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    188宝金博页面版: Generalized Baire category and differential inclusions in Banach spaces

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    内容提示： JOURNAL OF DIFFERENTIAL EQUATIONS 76, 135-158 (1988) Generalized Baire Category and Differential Inclusions in Banach Spaces A. BRESSAN Department of Mathematics, University of Colorado, Boulder. Colorado 80309 AND G. COLOMBO International School for Advanced Studies (SISSA), Str. Costiera II, 34014 Trieste, Italy Received July 31, 1987 1. INTRODUCTION Let E be a separable reflexive Banach space and consider the Cauchy problem i(f) E&(t)), x(0) = x0 E E, (1.1) where F is a Hausdorff continuous multifunctio...

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    JOURNAL OF DIFFERENTIAL EQUATIONS 76, 135-158 (1988) Generalized Baire Category and Differential Inclusions in Banach Spaces A. BRESSAN Department of Mathematics, University of Colorado, Boulder. Colorado 80309 AND G. COLOMBO International School for Advanced Studies (SISSA), Str. Costiera II, 34014 Trieste, Italy Received July 31, 1987 1. INTRODUCTION Let E be a separable reflexive Banach space and consider the Cauchy problem i(f) E&(t)), x(0) = x0 E E, (1.1) where F is a Hausdorff continuous multifunction with closed, bounded values. In this paper we prove the local existence of a solution of (1.1 ), assuming that the convex closure of F(x,,) has finite codimension. More precisely, we assume the existence of a closed affine subspace E, G E with finite codimension, such that the interior of E. n E6 F(x,) relative to E,, is nonempty. Two special cases deserve mention. If the interior of W F(x,) is nonempty, the above condition holds with E. = E. On the other hand, if E is finite dimensional, every continuous multifunction F satisfies our con- dition. Indeed, one can always select an element you 4x0) and set E,,= {y,}. The present result therefore contains the theorems of De Blasi and Pianigiani [7] and of Filippov [S], both as special cases. For a map F whose values are convex sets with finite codimension, the Cauchy problem (1.1) was recently studied by A. Cortesi [4]. To remove the convexity assumption, we rely on a generalized version of Baire’s category theorem, which will also be proved in this paper. Together with ( 1.1 ), we consider the problem i(t)~W F(x(t)), x(0)=x0. (1.2) 135 OO22-0396/88 $3.00 Copyright 0 1988 by Academic Press, Inc. All rights of reproduction in any form reserved.

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