# Article

 Title: Topological properties of the solution set of a class of nonlinear evolutions inclusions (English) Author: Papageorgiou, Nikolaos S. Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 47 Issue: 3 Year: 1997 Pages: 409-424 Summary lang: English . Category: math . Summary: In the paper we study the topological structure of the solution set of a class of nonlinear evolution inclusions. First we show that it is nonempty and compact in certain function spaces and that it depends in an upper semicontinuous way on the initial condition. Then by strengthening the hypothesis on the orientor field $F(t,x)$, we are able to show that the solution set is in fact an $R_\delta$-set. Finally some applications to infinite dimensional control systems are also presented. (English) Keyword: $R_\delta$-set Keyword: homotopic Keyword: contractible Keyword: evolution triple Keyword: evolution inclusion Keyword: compact embedding Keyword: optimal control MSC: 34G20 MSC: 34H05 MSC: 35B30 MSC: 35B37 MSC: 35R45 MSC: 49A20 MSC: 49J24 idZBL: Zbl 0898.35011 idMR: MR1461421 . Date available: 2009-09-24T10:06:41Z Last updated: 2020-07-03 Stable URL: http://hdl.handle.net/10338.dmlcz/127366 . Reference: [1] K. C. Chang: The obstacle problem and partial differential equations with discontinuous nonlinearities.Comm. Pure and Appl. Math. 33 (1980), 117–146. Zbl 0405.35074, MR 0562547, 10.1002/cpa.3160330203 Reference: [2] F. S. DeBlasi, J. Myjak: On the solution sets for differential inclusions.Bull. Polish. Acad. Sci. 33 (1985), 17–23. MR 0798723 Reference: [3] K. Deimling, M. R. M. Rao: On solution sets of multivalued differential equations.Applicable Analysis 30 (1988), 129–135. MR 0967566, 10.1080/00036818808839797 Reference: [4] J. Dugundji: Topology.Allyn and Bacon, Inc., Boston, 1966. Zbl 0144.21501, MR 0193606 Reference: [5] C. Himmelberg: Precompact contractions of metric uniformities and the continuity of $F(t,x)$.Rend. Sem. Matematico Univ. Padova 50 (1973), 185–188. MR 0355958 Reference: [6] C. Himmelberg, F. Van Vleck: A note on the solution sets of differential inclusions.Rocky Mountain J. Math 12 (1982), 621–625. MR 0683856, 10.1216/RMJ-1982-12-4-621 Reference: [7] D. M. Hyman: On decreasing sequences of compact absolute retracts.Fund. Math. 64 (1969), 91–97. Zbl 0174.25804, MR 0253303, 10.4064/fm-64-1-91-97 Reference: [8] A. Lasota, J. Yorke: The generic property of existence of solutions of differential equations on Banach spaces.J. Diff. Equations 13 (1973), 1–12. MR 0335994, 10.1016/0022-0396(73)90027-2 Reference: [9] N. S. Papageorgiou: Optimal control of nonlinear evolution inclusions.J. Optim. Theory Appl. 67 (1990), 321–357. Zbl 0697.49007, MR 1080139, 10.1007/BF00940479 Reference: [10] N. S. Papageorgiou: Convergence theorems for Banach space valued integrable multifunctions.Intern. J. Math and Math.Sci. 10 (1987), 433–442. Zbl 0619.28009, MR 0896595, 10.1155/S0161171287000516 Reference: [11] N. S. Papageorgiou: On the solution set of differential inclusions in Banach spaces.Applicable Anal. 25 (1987), 319–329. MR 0912190, 10.1080/00036818708839695 Reference: [12] N. S. Papageorgiou: Relaxability and well-posedness for infinite dimensional optimal control problems.Problems of Control and information Theory 20 (1991), 205–218. Zbl 0741.49001, MR 1119038 Reference: [13] L. Rybinski: On Caratheodory type selections.Fund. Math. CXXV (1985), 187–193. Zbl 0614.28005, MR 0813756 Reference: [14] D. Wagner: Survey of measurable selection theorems.SIAM J. Control and Optim. 15 (1977), 859–903. Zbl 0407.28006, MR 0486391, 10.1137/0315056 Reference: [15] J. Yorke: Spaces of solutions.Lecture Notes on Operations Research and Math. Economics 12 (1969), Springer, New York, 383–403. Zbl 0188.15502, MR 0361294 Reference: [16] E. Zeidler: Nonlinear Functional Analysis and its Applications II.Springer, New York, 1990. Zbl 0684.47029, MR 0816732 .

## Files

Files Size Format View
CzechMathJ_47-1997-3_3.pdf 1.561Mb application/pdf View/Open

Partner of