Generalized Baire Category and Differential Inclusions in Banach Spaces

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 condition. 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