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Journal of Convex Analysis, Vol. 5, No. 2, pp. 381-395 (1998)
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A Viability Result in the Upper Semicontinuous Case

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Andrea Gavioli

Dipartimento di Matematica Pura ed Applicata, Universita di Modena, Via G. Campi 213/B, 41100 Modena, Italy, gavioli@unimo.it

**Abstract:** We prove the existence of solutions of a differential inclusion $u'\in F(t,u)$ in a separable Banach space $X$ with constraint $u(t)\in D(t)$. $F$ is globally measurable, weakly upper semicontinuous with respect to $u$ and takes convex, weakly compact values. $D$ is upper semicontinuous from the left, and, for every $r>0$, the sets $D(t)\cap rB$ are compact. $F$ and $D$ fulfil a well-known tangential condition, which is expressed by means of the Bouligand cone.

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