  We prove some new results on existence of solutions to first--order ordinary
differential equations with deviating arguments. Delay differential equations
are included in our general framework, which even allows deviations to depend
on the unknown solutions. Our existence results lean on new definitions of
lower and upper solutions introduced in this paper, and we show with an example
that similar results with the classical definitions are false. We also
introduce an example showing that the problems considered need not have the
least (or the greatest) solution between given lower and upper solutions, but
we can prove that they do have minimal and maximal solutions in the usual
set--theoretic sense. Sufficient conditions for the existence of lower and
upper solutions, with some examples of application, are provided too.
