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Topological approach to differential inclusions. (English) Zbl 0834.34022
Granas, Andrzej (ed.) et al., Topological methods in differential equations and inclusions. Proceedings of the NATO Advanced Study Institute and séminaire de mathématiques supérieures on topological methods in differential equations and inclusions, Montréal, Canada, July 11-22, 1994. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 472, 129-190 (1995).
Summary: The purpose of these lectures is to show how the topological degree theory for (non-convex) multivalued mappings can be usefully applied to differential inclusions. Namely, we shall apply it to get a topological characterization of the set of solutions and periodic solutions for some differential inclusions. We discuss these problems in the case when the considered differential inclusions are defined on Banach spaces or on proximate retracts.
Recall that a proximate retract is a compact subset \(A\) of the Euclidean space \(\mathbb{R}^n\) such that there exists an open neighbourhood \(U\) of \(A\) in \(\mathbb{R}^n\) and a metric retraction \(r: U\to A\). It is well-known that any compact convex subset \(A\) of \(\mathbb{R}^n\) or any compact \(C^2\)-manifold \(M\subset \mathbb{R}^n\) is a proximate retract. Moreover, a topological degree method for implicit differential equations and differential inclusions is presented.
For the entire collection see [Zbl 0829.00024].

34A60 Ordinary differential inclusions
47H11 Degree theory for nonlinear operators