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Rates of approximation and ergodic limits of regularized operator families. (English) Zbl 1032.47024
If $a$ and $k$ are Laplace transformable functions on $[0,\infty)$ and $A$ is a closed linear operator, the $(a,k)$ regularized family generated by $A$ is the strongly continuous function $R$ satisfying $$ \widehat k(\lambda)(I-\widehat a(\lambda)A)^{-1} = \int_0^\infty e^{-\lambda s}R(s) ds. $$ The authors study the behavior as $t\to \infty$ of the family of operators $$ A_tx = {{1}\over{(k*a)(t)}} \int_0^t a(t-s)R(s)x ds. $$ By choosing $a$ and $k$ appropriately, the family $\{R(t)\}_{t\geq 0}$ corresponds to an $n$-times integrated semigroup, resolvent family, or cosine family, etc. The basic assumptions on $a$ and $k$ that make it possible to draw conclusions on the asymptotic behavior of $A_t$ is that $a(t)$ is positive, $k(t)$ is positive and decreasing; the most important additional hypotheses employed are $$ \lim_{t\to\infty}{k(t) \over (k*a)(t)} = 0,\quad \sup_{t> 0} {k(t)(1*a)(t) \over (k*a)(t)} < \infty,\quad \sup_{t> 0} {(a*a*k)(t)\over (a*k)(t)} = \infty.$$

47D06One-parameter semigroups and linear evolution equations
45J05Integro-ordinary differential equations
47D62Integrated semigroups
Full Text: DOI
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