Abstract
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 δ-set. Finally some applications to infinite dimensional control systems are also presented.
Similar content being viewed by others
References
K. C. Chang: The obstacle problem and partial differential equations with discontinuous nonlinearities. Comm. Pure and Appl. Math. 33 (1980), 117–146.
F. S. DeBlasi, J. Myjak: On the solution sets for differential inclusions. Bull. Polish. Acad. Sci. 33 (1985), 17–23.
K. Deimling, M. R. M. Rao: On solution sets of multivalued differential equations. Applicable Analysis 30 (1988), 129–135.
J. Dugundji: Topology. Allyn and Bacon, Inc., Boston, 1966.
C. Himmelberg: Precompact contractions of metric uniformities and the continuity of F(t,x). Rend. Sem. Matematico Univ. Padova 50 (1973), 185–188.
C. Himmelberg, F. Van Vleck: A note on the solution sets of differential inclusions. Rocky Mountain J. Math 12 (1982), 621–625.
D. M. Hyman: On decreasing sequences of compact absolute retracts. Fund. Math. 64 (1969), 91–97.
A. Lasota, J. Yorke: The generic property of existence of solutions of differential equations on Banach spaces. J. Diff. Equations 13 (1973), 1–12.
N. S. Papageorgiou: Optimal control of nonlinear evolution inclusions. J. Optim. Theory Appl. 67 (1990), 321–357.
N. S. Papageorgiou: Convergence theorems for Banach space valued integrable multifunctions. Intern. J. Math and Math.Sci. 10 (1987), 433–442.
N. S. Papageorgiou: On the solution set of differential inclusions in Banach spaces. Applicable Anal. 25 (1987), 319–329.
N. S. Papageorgiou: Relaxability and well-posedness for infinite dimensional optimal control problems. Problems of Control and information Theory 20 (1991), 205–218.
L. Rybinski: On Caratheodory type selections. Fund. Math. CXXV (1985), 187–193.
D. Wagner: Survey of measurable selection theorems. SIAM J. Control and Optim. 15 (1977), 859–903.
J. Yorke: Spaces of solutions. Lecture Notes on Operations Research and Math. Economics 12 (1969), 383–403. Springer, New York.
E. Zeidler: Nonlinear Functional Analysis and its Applications II. Springer, New York, 1990.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Papageorgiou, N.S. Topological properties of the solution set of a class of nonlinear evolutions inclusions. Czechoslovak Mathematical Journal 47, 409–424 (1997). https://doi.org/10.1023/A:1022451115373
Issue Date:
DOI: https://doi.org/10.1023/A:1022451115373