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On the convergence of hyperbolic semigroups in variable Hilbert spaces. (English. Russian original) Zbl 1126.47038

J. Math. Sci., New York 127, No. 5, 2263-2283 (2005); translation from Tr. Semin. Im. I. G. Petrovskogo 24, 215-249 (2004).
Resolvent convergence is considered for nonnegative selfadjoint operators acting in variable Hilbert spaces \(H_{\varepsilon}\). The limit of the resolvents is, generally, a “pseudoresolvent” and not a resolvent. The situation may occur even for \(H_{\varepsilon}\equiv H\). This convergence is used for passing to the limit in the corresponding hyperbolic differential-operator equations in \(H_{\varepsilon}\) considered from the semigroup point of view. The scheme studied in the paper can be applied to the homogenization of nonstationary problems of elasticity for thin structures.

MSC:

47D06 One-parameter semigroups and linear evolution equations
47A10 Spectrum, resolvent
34G10 Linear differential equations in abstract spaces
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