Papageorgiou, Nikolaos S. On the existence of \(\psi\)-minimal viable solutions for a class of differential inclusions. (English) Zbl 0759.34014 Arch. Math., Brno 27b, 175-182 (1991). 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 \(F(x)\cap T_ K(x)\neq\emptyset\). Reviewer: T.Rzezuchowski (Warszawa) Cited in 1 Document MSC: 34A60 Ordinary differential inclusions Keywords:differential inclusion; Banach space; continuous multifunction; measure of noncompactness; Bouligand’s tangent cone; tangency condition PDF BibTeX XML Cite \textit{N. S. Papageorgiou}, Arch. Math., Brno 27, 175--182 (1991; Zbl 0759.34014) Full Text: EuDML OpenURL