On Carathéodory type selectors in a Hilbert space
Abstract
In this paper we consider a set-valued function of two variables, measurable in the first and continuous in the second variable. Using metric projections we construct for this function a family of selectors which are Carathéodory maps. The existence of Carathéodory selectors was studied by Castaing [2], [3], Cellina [4], Fryszkowski [9] and the first author [11].
References
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12. K. Kuratowski, Topology, vol. 1, Academic Press, New York, 1966.
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14. A.A. Tolstonogov, The topological structure of continuous multivalued mappings, Sibirsk. Mat. Zh. 16 (1975), 837-852 (in Russian).
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16. D.H. Wagner, Survey of measurable selection theorems: An update, in: Measure Theory, Oberwolfach 1979, Lecture Notes in Mathematics 794, Springer, 1980.
2. C. Castaing, Sur l'existence des sections séparément mesurables et séparément continues d'une multi-application, Séminaire d'Analyse Convexe, Montpellier 14 (1975).
3. C. Castaing, À propos de l'existence des sections séparément mesurables et séparément continues d'une multi-application séparément mesurable et séparément semi-continue inferieurement, Séminaire d'Analyse Convexe, Montpellier 6 (1976).
4. A. Cellina, A selection theorem, Rend. Sem. Mat. Univ. Padova 55 (1976), 143-149.
5. J.W. Daniel, The continuity of metric projections as functions of the date, J. Approx. Theory 12 (1974), 234-239.
6. I. Ekeland, M. Valadier, Representation of set-valued mappings, J. Math. Anal. Appl. 35 (1971), 621-629.
7. H.W. Engl, M.Z. Nashed, Stochastic projectional schemes for random linear operator equations of the first and the second kind, Numer. Funct. Anal. Optim. 1 (1979), 451-473.
8. A.F. Filippov, Classical solutions of differential equations with multivalued right-hand side, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 22 (1967), 16-26. English transl.: SIAM J. Control Optim. 5 (1967), 609-621.
9. A. Fryszkowski, Carathéodory type selectors of set-valued maps of two variables, Bull. Acad. Polon. Sci. Sér. Sci. Math. 25 (1977), 41-46.
10. C.J. Himmelberg, Measurable relations, Fund. Math. 87 (1975), 53-72.
11. A. Kucia, On the existence of Carathéodory selectors, Bull. Acad. Polon. Sci. Sér. Sci. Math. 32 (1984), 233-241.
12. K. Kuratowski, Topology, vol. 1, Academic Press, New York, 1966.
13. P.-J. Laurent, Approximation et Optimisation, Hermann, Paris, 1972.
14. A.A. Tolstonogov, The topological structure of continuous multivalued mappings, Sibirsk. Mat. Zh. 16 (1975), 837-852 (in Russian).
15. D.H. Wagner, Survey of measurable selection theorems, SIAM J. Control Optim. 15 (1977), 859-903.
16. D.H. Wagner, Survey of measurable selection theorems: An update, in: Measure Theory, Oberwolfach 1979, Lecture Notes in Mathematics 794, Springer, 1980.
KuciaA., & NowakA. (1986). On Carathéodory type selectors in a Hilbert space. Annales Mathematicae Silesianae, 2, 47-52. Retrieved from https://www.journals.us.edu.pl/index.php/AMSIL/article/view/14320
Anna Kucia
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
Andrzej Nowak
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
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