Continuous selections of solution sets to evolution equations
HTML articles powered by AMS MathViewer
- by Vasile Staicu
- Proc. Amer. Math. Soc. 113 (1991), 403-413
- DOI: https://doi.org/10.1090/S0002-9939-1991-1076580-7
- PDF | Request permission
Abstract:
We prove the existence of a continuous selection of the multivalued map $\xi \to \mathcal {T}(\xi )$, where $\mathcal {T}(\xi )$ is the set of all weak (resp. mild) solutions of the Cauchy problem \[ \dot x(t) \in Ax(t) + F(t,x(t)),\quad x(0) = \xi \], assuming that $F$ is Lipschitzian with respect to $x$ and $- A$ is a maximal monotone map (resp. $A$ is the infinitesimal generator of a ${C_0}$-semigroup). We also establish an analog of Michael’s theorem for the solution sets of the Cauchy problem $\dot x(t) \in F(t,x(t)),\;x(0) = \xi$.References
- Jean-Pierre Aubin and Arrigo Cellina, Differential inclusions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 264, Springer-Verlag, Berlin, 1984. Set-valued maps and viability theory. MR 755330, DOI 10.1007/978-3-642-69512-4
- Alberto Bressan and Giovanni Colombo, Extensions and selections of maps with decomposable values, Studia Math. 90 (1988), no. 1, 69–86. MR 947921, DOI 10.4064/sm-90-1-69-86
- Alberto Bressan, Arrigo Cellina, and Andrzej Fryszkowski, A class of absolute retracts in spaces of integrable functions, Proc. Amer. Math. Soc. 112 (1991), no. 2, 413–418. MR 1045587, DOI 10.1090/S0002-9939-1991-1045587-8 H. Brezis, Operateurs maximaux monotones et semigroups de contractions dans les espaces de Hilbert, North-Holland, Amsterdam, 1973.
- Arrigo Cellina, On the set of solutions to Lipschitzian differential inclusions, Differential Integral Equations 1 (1988), no. 4, 495–500. MR 945823
- Arrigo Cellina and António Ornelas, Representation of the attainable set for Lipschitzian differential inclusions, Rocky Mountain J. Math. 22 (1992), no. 1, 117–124. MR 1159946, DOI 10.1216/rmjm/1181072798
- R. M. Colombo, A. Fryszkowski, T. Rzeżuchowski, and V. Staicu, Continuous selections of solution sets of Lipschitzean differential inclusions, Funkcial. Ekvac. 34 (1991), no. 2, 321–330. MR 1130468
- F. S. De Blasi and G. Pianigiani, Non-convex-valued differential inclusions in Banach spaces, J. Math. Anal. Appl. 157 (1991), no. 2, 469–494. MR 1112329, DOI 10.1016/0022-247X(91)90101-5
- A. F. Filippov, Classical solutions of differential equations with multi-valued right-hand side, SIAM J. Control 5 (1967), 609–621. MR 0220995
- Halina Frankowska, A priori estimates for operational differential inclusions, J. Differential Equations 84 (1990), no. 1, 100–128. MR 1042661, DOI 10.1016/0022-0396(90)90129-D
- Fumio Hiai and Hisaharu Umegaki, Integrals, conditional expectations, and martingales of multivalued functions, J. Multivariate Anal. 7 (1977), no. 1, 149–182. MR 507504, DOI 10.1016/0047-259X(77)90037-9
- Ernest Michael, Continuous selections. I, Ann. of Math. (2) 63 (1956), 361–382. MR 77107, DOI 10.2307/1969615 A. A. Tolstonogov, On the properties of solutions of evolutions equations of subdifferential type in a Hilbert space, Proceedings of EQUADIFF 1989, Praha.
Bibliographic Information
- © Copyright 1991 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 113 (1991), 403-413
- MSC: Primary 49J24; Secondary 34A60, 47H04, 54C65
- DOI: https://doi.org/10.1090/S0002-9939-1991-1076580-7
- MathSciNet review: 1076580