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On the properties of the solution set of nonconvex evolution inclusions of the subdifferential type. (English) Zbl 0792.34014

Extending previous work [V. Staicu, Preprint SISSA No. 42M/1990, Trieste, Italy; the author Kôdai Math. J. 15, No. 3, 387-402 (1992; Zbl 0774.34011)], the author proves the existence of a continuous selection \(u(\xi) \in S(\xi)\), \(\xi \in K\), \(u(\xi_ 0)=w \in S(\xi_ 0)\), of the set, \(S(\xi)\), of solutions of the differential inclusion \(- x'\in\partial\varphi(t,x)+F(t,x)\), \(t \in T=[0,b]\), \(x(0)=\xi \in H\); here \(H\) is a separable Hilbert space, \(\partial \varphi(t,x)\) denotes the subdifferential of the proper convex function \(\varphi (t,.)\) and \(K \subset \text{dom} \partial \varphi(t,0)\) is a compact subset. Under similar hypotheses, the set \(S(\xi)\) of solutions of the autonomous differential inclusion \(-x' \in \partial \varphi (x)+F(x)\), \(x(0)=\xi \in H\), is shown to be path connected in the space \(C(T;H)\). As applications, the path connectedness of the reachable set of a parabolic control system and of the solution set of a differential variational inequality are obtained.

MSC:

34A60 Ordinary differential inclusions
35K55 Nonlinear parabolic equations
34G20 Nonlinear differential equations in abstract spaces

Citations:

Zbl 0774.34011
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