Plotnikova, N. V. 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. Reviewer: Valerii V. Obukhovskij (Voronezh) Cited in 4 Documents MSC: 34A60 Ordinary differential inclusions 34C29 Averaging method for ordinary differential equations 34D35 Stability of manifolds of solutions to ordinary differential equations Keywords:differential inclusion; continuous dependence; averaging; \(R\)-solution Citations:Zbl 0064.33901 PDFBibTeX XMLCite \textit{N. V. Plotnikova}, Differ. Equ. 41, No. 7, 1049--1053 (2005; Zbl 1109.34306); translation from Differ. Uravn. 41, No. 7, 997--1000 (2005) Full Text: DOI References: [1] Vasil’ev, A.B., Ukr. Mat. Zh., 1983, vol. 35, no.5, pp. 607–611. [2] Plotnikov, V.A., Plotnikov, A.V., and Vityuk, A.N., Differentsial’nye uravneniya s mnogoznachnoi pravoi chast’yu. Asimptoticheskie metody (Differential Equations with Multivalued Right-Hand Side. Asymptotic Methods), Odessa, 1999. [3] Filatov, O.P. and Khapaev, M.M., Usrednenie sistem differentsial’nykh vklyuchenii (Averaging of Systems of Differential Inclusions), Moscow, 1988. [4] Janiak, T. and Luczak-Kumorek, E., Discuss. Math., 1991, no. 11, pp. 63–73. [5] Krasnosel’skii, M.A. and Krein, S.G., Uspekhi Mat. Nauk, 1955, vol. 10, no.3(65), pp. 147–152. [6] Dawidowski, M., Funct. et Approxim. (PRL), 1979, no. 7, pp. 55–70. [7] Panasyuk, A.I. and Panasyuk, V.I., Asimptoticheskaya magistral’naya optimizatsiya upravlyaemykh sistem (Asymptotic Turnpike Optimization of Control Systems), Minsk, 1986. · Zbl 0613.49003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.