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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 $\left[0,\infty \right)$ and $A$ is a closed linear operator, the $\left(a,k\right)$ regularized family generated by $A$ is the strongly continuous function $R$ satisfying

$\stackrel{^}{k}\left(\lambda \right){\left(I-\stackrel{^}{a}\left(\lambda \right)A\right)}^{-1}={\int }_{0}^{\infty }{e}^{-\lambda s}R\left(s\right)ds·$

The authors study the behavior as $t\to \infty$ of the family of operators

${A}_{t}x=\frac{1}{\left(k*a\right)\left(t\right)}{\int }_{0}^{t}a\left(t-s\right)R\left(s\right)xds·$

By choosing $a$ and $k$ appropriately, the family ${\left\{R\left(t\right)\right\}}_{t\ge 0}$ 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 $a\left(t\right)$ is positive, $k\left(t\right)$ is positive and decreasing; the most important additional hypotheses employed are

$\underset{t\to \infty }{lim}\frac{k\left(t\right)}{\left(k*a\right)\left(t\right)}=0,\phantom{\rule{1.em}{0ex}}\underset{t>0}{sup}\frac{k\left(t\right)\left(1*a\right)\left(t\right)}{\left(k*a\right)\left(t\right)}<\infty ,\phantom{\rule{1.em}{0ex}}\underset{t>0}{sup}\frac{\left(a*a*k\right)\left(t\right)}{\left(a*k\right)\left(t\right)}=\infty ·$

##### MSC:
 47D06 One-parameter semigroups and linear evolution equations 45J05 Integro-ordinary differential equations 47D62 Integrated semigroups