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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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##### References:
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