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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

Felipe Alvarez 1, and  Juan Peypouquet 2,

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.
Keywords: proximal iterations, Monotone operators, nonautonomous differential inclusions..
Mathematics Subject Classification: Primary: 47H05; Secondary: 34A60, 49M2.
Citation: Felipe Alvarez, Juan Peypouquet. Asymptotic equivalence and Kobayashi-type estimates for nonautonomous monotone operators in Banach spaces. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1109-1128. doi: 10.3934/dcds.2009.25.1109
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