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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.

##### MSC:
 47D09 Operator sine and cosine functions and higher-order Cauchy problems
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##### References:
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