×

zbMATH — the first resource for mathematics

Regularization of closed-valued multifunctions in a non-metric setting. (English) Zbl 0816.28009
Let \(F: T\times X\to Y\) be a Carathéodory type closed-valued multifunction. Scorza-Dragoni type theorems for \(F\) are useful in the study of solutions to the differential inclusion \(\dot x\in F(t, x)\). Suppose that \(F\) is only upper semicontinuous in \(x\) for almost every fixed \(t\). In this case J. Jarník and J. Kurzweil [Časopis Pěst. Mat. 102, 334-349 (1977; Zbl 0369.34002)] used another tool, namely the regularization of \(F\). The multifunction \(F\) is replaced by a more regular map so that the set of solutions of the differential inclusion remains unchanged. T. Rzeżuchowski [Bull. Acad. Pol. Sci., Ser. Sci. Math. 28, 61-66 (1980; Zbl 0459.28007)] worked along this line of ideas assuming \(T\), \(X\), and \(Y\) to be metric spaces. In the present paper the Rzeżuchowski existence and uniqueness theorems for regularizations are extended and proved in a more general framework.
Reviewer: H.-A.Klei (Paris)

MSC:
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
54C60 Set-valued maps in general topology
26E25 Set-valued functions
34A60 Ordinary differential inclusions
PDF BibTeX XML Cite
Full Text: EuDML
References:
[1] ARTSTEIN Z.: Weak convergence of set-valued functions and control. SIAM J. Control Optim. 13 (1975), 865-878. · Zbl 0276.93015
[2] AVERNA D.: Separation properties in X and 2X. Measurable multifunctions and graphs. Math. Slovaca 41 (1991), 51-60. · Zbl 0759.28009
[3] AVERNA D.: Lusin type theorems for multifunctions, Scorza Dragoni’s property and, Carathéodory selections. Boll. Un. Mat. Ital. (7) · Zbl 0817.28007
[4] JARNÍK J., KURZWEIL J.: On conditions on right-hand sides of differential relations. Časopis Pěst. Mat. 102 (1977), 334-349. · Zbl 0369.34002
[5] RZEŹUCHOWSKI T.: Scorza-Dragoni type theorem for upper semicontinuous multivalued functions. Bull. Polish Acad. Sci. Math. 28 (1980), 61-66. · Zbl 0459.28007
[6] SAINTE-BEUVE M. F.: On the extension of Von Neumann-Aumann’s theorem. J. Funct. Anal. 17 (1974), 112-129. · Zbl 0286.28005
[7] SCHWARTZ L.: Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures. Oxford U.P., Oxford, 1973. · Zbl 0298.28001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.