**
ACTA MATHEMATICA UNIVERSITATIS COMENIANAE **

Vol. 65, 1 (1996)

pp. 33-51

ON THE STRUCTURE OF THE SOLUTION SET OF EVOLUTION INCLUSIONS WITH TIME-DEPENDENT SUBDIFFERENTIALS

N. S. PAPAGEORGIOU and F. PAPALINI

**Abstract**.
In this paper we consider evolution inclusions driven by a time dependent subdifferential operator and a set-valued perturbation term. First we show that the problem with a convex-valued, $h$-u.s.c. orientor field (i.e. perturbation term) has a nonempty solution set which is an $R_\delta $-set in $C(T, H)$, in particular then compact and acyclic. For the non convex problem (i.e. the orientor field is non convex-valued), without assuming that the functional $\varphi(t, x)$ of the subdifferential is of compact type, we show that for every initial datum $\xi\in \dm \varphi(0, \cdot)$ the solution set $S(\xi)$ is nonempty and we also produce a continuous selector for the multifunctions $\xi\to S(\xi)$. Some examples of distributed parameter systems are also included.

**AMS subject classification**.

**Keywords**.
$h$-upper semicontinuity, Hausdorff metric, subdifferential, function of compact type, parabolic problem, compact embedding

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