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Title

Multifunctions of two variables: examples and counterexamples

Authors 1

Affiliations

  1. Department of Mathematics, University of Würzburg, Am Hubland, D-97074 Würzburg, Germany

Abstract

A brief account of the connections between Carathéodory multifunctions, Scorza-Dragoni multifunctions, product-measurable multifunctions, and superpositionally measurable multifunctions of two variables is given.

Bibliography

  1. J. Appell, E. De Pascale, H. T. Nguyêñ and P. P. Zabreĭko, Multi-valued superpositions, Dissertationes Mathematicae 345 (1995).
  2. J. Appell, H. T. Nguyêñ and P. P. Zabreĭko, Multivalued superposition operators in ideal spaces of vector functions I, Indag. Math. 2 (4) (1991), 385-395; Zbl. 748.47050.
  3. J. Appell, H. T. Nguyêñ and P. P. Zabreĭko, Multivalued superposition operators in ideal spaces of vector functions II, Indag. Math. 2 (4) (1991), 397-409; Zbl. 748.47051.
  4. Z. Artstein and K. Prikry, Carathéodory selections and the Scorza-Dragoni property, J. Math. Anal. Appl. 127 (1987), 540-547; Zbl. 649.28011.
  5. P. Brunovsky, Scorza-Dragoni's theorem for unbounded set-valued functions, Mat. Stručno-metod. Časopis 20 (1970), 205-213; Zbl. 215.216.
  6. C. Castaing, Sur les équations différentielles multivoques, C. R. Acad. Sci. Paris 263 (1966), 63-66; Zbl. 143.311.
  7. K. Deimling, Multivalued Differential Equations, de Gruyter, Berlin 1992.
  8. V. V. Filippov, On Luzin's theorem and right-hand sides of differential inclusions (Russian), Mat. Zametki 37 (1985), 93-98 [= Math. Notes 37 (1985), 53-56]; Zbl. 588.34011.
  9. V. V. Filippov, On Luzin's and Scorza-Dragoni's theorem (Russian), Vestnik Mosk. Univ. 42 (1987), 66-68 [= Mosc. Univ. Math. Bull., 42 (1987), 61-63]; Zbl. 617.28005.
  10. A. Fryszkowski, Carathéodory type selectors of set-valued maps of two variables, Bull. Pol. Acad. Sci. Math. 25 (1977), 41-46; Zbl. 358.54002.
  11. C. J. Himmelberg, Measurable relations, Fundamenta Math. 87 (1975), 53-72; Zbl. 296.28003.
  12. C. J. Himmelberg and F. S. van Vleck, An extension of Brunovsky's Scorza-Dragoni type theorem for unbounded set-valued functions, Math. Slovaca 26 (1976), 47-52; Zbl. 32.28004.
  13. J. Janus, A remark on Carathéodory-type selections, Matematiche 41 (1986), 3-13; Zbl. 678.28005.
  14. T. Kaczynski, Unbounded multivalued Nemytskii operators in Sobolev spaces and their applications to discontinuous nonlinearity, Rocky Mountain J. Math. 22 (1992), 635-643; Zbl. 810.35172.
  15. N. Kikuchi, On contingent equations satisfying the Carathéodory-type conditions, Publ. Res. Inst. Math., Kyoto Univ. 3,3 (1968), 361-371; Zbl. 197.131.
  16. M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, Systems with Hysteresis (Russian), Nauka, Moscow 1983 [Engl. transl.: Springer, Berlin 1988]; Zbl. 665.47038.
  17. A. Kucia, On the existence of Carathéodory selectors, Bull. Pol. Acad. Sci. Math. 32 (1984), 233-241; Zbl. 562.28004.
  18. A. Kucia, Extending Carathéodory functions, Bull. Pol. Acad. Sci. Math. 36 (1988), 593-601; Zbl. 771.28008.
  19. A. Kucia, Scorza Dragoni type theorems, Fundamenta Math. 138 (1991), 197-203; Zbl. 744.28011.
  20. A. Kucia and A. Nowak, On Carathéodory type selectors in a Hilbert space, Ann. Mat. Sil. 14 (1986), 47-52; Zbl. 593.54018.
