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.
\[ (3)\quad (\partial /\partial x)F(x,t)\cdot G(x)=G[F(x,t)] \] 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 |