zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 $$ \lim_{t\to\infty}{k(t) \over (k*a)(t)} = 0,\quad \sup_{t> 0} {k(t)(1*a)(t) \over (k*a)(t)} < \infty,\quad \sup_{t> 0} {(a*a*k)(t)\over (a*k)(t)} = \infty.$$

MSC:
47D06One-parameter semigroups and linear evolution equations
45J05Integro-ordinary differential equations
47D62Integrated semigroups
WorldCat.org
Full Text: DOI
References:
[1] Arendt, W.; Batty, C. J. K.: A complex Tauberian theorem and mean ergodic semigroups. Semigroup forum 50, No. 3, 351-366 (1995) · Zbl 0836.47032
[2] Arendt, W.; Kellerman, H.: Integrated solutions of Volterra integrodifferential equations and applications. Pitman research notes in mathematics 190, 21-51 (1989)
[3] Butzer, P. L.; Berens, H.: Semi-groups of operators and approximation. (1967) · Zbl 0164.43702
[4] Chan, J. -C.; Shaw, S. -Y.: Rates of approximation and ergodic limits of resolvent families. Arch. math. 66, 320-330 (1996) · Zbl 0859.47027
[5] Cioranescu, I.: On twice differentiable functions. Resultate math. 16, No. 1/2, 49-53 (1989) · Zbl 0691.46026
[6] I. Cioranescu, Local convoluted semigroups, Lecture Notes in Pure and Applied Mathematics, Vol. 168, Marcel Dekker, New York, 1995, pp. 107--122. · Zbl 0818.47038
[7] I. Cioranescu, G. Lumer, On K(t)-convoluted semigroups, Pitman Research Notes in Mathematics, Vol. 324, Longman Sci. Tech., Harlow, 1995, pp. 86--93. · Zbl 0828.34046
[8] K.-J. Engel, R. Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, Vol. 194, Springer, New York--Berlin--Heidelberg, 2000. · Zbl 0952.47036
[9] Gripenberg, G.; Londen, S. O.; Staffans, O. J.: Volterra integral and functional equations. (1990) · Zbl 0695.45002
[10] M. Hieber, Integrated semigroups and differential operators on Lp spaces, Math. Ann. (1991) 1--16. · Zbl 0724.34067
[11] Liu, H.; Shaw, S. -Y.: Rates of local ergodic limits of n-times integrated solution families. Progress in nonlinear differential equations applications 42 (2000) · Zbl 0961.45009
[12] Lizama, C.: A mean ergodic theorem for resolvent operators. Semigroup forum 47, 227-230 (1993) · Zbl 0799.47024
[13] Lizama, C.: Regularized solutions for abstract Volterra equations. J. math. Anal. appl. 243, 278-292 (2000) · Zbl 0952.45005
[14] Lizama, C.: On approximation and representation of k-regularized resolvent families. Integr. equations oper. Theory 41, 223-229 (2001) · Zbl 1011.45006
[15] C. Lizama, J. Sanchez, On perturbation of k-regularized resolvent families, Taiwanese J. Math, to appear.
[16] Oka, H.: Linear Volterra equations and integrated solution families. Semigroup forum 53, 278-297 (1996) · Zbl 0862.45017
[17] Prüss, J.: Evolutionary integral equations and applications. (1993) · Zbl 0784.45006
[18] Shaw, S. -Y.: Mean ergodic theorems and linear functional equations. J. funct. Anal. 87, 428-441 (1989) · Zbl 0704.47006
[19] Shaw, S. -Y.: Uniform convergence of ergodic limits and approximate solutions. Proc. amer. Math. soc. 114, No. 2, 405-411 (1992) · Zbl 0764.47006
[20] Shaw, S. -Y.: Convergence rates of ergodic limits and approximate solutions. J. approx. Theory 75, 157-166 (1993) · Zbl 0792.41024
[21] S.-Y. Shaw, Ergodic properties of integrated semigroups and resolvent families, in: Proceedings ICM’94, World Scientific Press, Singapore, 1996, pp. 171--178. · Zbl 0928.47034
[22] Shaw, S. -Y.: Non-optimal rates of ergodic limits and approximate solutions. J. approx. Theory 94, 285-299 (1998) · Zbl 0914.47010
[23] S.-Y. Shaw, Ergodic theorems with rates for r-times integrated solution families, Operator Theory and Related Topics, Vol. II, Operator Theory Advances and Applications, Vol. 118, Birkhäuser, Basel, 2000, pp. 359--371. · Zbl 0946.45005
[24] Yosida, K.: Functional analysis. (1980) · Zbl 0435.46002