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Invariant measures of differential inclusions applied to singular perturbations. (English) Zbl 0923.34013

Generalizing the well-known classical concepts and results concerning invariant measures for dynamical systems in [N. Kryloff and N. Bogoliouboff, Ann. Math., II. Ser. 38, 65-113 (1937; Zbl 0016.08604); V. V. Nemytskii and V. V. Stepanov, Qualitative theory of differential equations, Princeton Univ. Press (1960; Zbl 0089.29502)], the author introduces two concepts of invariant measure (“occupational” and, respectively, “projectional”) for the solutions to an autonomous compact and convex-valued differential inclusion of the form: \(y'\in G(y)\subset \mathbb{R}^n\), in particular for differential equations without uniqueness.
This very consistent and clearly written article contains a large number of results concerning the equivalence of two concepts of invariant measures, their basic properties and relevant applications to singularly perturbed differential inclusions of the form: \((x',\varepsilon y')\in G(x,y)\).

MSC:

34A60 Ordinary differential inclusions
34E15 Singular perturbations for ordinary differential equations
34C29 Averaging method for ordinary differential equations
34D15 Singular perturbations of ordinary differential equations
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