American Institute of Mathematical Sciences

Advanced
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 and Continuous Dynamical Systems, 2009, 25 (4) : 1109-1128. doi: 10.3934/dcds.2009.25.1109
[1]

Radu Ioan Boţ, Christopher Hendrich. Solving monotone inclusions involving parallel sums of linearly composed maximally monotone operators. Inverse Problems and Imaging, 2016, 10 (3) : 617-640. doi: 10.3934/ipi.2016014
[2]

Jacson Simsen, Mariza Stefanello Simsen, José Valero. Convergence of nonautonomous multivalued problems with large diffusion to ordinary differential inclusions. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2347-2368. doi: 10.3934/cpaa.2020102
[3]

Xiao Tang, Yingying Zeng, Weinian Zhang. Interval homeomorphic solutions of a functional equation of nonautonomous iterations. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6967-6984. doi: 10.3934/dcds.2020214
[4]

Elimhan N. Mahmudov. Optimal control of evolution differential inclusions with polynomial linear differential operators. Evolution Equations and Control Theory, 2019, 8 (3) : 603-619. doi: 10.3934/eect.2019028
[5]

Dag Lukkassen, Annette Meidell, Peter Wall. Multiscale homogenization of monotone operators. Discrete and Continuous Dynamical Systems, 2008, 22 (3) : 711-727. doi: 10.3934/dcds.2008.22.711
[6]

Augusto VisintiN. On the variational representation of monotone operators. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 909-918. doi: 10.3934/dcdss.2017046
[7]

Mariusz Michta. On solutions to stochastic differential inclusions. Conference Publications, 2003, 2003 (Special) : 618-622. doi: 10.3934/proc.2003.2003.618
[8]

Thomas Lorenz. Mutational inclusions: Differential inclusions in metric spaces. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 629-654. doi: 10.3934/dcdsb.2010.14.629
[9]

Nils Svanstedt. Multiscale stochastic homogenization of monotone operators. Networks and Heterogeneous Media, 2007, 2 (1) : 181-192. doi: 10.3934/nhm.2007.2.181
[10]

Robert J. Kipka, Yuri S. Ledyaev. Optimal control of differential inclusions on manifolds. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4455-4475. doi: 10.3934/dcds.2015.35.4455
[11]

Ovidiu Carja, Victor Postolache. A Priori estimates for solutions of differential inclusions. Conference Publications, 2011, 2011 (Special) : 258-264. doi: 10.3934/proc.2011.2011.258
[12]

Gheorghe Craciun, Abhishek Deshpande, Hyejin Jenny Yeon. Quasi-toric differential inclusions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2343-2359. doi: 10.3934/dcdsb.2020181
[13]

Andrej V. Plotnikov, Tatyana A. Komleva, Liliya I. Plotnikova. The averaging of fuzzy hyperbolic differential inclusions. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1987-1998. doi: 10.3934/dcdsb.2017117
[14]

Svetlana Pastukhova, Valeria Chiadò Piat. Homogenization of multivalued monotone operators with variable growth exponent. Networks and Heterogeneous Media, 2020, 15 (2) : 281-305. doi: 10.3934/nhm.2020013
[15]

Piermarco Cannarsa, Peter R. Wolenski. Semiconcavity of the value function for a class of differential inclusions. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 453-466. doi: 10.3934/dcds.2011.29.453
[16]

Janosch Rieger. The Euler scheme for state constrained ordinary differential inclusions. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2729-2744. doi: 10.3934/dcdsb.2016070
[17]

Tomás Caraballo, José A. Langa, José Valero. Stabilisation of differential inclusions and PDEs without uniqueness by noise. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1375-1392. doi: 10.3934/cpaa.2008.7.1375
[18]

Thomas Lorenz. Partial differential inclusions of transport type with state constraints. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1309-1340. doi: 10.3934/dcdsb.2019018
[19]

Mieczysław Cichoń, Bianca Satco. On the properties of solutions set for measure driven differential inclusions. Conference Publications, 2015, 2015 (special) : 287-296. doi: 10.3934/proc.2015.0287
[20]

Volodymyr Pichkur. On practical stability of differential inclusions using Lyapunov functions. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1977-1986. doi: 10.3934/dcdsb.2017116

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (100)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy