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On evolution inclusions with nonconvex valued orientor fields. (English) Zbl 0839.34021

Using a continuous selection theorem of Fryszkowski for the multifunction \(x \mapsto S^2_{F (\cdot, x(\cdot))}\) (where \(S^2_F\) denote the set of measurable selectors of \(F\) that belong to the Lebesgue-Bochner space \(L^2)\) the authors establish the existence of solutions for an evolution inclusion driven by a demicontinuous monotone operator having a nonconvex valued orientor field, \(\dot x(t) + A(t(x(t)) \in F(t,x(t))\), \(x(0) = x^0\). A relaxation theorem and an example of a control system governed by a nonlinear parabolic differential equation are also given.

MSC:

34A60 Ordinary differential inclusions
47J05 Equations involving nonlinear operators (general)
34G20 Nonlinear differential equations in abstract spaces
93C20 Control/observation systems governed by partial differential equations
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References:

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