ASYMPTOTIC EQUIVALENCE AND KOBAYASHI-TYPE ESTIMATES FOR NONAUTONOMOUS MONOTONE OPERATORS IN BANACH SPACES

Abstract

We provide a sharp generalization to the nonautonomous case of the well-known Kobayashi estimate for proximal iterates associated with maximal monotone operators. We then derive a bound for the distance between a continuous-in-time trajectory, namely the solution to the differential inclusion x˙ + A(t)x ∋ 0, and the corresponding proximal iterations. We also establish continuity properties with respect to time of the nonautonomous flow under simple assumptions by revealing their link with the function t 7→ A(t). Moreover, our sharper estimations allow us to derive equivalence results which are useful to compare the asymptotic behavior of the trajectories defined by different evolution systems. We do so by extending a classical result of Passty to the nonautonomous setting.