Abstract
Simple estimates are obtained for the spectrum of the operator bundle\(R(\lambda ) = \sum\nolimits_{i = 0}^n {A_{n - i} \lambda ^i }\) in terms of estimates of the maximum and minimum eigenvalues of the operators\(\frac{1}{2}(A_{n - i} - A_{n - i}^* )(i = 0,1,2, \ldots n)\) and the norms of the operators\(\frac{1}{2}(A_{n - i} - A_{n - i}^* )(i = 0,1,2, \ldots n)\) We formulate a criterion of the asymptotic stability of the differential equations
We present examples of the stability conditions for equations with n=2 and n=3.
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A. I. Plesner, Spectral Theory of Linear Operators, F. Unger Publ. Co. (1969).
Yu. L. Daletskii and M. G. Krein, Stability of Solutions of Differential Equations in a Banach Space [in Russian], Nauka, Moscow (1970).
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Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 611–614, April, 1976.
The author thanks M. A. Krasnosel'skii for attention to the work and useful discussion.
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Frolov, V.I. Estimate for the spectrum of an operator bundle and its application to stability problems. Mathematical Notes of the Academy of Sciences of the USSR 19, 369–371 (1976). https://doi.org/10.1007/BF01156800
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DOI: https://doi.org/10.1007/BF01156800