|
Katznelson-Tzafriri type theorems for individual solutions of evolution equations
Author:
Nguyen Van Minh
Journal:
Proc. Amer. Math. Soc. 136 (2008), 1749-1755
MSC (2000):
Primary 34G10; Secondary 47D06
Posted:
January 28, 2008
Corrigendum:
Proc. Amer. Math. Soc. 138 (2010), 2263-2263.
MathSciNet review:
2373605
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: In this paper we present an extension of the Katznelson-Tzafriri Theorem to the asymptotic behavior of individual solutions of evolution equations  . The obtained results do not require the uniform continuity of solutions as well as the well-posedness of the equations. The method of study is based on a recently developed approach to the spectral theory of functions that is direct and free of  -semigroups.
- 1.
Wolfgang
Arendt, Charles
J. K. Batty, Matthias
Hieber, and Frank
Neubrander, Vector-valued Laplace transforms and Cauchy
problems, Monographs in Mathematics, vol. 96, Birkhäuser
Verlag, Basel, 2001. MR 1886588
(2003g:47072)
- 2.
Wolfgang
Arendt and Jan
Prüss, Vector-valued Tauberian theorems and asymptotic
behavior of linear Volterra equations, SIAM J. Math. Anal.
23 (1992), no. 2, 412–448. MR 1147871
(92m:47150), http://dx.doi.org/10.1137/0523021
- 3.
Bolis
Basit, Harmonic analysis and asymptotic behavior of solutions to
the abstract Cauchy problem, Semigroup Forum 54
(1997), no. 1, 58–74. MR 1418374
(98f:47049), http://dx.doi.org/10.1007/BF02676587
- 4.
Bolis
Basit and A.
J. Pryde, Ergodicity and differences of functions on
semigroups, J. Austral. Math. Soc. Ser. A 64 (1998),
no. 2, 253–265. MR 1619808
(2000g:43007)
- 5.
Charles
J. K. Batty, Jan
van Neerven, and Frank
Räbiger, Local spectra and individual stability
of uniformly bounded 𝐶₀-semigroups, Trans. Amer. Math. Soc. 350 (1998), no. 5, 2071–2085. MR 1422890
(98h:47054), http://dx.doi.org/10.1090/S0002-9947-98-01919-9
- 6.
Ralph
Chill and Yuri
Tomilov, Stability of operator semigroups: ideas and results,
Perspectives in operator theory, Banach Center Publ., vol. 75, Polish
Acad. Sci., Warsaw, 2007, pp. 71–109. MR 2336713
(2008m:47054), http://dx.doi.org/10.4064/bc75-0-6
- 7.
J.
Esterle, E.
Strouse, and F.
Zouakia, Stabilité asymptotique de certains semi-groupes
d’opérateurs et idéaux primaires de
𝐿¹(𝑅⁺), J. Operator Theory
28 (1992), no. 2, 203–227 (French). MR 1273043
(95f:43001)
- 8.
Y.
Katznelson and L.
Tzafriri, On power bounded operators, J. Funct. Anal.
68 (1986), no. 3, 313–328. MR 859138
(88e:47006), http://dx.doi.org/10.1016/0022-1236(86)90101-1
- 9.
James
Liu, Gaston
N’Guérékata, Nguyen
Van Minh, and Quoc
Phong Vu, Bounded solutions of parabolic equations in continuous
function spaces, Funkcial. Ekvac. 49 (2006),
no. 3, 337–355. MR 2297943
(2008f:34147), http://dx.doi.org/10.1619/fesi.49.337
- 10.
Nguyen Van Minh, A new approach to the spectral theory of functions and the Loomis-Arendt-Batty-Vu Theory. In ArXiv.org at the URL: http://arxiv.org/abs/math.FA/0609652
- 11.
Heybetkulu
Mustafayev, The Banach algebra generated by a
𝐶₀-semigroup, C. R. Math. Acad. Sci. Paris
342 (2006), no. 8, 575–578 (English, with
English and French summaries). MR 2217918
(2007a:47050), http://dx.doi.org/10.1016/j.crma.2006.02.017
- 12.
Jan
van Neerven, The asymptotic behaviour of semigroups of linear
operators, Operator Theory: Advances and Applications, vol. 88,
Birkhäuser Verlag, Basel, 1996. MR 1409370
(98d:47001)
- 13.
