Taiwanese Journal of Mathematics

On Approximation of Integrated Semigroups

Miao Li and Sergey Piskarev

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This paper is devoted to approximation of integrated semigroups in space and in time variables. The presentation is given in the abstract framework of discrete approximation scheme, which includes finite element methods, finite difference schemes and projection method.

Article information

Taiwanese J. Math., Volume 14, Number 6 (2010), 2137-2161.

First available in Project Euclid: 18 July 2017

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Zentralblatt MATH identifier

Primary: 34G10: Linear equations [See also 47D06, 47D09] 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30] 65J20: Improperly posed problems; regularization 65J22: Inverse problems

abstract differential equations in Banach spaces $C_0$-semigroups integrated semigroups Trotter-Kato theorem discretization methods difference schemes stability of difference schemes discrete semigroups Banach spaces


Li, Miao; Piskarev, Sergey. On Approximation of Integrated Semigroups. Taiwanese J. Math. 14 (2010), no. 6, 2137--2161. doi:10.11650/twjm/1500406067. https://projecteuclid.org/euclid.twjm/1500406067

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  • W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Monorgraphs Math., 96, Birkhäuser, 2001.
  • A. Bobrowski, Integrated semigroups and the Trotter-Kato theorem, Bull. Polish Acad. Sci. Math., 42 (1994), 297-304.
  • A. Bobrowski, Generalized telegraph equation and the Sova-Kurtz version of the Trotter-Kato theorem, Ann. Polon. Math., 64 (1996), 37-45.
  • A. Bobrowski, On the Yosida approximation and the Widder-Arendt representation theorem, Studia Math., 124 (1997), 281-290.
  • A. Bobrowski, On approximation of (1.$A$) semigroups by disrete semigroups, Bull. Polish Acad. Sci., 46 (1998), 142-154.
  • A. Bobrowski, On the generation of non-continuous semigroups, Semigroup Forum, 54 (1997), 237-252.
  • A. Bobrowski, On limitations and insufficiency of the Trotter-Kato theorem, Semigroup Forum, 75(2) (2007), 317-336.
  • P. Brenner and V. Thomée, On rational approximations of semigroups, SIAM J. Numer. Anal., 16 (1979), 683-694.
  • S. Busenberg and B. H. Wu, Convergence theorems for integrated semigroups, Differential Integral Equations, 5 (1992), 509-520.
  • V. Cachia, Convergence at the origin of integrated semigroups, Conference on operator semigroups, evolution equations and spectral theory in Mathematical Physics, Marselle-Luminy, 2005.
  • M. Crouzeix, S. Larsson, S. Piskarev and V. Thomee, The stability of rational approximations of analytic semigroups, BIT, 33 (1993), 74-84.
  • R. deLaubenfels, Inverses of generators of integrated or regularized semigroups, Semigroup Forum, 75 (2007), 457-463.
  • B. Eberhardt and G. Greiner, Baillon's theorem on maximal regularity, Acta Appl. Math., 27 (1992), 47-54.
  • H. Fujita and A. Mizutani, On the finite element method for parabolic equations. I. Approximation of holomorphic semi-groups, J. Math. Soc. Japan, 28(4) (1976), 749-771.
  • D. Guidetti, B. Karasozen and S. Piskarev, Approximation of abstract differential equations, Journal of Mathematical Sciences, 122 (2004), 3013-3054.
  • H. Kellerman and M. Hieber, Integrated semigroups, J. Funct. Anal., 84 (1989), 160-180.
  • T. Kato, Perturbation theory for linear operators, Classics in Mathematics. Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition.
  • S. G. Krein, Linear differential equations in Banach space, American Mathematical Society, Providence, R.I., Translated from the Russian by J. M. Danskin, Translations of Mathematical Monographs, Vol. 29, 1971.
  • C.-C. Kuo, On $\alpha$-times integrated $C$-semigroups and the abstract Cauchy problem, Studia Math., 142 (2000), 201-217.
  • T. G. Kurtz, Extensions of Trotter's operator semigroup approximation theorems, J. Funct. Anal., 3 (1969), 354-375.
  • T. G. Kurtz, A general theorem on the convergence of operator semigroups, Tran. Amer. Math. Soc., 148 (1970), 23-32.
  • Y.-C. Li and S.-Y. Shaw, $N$-times integrated $C$-semigroups and the abstract Cauchy problem, Taiwanese J. Math., 1 (1997), 75-102.
  • C. Lizama, On the convergence and approximation of integrated semigroups, J. Math. Anal. Appl., 181 (1994), no. 1, 89-103.
  • S. Piskarev, Approximation of holomorphic semigroups, Tartu Riikl. Ül. Toimetised, 492 (1979), 3-14.
  • S. Piskarev, Error estimates in the approximation of semigroups of operators by Padé fractions, Izv. Vuzov Math., 4 (1979), 33-38.
  • S. Piskarev, Differential equations in Banach space and their approximation, Moscow, Moscow State University Publish House, 2005, (in Russian).
  • N. Tanaka, Approximation of integrated semigroups by “integrated" discrete parameter semigroups, Semigroup Forum, 55 (1997), 57-67.
  • T. Ushijima, Approximation theory for semi-groups of linear operators and its application to approximation of wave equations, Japan J. Math., 1 (1975/76), 185-224.
  • G. Vainikko, Approximative methods for nonlinear equations (two approaches to the convergence problem), Nonlinear Anal., 2 (1978), 647-687.
  • V. V. Vasilev, S. G. Krein and S. Piskarev, Operator semigroups, cosine operator functions, and linear differential equations, In: Mathematical analysis, 28 (Russian), Itogi Nauki i Tekhniki, pages 87-202, 204. Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990, Translated in J. Soviet Math., 54 (1991), 1042-1129.
  • T.-J. Xiao and J. Liang, Approximations of Laplace transforms and integrated semigroups, J. Funct. Anal., 172 (2000), 202-220.
  • Q. Zheng and Y. S. Lei, Exponentially bounded $C$-semigroup and integrated semigroup with nondensely defined generators, I. Approximation. Acta Math. Sci., (English 4th. Ed.) 13 (1993), 251-260.
  • Q. Zheng, Perturbations and approximations of integrated semigroups, Acta Math. Sinica, 9 (1993), 252-260.