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Semicontinuous differential inclusions. (English) Zbl 0936.34010
The author deals with the Cauchy problem for the differential inclusion $x'\in F(t,x)+ G(t,x)\quad \text{a.e. on}\quad x(0)= x_0\tag{$$*$$}$ in a Banach space $$E$$ with uniformly convex dual, where $$F$$ is an almost upper-semicontinuous multifunction with compact convex values and $$G$$ is an almost lower-semicontinuous multifunction with compact values. A relaxed problem, i.e. the problem $$(*)$$ with the function $$G$$ replaced by a convexified upper-semicontinuous regularization of $$G$$, is considered, too. Supposing a suitable growth and suitable Lipschitz type conditions, the author proves that: (i) the problem $$(*)$$ admits at least one solution; (ii) the solution set to problem $$(*)$$ is dense in the solution set of the relaxed problem; (iii) the solution set of the relaxed problem is an $$R_\delta$$ set; and finally, under some additional condition, that: (iv) the solution set to $$(*)$$ is connected.

##### MSC:
 34A60 Ordinary differential inclusions 34G25 Evolution inclusions
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##### References:
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