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The Krasnosel’skii-Krein theorem for differential inclusions. (English. Russian original) Zbl 1109.34306
Differ. Equ. 41, No. 7, 1049-1053 (2005); translation from Differ. Uravn. 41, No. 7, 997-1000 (2005).
A family of differential inclusions in a finite-dimensional space \[ \dot{x} \in F(t,x,\lambda) \qquad x(0,\lambda) = x_0, \] is considered under the assumption that the convex-valued right-hand side is integrally continuous with respect to \(\lambda\) at the point \(\lambda_0\). Extending the result of M. A. Krasnoselskii and S. G. Krein [Usp. Mat. Nauk 10, No. 3(65), 147–152 (1955; Zbl 0064.33901)], the author proves that under some additional conditions, for each \(\eta >0\), there exists a neighborhood \(U(\lambda_0)\) of \(\lambda_0\) such that for each solution \(x(t,\lambda),\) \(\lambda \in U(\lambda_0)\), of the above problem defined for \(0 \leq t \leq T\) there exists a solution \(x(t,\lambda_0)\) such that \(\| x(t,\lambda) - x(t,\lambda_0)\| < \eta,\) \(0 \leq t \leq T.\) These results are applied to obtain the analogs of the Bogolyubov average theorem for differential inclusions.

MSC:
34A60 Ordinary differential inclusions
34C29 Averaging method for ordinary differential equations
34D35 Stability of manifolds of solutions to ordinary differential equations
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