  21. A. Kucia and A. Nowak, On Filippov type theorems and measurability of inverses of random operators, Bull. Pol. Acad. Sci. Math. 38 (1990), 151-158; Zbl. 771.28009.
  22. A. Kucia and A. Nowak, A note on Carathéodory type functions, Acta Univ. Carol. Math. Phys. 34, 2 (1993), 71-74; Zbl. 809.28007.
  23. K. Kuratowski and C. Ryll-Nardzewski, A general theorem on selectors, Bull. Pol. Acad. Sci. Math. 13 (1965), 397-403; Zbl. 152.214.
  24. V. L. Levin, Convex Analysis on Spaces of Measurable Functions and Applications in Mathematics and Economics (Russian), Nauka, Moscow 1985; Zbl. 617.46035.
  25. S. Łojasiewicz, Some theorems of Scorza Dragoni type for multifunctions with applications to the problem of existence of solutions for multivalued differential equations, Math. Control Theory, Banach Center Publ. 14 (1985), 625-643; Zbl. 576.49024.
  26. E.A. Michael, Continuous selections I, Ann. Math. 63,2 (1956), 361-381; Zbl. 71.159.
  27. J. Myjak, A remark on Scorza-Dragoni theorem for differential inclusions, Čas. Pěstování Mat. 114 (1989), 294-298; Zbl. 701.34026.
  28. V. V. Obukhovskiĭ, On periodic solutions of differential equations with multivalued right-hand side (Russian), Trudy Mat. Fak. Voronezh. Gos. Univ. 10 (1973), 74-82.
  29. V. V. Obukhovskiĭ, Personal communication, 1989.
  30. T. Rzeżuchowski, Scorza-Dragoni type theorem for upper semicontinuous multi-valued functions, Bull. Pol. Acad. Sci. Math. 28 (1980), 61-66; Zbl. 459.28007.
  31. A. Spakowski, On superpositionally measurable multifunctions, Acta Univ. Carolinae 30 (1989), 149-151; Zbl. 705.28003.
  32. W. Zygmunt, On the Scorza-Dragoni's type property of the real function semicon-tinuous to the second variable, Rend. Accad. Naz. Sci. XL 11 (1987), 53-63; Zbl. 654.28006.
  33. W. Zygmunt, The Scorza-Dragoni's type property and product measurability of a multifunction of two variables, Rend. Accad. Naz. Sci. XL 12 (1988), 109-115; Zbl. 677.28004.
  34. W. Zygmunt, Product measurability and Scorza-Dragoni's property, Rend. Sem. Mat. Univ. Padova 79 (1988), 301-304; Zbl. 649.28012.
  35. W. Zygmunt, On the approximation of semicontinuous Scorza-Dragonians by the multifunctions of Carathéodory type, Rend. Accad. Naz. Sci. XL 13 (1989), 23-30; Zbl. 692.28005.
  36. W. Zygmunt, A note concerning the Scorza-Dragoni's type property of the compact multivalued multifunctions, Rend. Accad. Naz. Sci. XL 13 (1989), 31-33; Zbl. 697.28005.
  37. W. Zygmunt, Remarks on superpositionally measurable multifunctions (Russian), Mat. Zametki 48 (1990), 70-72 [= Math. Notes, 48 (1990), 923-924]; Zbl. 728.28010.
  38. W. Zygmunt, The Scorza-Dragoni property (Polish), Thesis, M. Curie Univ. Lublin 1990; Zbl. 734.28013.
  39. W. Zygmunt, On superpositionally measurable semi-Carathéodory multifunctions, Comm. Math. Univ. Carol. 33, 1 (1992), 73-77; Zbl. 756.28008.
  40. W. Zygmunt, Personal communication 1994.
  41. W. Zygmunt, On superpositional measurability of semi-Carathéodory multifunctions, Comm. Math. Univ. Carol. 35, 4 (194), 741-744.
Banach Center Publications
1996/35/1
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Pages:
119-128
Main language of publication
English
Published
1996
Exact and natural sciences