  Sufficient conditions for the invariance of evolution problems governed by
perturbations of (possibly nonlinear) $m$-accretive operators are provided. The
conditions for the invariance with respect to sublevel sets of a constraint
functional are expressed in terms of the Dini derivative of that functional,
outside the considered sublevel set in directions determined by the governing
$m$-accretive operator. An approach for non-reflexive Banach spaces is
developed and some result improving a recent paper [P. Cannarsa, G. Da Prato,
H. Frankowska, Invariance of quasi-dissipative systems in Banach spaces. J.
Math. Anal. App. 457 (2018), 1173-1187] is presented. Applications to nonlinear
obstacle problems and age-structured population models are presented in spaces
of continuous functions where advantages of that approach are taken. Moreover,
some new abstract criteria for the so-called strict invariance are derived and
their direct applications to problems with barriers are shown.
