Invariance and Strict Invariance for Nonlinear Evolution Problems with Applications

Abstract

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 [7] is presented. Applications to nonlinear obstacle problems and agestructured 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.