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On the existence of \(\psi\)-minimal viable solutions for a class of differential inclusions. (English) Zbl 0759.34014

Let \(X\) be a Banach space, \(\psi: X\to R\) a convex, continuous function, \(K\subset X\), \(\text{int}(K)\neq\emptyset\). \(F\) is a continuous multifunction on \(K\) with convex values in \(X\) and satisfies standard conditions required for the existence of solutions of (1) \(\dot x\in F(x)\), including \(\alpha(F(B))\leq k'\alpha(B)\), where \(\alpha\) denotes the measure of noncompactness. In his main result the author proves that if \(F(x)\cap\text{int}(T_ K(x))\neq\emptyset\) (\(T_ K\) — Bouligand’s tangent cone) then there is a solution \(x(\cdot)\) of (1) for which \(\psi(\dot x(t))=\inf\{\psi(z)\): \(z\in F(x(t))\cap T_ K(x(t))\}\). If \(X\) is finite dimensional then the tangency condition is weakened to \(F(x)\cap T_ K(x)\neq\emptyset\).

MSC:

34A60 Ordinary differential inclusions
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