Measurability and Continuity Conditions for Evolutionary Processes

Abstract

This chapter describes the measurability and continuity conditions for evolutionary processes. Evolutionary processes arise in the mathematical modeling of nonautonomous systems, when U(t, s)x represents the position (or state) at time t + 5 of the point that at time s was at x. In certain situations, measurability and continuity properties known to be satisfied for semigroups could be strengthened using the semigroup properties. These results directly generalize to nonlinear semigroups those known for semigroups of continuous linear operators on a Banach space. This work is extended to evolutionary processes. It is assumed that X is an arbitrary. The chapter presents a theorem that states that for each s ∈ R, x ∈ X, the map t1→ U (t, s)x is Baire continuous on (0, ∞) and, when restricted to the complement of some first category set, has second countable range. Then the map (t, s, x) → U (t, s)x is sequentially continuous on (0, ∞) × ℝ × X.