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BY-NC-ND 3.0 license Open Access Published by De Gruyter May 8, 2014

On the Carathéodory superposition of multifunctions and an existence theorem

  • Grażyna Kwiecińska EMAIL logo
From the journal Mathematica Slovaca

Abstract

Let I ⊂ ℝ be an interval and Y a reflexive Banach space. We introduce the (H) property of a multifunction F from I × Y to Y and prove that the Carathéodory superposition of F with each continuous function f from I to Y is a derivative provided that F has the (H) property. Some application of this theorem to the existence of solutions of differential inclusions f′(x) ∈ F(x, f(x)) is given.

[1] ANTOSIEWICZ, H. A.— CELLINA, A.: Continuous selectios and differential relations, J. Differential Equations 19 (1975), 386–398. http://dx.doi.org/10.1016/0022-0396(75)90011-X10.1016/0022-0396(75)90011-XSearch in Google Scholar

[2] AUMANN, J. R.: Integrals of set-valued functions, J. Math. Appl. 12 (1965), 1–12. Search in Google Scholar

[3] BANKS, H. T.— JACOBS, M. Q.: A differential Calculus for Multifunctions, J. Math. Anal. Appl. 29 (1970), 246–272. http://dx.doi.org/10.1016/0022-247X(70)90078-810.1016/0022-247X(70)90078-8Search in Google Scholar

[4] BRIDGLAND, T. F.: Trajectory integrals of set valued functions, Pacific J. Math. 33 (1970), 43–68. http://dx.doi.org/10.2140/pjm.1970.33.4310.2140/pjm.1970.33.43Search in Google Scholar

[5] CASTAING, C.: Sur les équations différentielles multivoques, C. R. Math. Acad. Sci. Paris 263 (1966), 63–66. Search in Google Scholar

[6] CASTAING, C.— VALADIER, M.: Convex analysis and measurable multivalued functions. Lecture Notes in Math., Springer, Berlin-Heidelberg-New York 1977. http://dx.doi.org/10.1007/BFb008768510.1007/BFb0087685Search in Google Scholar

[7] DEBREU, G.: Integration of correspondences. In: Proc. Fifth Berkeley Symposium of Mathematical Statistics and Probability, University of California Press, 1966, pp. 351–372. Search in Google Scholar

[8] DEIMLING, K.: Multivalued Differential Equations, de Gruyter, Berlin, 1992. http://dx.doi.org/10.1515/978311087422810.1515/9783110874228Search in Google Scholar

[9] DUNFORD, N.— SCHWARTZ, J. T.: Linear Operators, Interscince Publishers, INC, New York, 1958. Search in Google Scholar

[10] FILIPPOV, A. F.: The existence of solutions of generalized differential equations, Math. Notes 10 (1971), 608–611. http://dx.doi.org/10.1007/BF0146472210.1007/BF01464722Search in Google Scholar

[11] GRANDE, Z.: A theorem about Carathéodory’s superposition, Math. Slovaca 42 (1992), 443–449. Search in Google Scholar

[12] HARTMAN, P.: Ordinary Differential Equatins, J. Wiley&Sons, New York, 1964. Search in Google Scholar

[13] HERMES, H.: On continuous and measurable selections and the existence of solutions of generalized differential equations, Proc. Amer. Math. Soc. 29 (1971), 535–542. http://dx.doi.org/10.1090/S0002-9939-1971-0277794-310.1090/S0002-9939-1971-0277794-3Search in Google Scholar

[14] HILL, E.— PHILLIPS, R. S.: Functional Analysis and Semi-groups. Amer. Math. Soc. Colloq. Publ. 31, Amer. Math. Soc., Providence, RI, 1957. Search in Google Scholar

[15] HUKUHARA, M.: Intégration des applications mesurables dont la valuer est un compact convexe, Funkcial. Ekvac. 10 (1967), 205–223. Search in Google Scholar

[16] KACZYŃSKI, H.— OLECH, C.: Existence of solutions of orientor fields with nonconvex right-hand side, Ann. Polon. Math. 29 (1974), 61–66. 10.4064/ap-29-1-61-66Search in Google Scholar

[17] KURATOWSKI, K.— RYLL-NARDZEWSKI, C.: A general theorem on selectors, Bull. Pol. Acad. Sci. Math. 13 (1965), 397–403. Search in Google Scholar

[18] KWIECIŃSKA, G.: On the intermediate value property of multivalued functions, Real Anal. Exchange 26 (2000–2001), 245–260. 10.2307/44153161Search in Google Scholar

[19] KWIECIŃSKA, G.: Measurability of multifunctions of two variables, Dissertationes Math. (Rozprawy Mat.) 452 (2008), 1–67. http://dx.doi.org/10.4064/dm452-0-110.4064/dm452-0-1Search in Google Scholar

[20] RÅDSTRÖM, H.: An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3 (1952), 165–169. http://dx.doi.org/10.2307/203247710.2307/2032477Search in Google Scholar

[21] WAŻEWSKI, T.: Sur une condition équivalente `a l’équation au contingent, Bull. Pol. Acad. Sci. Math. 9 (1961), 865–867. Search in Google Scholar

[22] ZYGMUNT, W.: Remarks on superpositionally measurable multifunctions, Mat. Zametki 48 (1990), 70–72 (Russian). Search in Google Scholar

Published Online: 2014-5-8
Published in Print: 2014-4-1

© 2014 Mathematical Institute, Slovak Academy of Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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