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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,) and A is a closed linear operator, the (a,k) regularized family generated by A is the strongly continuous function R satisfying

k ^(λ)(I-a ^(λ)A) -1 = 0 e -λs R(s)ds·

The authors study the behavior as t of the family of operators

A t x=1 (k*a)(t) 0 t a(t-s)R(s)xds·

By choosing a and k appropriately, the family {R(t)} t0 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 k(t) (k*a)(t)=0,sup t>0 k(t)(1*a)(t) (k*a)(t)<,sup t>0 (a*a*k)(t) (a*k)(t)=·

47D06One-parameter semigroups and linear evolution equations
45J05Integro-ordinary differential equations
47D62Integrated semigroups