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Perturbation and comparison of cosine operator functions. (English) Zbl 0828.47037
Summary: Let \(A\) be a closed linear operator such that the abstract Cauchy problem \(u''(t) = Au(t)\), \(t \in\mathbb{R}\), \(u(0) = x\), \(u'(0) = y\), is well-posed. We present some multiplicative perturbation theorems which give conditions on an operator \(C\) so that the abstract Cauchy problems for differential equations \(u''(t) = AC u(t)\) and \(u''(t) = C Au(t)\) also are well-posed. Some new or known additive perturbation theorems and mixed- type perturbation theorems are deduced as corollaries. Applications to characterization of the infinitesimal comparison of two cosine operator functions are also discussed.

47D09 Operator sine and cosine functions and higher-order Cauchy problems
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