x,t)] \] follow from the translation equation \[ (T)\quad F[F(x,s),t]=F(x,s+t) \] (satisfied by the t-th iterate f \(t(x)=F(x,t)\) of a function f) and from the initial conditions \[ (I)\quad F(x,0)=x,\quad (4)\quad (\partial /\partial t)F(x,0)=G(x). \] Disproving conjectures, we give counterexamples which show that (4) and (1), (2), (3) neither individually, nor all together imply (T). In addition, several positive results are proved (under somewhat stronger conditions). Also the generalization of (T), \(F[F(x,s),t]=F[F(x,t),s]\), satisfied by commuting functions is considered. The underlying spaces are \({\mathbb{R}}\), \({\mathbb{C}}\) and, in general, Banach spaces.
Reviewer: J.Aczél

MSC:

35F25 Initial value problems for nonlinear first-order PDEs
35R10 Partial functional-differential equations
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