Vũ
Qu\cfac{o}c Phóng, Theorems of Katznelson-Tzafriri type for
semigroups of operators, J. Funct. Anal. 103 (1992),
no. 1, 74–84. MR 1144683
(93e:47050), http://dx.doi.org/10.1016/0022-1236(92)90135-6
- 14.
Jan
Prüss, Evolutionary integral equations and applications,
Monographs in Mathematics, vol. 87, Birkhäuser Verlag, Basel,
1993. MR
1238939 (94h:45010)
- 1.
- W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander, ``Vector-valued Laplace transforms and Cauchy problems'', Monographs in Mathematics, 96, Birkhäuser Verlag, Basel, 2001. MR 1886588 (2003g:47072)
- 2.
- W. Arendt, J. Pruss, Vector-valued Tauberian theorems and asymptotic behavior of linear Volterra equations, SIAM J. Math. Anal. 23 (1992), 412-448. MR 1147871 (92m:47150)
- 3.
- B. Basit, Harmonic analysis and asymptotic behavior of solutions to the abstract Cauchy problem, Semigroup Forum 54 (1997), 58-74. MR 1418374 (98f:47049)
- 4.
- B. Basit, J. Pryde, Ergodicity and differences of functions on semigroups, J. Austral. Math. Soc. (Series A) 64 (1998), 253-265. MR 1619808 (2000g:43007)
- 5.
- C. J. K. Batty, Jan van Neerven, Frank Rabiger, Local spectra and individual stability of uniformly bounded
 -semigroups, Trans. Amer. Math. Soc. 350 (1998), 2071-2085. MR 1422890 (98h:47054)
- 6.
- R. Chill, Y. Tomilov, Stability of operators semigroups: ideas and results. In ``Perspectives in Operator Theory'', Banach Center Publications, Vol. 75 (2007), pp. 71-109. MR 2336713
- 7.
- J. Esterle, E. Strouse, F. Zouakia, Stabilité asymptotique de certains semi-groupes d'opérateurs et idéaux primaires de
 , J. Operator Theory 28 (1992), 203-227. MR 1273043 (95f:43001)
- 8.
- Y. Katznelson, L. Tzafriri, On power bounded operators, J. Funct. Anal. 68 (1986), 313-328. MR 859138 (88e:47006)
- 9.
- J. Liu, G. Nguerekata, Nguyen Van Minh, Vu Quoc Phong, Bounded solutions of parabolic equations in continuous function spaces, Funkciolaj Ekvacioj 49 (2006), 337-355. MR 2297943
- 10.
- Nguyen Van Minh, A new approach to the spectral theory of functions and the Loomis-Arendt-Batty-Vu Theory. In ArXiv.org at the URL: http://arxiv.org/abs/math.FA/0609652
- 11.
- H. Mustafayev, The Banach algebra generated by a
 -semigroup, C. R. Math. Acad. Sci. Paris 342 (2006), no. 8, 575-578. MR 2217918 (2007a:47050)
- 12.
- J. M. A. M. van Neerven, ``The Asymptotic Behaviour of Semigroups of Linear Operators'', in Operator Theory: Advances and Applications, Vol. 88, Birkhäuser Verlag, Basel, Boston, Berlin, 1996. MR 1409370 (98d:47001)
- 13.
- Vu Quoc Phong, Theorems of Katznelson-Tzafriri type for semigroups of operators, J. Funct. Anal. 103 (1992), 74-84. MR 1144683 (93e:47050)
- 14.
- J. Pruss, ``Evolutionary Integral Equations and Applications'', Birkhäuser, Basel, 1993. MR 1238939 (94h:45010)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
34G10,
47D06
Retrieve articles in all journals
with MSC (2000):
34G10,
47D06
Additional Information
Nguyen Van Minh
Affiliation:
Department of Mathematics, University of West Georgia, Carrollton, Georgia 30118
Email:
vnguyen@westga.edu
DOI:
http://dx.doi.org/10.1090/S0002-9939-08-09330-1
PII:
S 0002-9939(08)09330-1
Keywords:
Katznelson-Tzafriri Type Theorem,
reduced spectrum of a function,
asymptotic behavior
Received by editor(s):
March 26, 2007
Posted:
January 28, 2008
Additional Notes:
The author thanks the referee for carefully reading the manuscript and for making useful remarks.
Communicated by:
Carmen C. Chicone
Article copyright:
© Copyright 2008 American Mathematical Society
|
