Mutational analysis with transitions whose domains vary in state and that are not necessarily Lipschitz continuous in time

Abstract

A pioneering work of Aubin for mutational analysis was improved by Lorenz in that global Lipschitz continuity of transitions in time was weakened to local Lipschitz continuity and contractivity in state was replaced by an additional structural inequality. This inequality was recently eliminated by Kobayashi and the second author. Local Lipschitz continuity assumption is still restrictive in its application to semilinear equations in general Banach spaces and mutational equations with delay, and the previous results cannot be directly applied to mixed problems because they use transitions whose domains contain a common core. To overcome such situations, we introduce a new class of transitions whose domains vary in state and that do not necessarily satisfy local Lipschitz continuity, and a new structural condition for transitions by a family of metrics depending on state and time. Under this structural condition and a growth condition for transitions, the main theorem asserts that the so-called subtangential condition and a dissipativity condition with respect to a metric-like functional ensure that a given mutational equation is well-posed.