Proceedings of the American Mathematical Society

Katznelson-Tzafriri type theorems for individual solutions of evolution equations

Author(s): Nguyen Van Minh
Journal: Proc. Amer. Math. Soc. 136 (2008), 1749-1755.
MSC (2000): Primary 34G10; Secondary 47D06
Posted: January 28, 2008
Retrieve article in: PDF

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.: 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 $C\sb 0$ -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 $L^1(\mathbb{R}_+)$ , 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 $C\sb 0$ -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)

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 34G10, 47D06

Nguyen Van Minh
Affiliation: Department of Mathematics, University of West Georgia, Carrollton, Georgia 30118
Email: vnguyen@westga.edu

DOI: 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
Copyright of article: Copyright 2008, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS