Flow invariance for nonlinear accretive evolutions under range conditions

Abstract

Let A be a nonlinear accretive operator in a real Banach space X and f : J × X → X be continuous or of Carathéodory type, where J = [0, T ] ⊂ R. We investigate the existence of mild solutions of the evolution systemin case A satisfies the range condition or weaker variants thereof. This requires a careful construction of approximate solutions since m-accretivity of A is not assumed, hence associated quasi-autonomous problems need not have mild solutions. Conditions are such that additional constraints like u(t) ∈ K on J for a given closed K ⊂ X are accounted for.

Flow invariance for nonlinear partial differential delay equations

“…[9]. For this special case, the references [2,9,10], [42,Ch. VIII], and [54,60] (and many others, compare the references in [9]) also use a weaker subtangential condition which, in particular cases, is also necessary for flow-invariance.…”

“…[2], [42,Ch. VIII], [54,56,60], and, for the most recent account as well as further references, [9,10]), and (b) of ordinary functional differential equations with B ≡ 0 ( [39,40,41,75]), while (c) the semilinear case of (PFDE), with B : D(B) ⊂ X → X (single-valued and) linear, and −B generating a C 0 −semigroup of bounded linear operators on X, has been solved in [43,45,55,56]. (For a rather complete account of the semilinear case of (PFDE) in general, we refer to [87].…”

Necessary and Sufficient Conditions for Viability for Nonlinear Evolution Inclusions

“…use "lim" instead of "lim inf" in the tangency conditions. Other sufficient conditions for viability referring to single-valued Carathéodory perturbations of dissipative operators are due to Bothe [8] and [9].…”