# American Institute of Mathematical Sciences

December  2009, 25(4): 1109-1128. doi: 10.3934/dcds.2009.25.1109

## Asymptotic equivalence and Kobayashi-type estimates for nonautonomous monotone operators in Banach spaces

 1 Departamento de Ingeniería Matemática, Centro de Modelamiento Matemático, Universidad de Chile, Av. Blanco Encalada 2120, 837-0459 Santiago 2 Departamento de Matemática, Universidad Técnica Federico Santa María, Av. España 1680, Valparaíso, Chile

Received  October 2008 Revised  June 2009 Published  September 2009

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 $\dot{x} + A(t)x$∋ $0$, 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 $t \mapsto A(t)$. 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.
Citation: Felipe Alvarez, Juan Peypouquet. Asymptotic equivalence and Kobayashi-type estimates for nonautonomous monotone operators in Banach spaces. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1109-1128. doi: 10.3934/dcds.2009.25.1109
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