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Upper semicontinuous differential inclusions without convexity. (English) Zbl 0698.34014

The present paper deals with the existence of absolutely continuous solutions to differential inclusions with upper semicontinuous and not necessarily convex right-hand sides. It is well known that generally in such case existence theorems are not true. It was proved here that in the case of cyclically monotone multifunctions, compactness and upper semicontinuity are sufficient for the existence of solutions. Recall that a multifunction A: \({\mathbb{R}}^ n\to {\mathbb{R}}^ n\) is called cyclically monotone if for every cyclical sequence \(x_ 0,x_ 1,...,x_ N=x_ 0\) (N-arbitrary) and every sequence \(y_ i\in A(x_ i)\), \(i=1,...,N\), we have \(\sum^{N}_{i=1}<x_ i-x_{i-1},y_ i>\geq 0\) where as usual \(<.,.>\) denotes inner product of \({\mathbb{R}}^ n\).
Reviewer: M.Kisielewicz

MSC:

34A60 Ordinary differential inclusions
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[1] Jean-Pierre Aubin and Arrigo Cellina, Differential inclusions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 264, Springer-Verlag, Berlin, 1984. Set-valued maps and viability theory. · Zbl 0538.34007
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