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1994-1995 | 146 | 3 | 239-249
Tytuł artykułu

Borel partitions of unity and lower Carathéodory multifunctions

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EN
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EN
We prove the existence of Carathéodory selections and representations of a closed convex valued, lower Carathéodory multifunction from a set A in $A(ℰ ⊗ ℬ(X))$ into a separable Banach space Y, where ℰ is a sub-σ-field of the Borel σ-field ℬ(E) of a Polish space E, X is a Polish space and A is the Suslin operation. As applications we obtain random versions of results on extensions of continuous functions and fixed points of multifunctions. Such results are useful in the study of random differential equations and inclusions and in mathematical economics.
  As a key tool we prove that if A is an analytic subset of E × X and if ${U_n : n ∈ w}$ is a sequence of Borel sets in A such that $A=∪_n U_n$ and the section $U_n(e)$ is open in A(e), e ∈ E, n ∈ w, then there exist Borel functions $p_n : A → [0,1]$, n ∈ w, such that for every e ∈ E, ${p_n(e,·) : n ∈ w}$ is a locally Lipschitz partition of unity subordinate to ${U_n(e) : n ∈ w}$.
Twórcy
Bibliografia
  • [AC] J.-P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin, 1984.
  • [AF] J.-P. Aubin and H. Frankowska, Set Valued Analysis, Systems and Control: Found. Appl., Birkhäuser, Boston, 1990.
  • [F] A. Fryszkowski, Carathéodory type selectors of set-valued maps of two variables, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), 41-46.
  • [J] J. Janus, A remark on Carathéodory type selections, Le Matematiche 25 (1986), 3-13.
  • [Kuc] A. Kucia, On the existence of Carathéodory selectors, Bull. Polish Acad. Sci. Math. 32 (1984), 233-241.
  • [Kur] K. Kuratowski, Topology, Vol. I, PWN, Warszawa, and Academic Press, New York, 1966.
  • [KPY] T. Kim, K. Prikry and N. C. Yannelis, Carathéodory-type selections and random fixed point theorems, J. Math. Anal. Appl. 122 (1987), 393-407.
  • [Łoj] S. Łojasiewicz, Jr., Parametrizations of convex sets, J. Approx. Theory, to appear.
  • [Lou] A. Louveau, A separation theorem for $Σ_1^1$ sets, Trans. Amer. Math. Soc. 260 (1980), 363-378.
  • [Mic] E. Michael, Continuous selections I, Ann. of Math. 63 (1956), 363-382.
  • [Mil] D. E. Miller, Borel selectors for separated quotients, Pacific J. Math. 91 (1980), 187-198.
  • [Mo] Y. N. Moschovakis, Descriptive Set Theory, North-Holland, Amsterdam, 1980.
  • [MR] A. Maitra and B. V. Rao, Generalizations of Castaing's theorem on selectors, Colloq. Math. 42 (1979), 295-300.
  • [BR] B. V. Rao and K. P. S. Bhaskara Rao, Borel spaces, Dissertationes Math. (Rozprawy Mat.) 190 (1981).
  • [Ri] B. Ricceri, Selections of multifunctions of two variables, Rocky Mountain J. Math. 14 (1984), 503-517.
  • [Ry] L. Rybiński, On Carathéodory type selections, Fund. Math. 125 (1985), 187-193.
  • [S1] S. M. Srivastava, A representation theorem for closed valued multifunctions, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), 511-514.
  • [S2] S. M. Srivastava, Approximations and approximate selections of upper Carathéodory multifunctions, Boll. Un. Mat. Ital. A (7) 8 (1994), 251-262.
  • [SS1] H. Sarbadhikari and S. M. Srivastava, Random theorems in topology, Fund. Math. 136 (1990), 65-72.
  • [SS2] H. Sarbadhikari and S. M. Srivastava, Random versions of extension theorems of Dugundji type and fixed point theorems, Boll. Un. Mat. Ital., to appear.
  • [Y] N. C. Yannelis, Equilibria in non-cooperative models of computations, J. Econom. Theory 41 (1987), 96-111.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-fmv146i3p239bwm
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