Conditional Lipschitz Shadowing for Ordinary Differential Equations

We introduce the notion of conditional Lipschitz shadowing, which does not aim to shadow every pseudo-orbit, but only those belong a certain prescribed set. establish two types sufficient conditions under non\-auto\-nomous ordinary differential equations have such property. The first criterion applies semilinear equation provided that its linear part is hyperbolic and nonlinearity small in neighborhood second requires logarithmic norm derivative right-hand side with respect state variable uniformly negative results are applicable important classes model including logistic equation, whose shadowing has recently been studied. Several examples constructed showing obtained optimal.