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Spectral conditions guaranteeing a nontrivial
solution of the abstract Cauchy problem


Authors: R. deLaubenfels and S. Wang
Journal: Proc. Amer. Math. Soc. 126 (1998), 3271-3278
MSC (1991): Primary 47D03, 34G10, 47D06, 47A60
DOI: https://doi.org/10.1090/S0002-9939-98-04572-9
MathSciNet review: 1469403
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Abstract | References | Similar Articles | Additional Information

Abstract: We characterize subsets, $\Omega $, of the complex plane, with the following property: If $A$ has spectrum contained in $\Omega $, with polynomially bounded resolvent outside $\Omega $, then the abstract Cauchy problem corresponding to $A$ has a nontrivial solution.


References [Enhancements On Off] (What's this?)

  • [1] Edward Brian Davies, One-parameter semigroups, London Mathematical Society Monographs, vol. 15, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980. MR 591851
  • [2] Ralph deLaubenfels, Existence families, functional calculi and evolution equations, Lecture Notes in Mathematics, vol. 1570, Springer-Verlag, Berlin, 1994. MR 1290783
  • [3] Ralph deLaubenfels, Guo Zheng Sun, and Sheng Wang Wang, Regularized semigroups, existence families and the abstract Cauchy problem, Differential Integral Equations 8 (1995), no. 6, 1477–1496. MR 1329851
  • [4] Ralph deLaubenfels, Automatic extensions of functional calculi, Studia Math. 114 (1995), no. 3, 237–259. MR 1338830
  • [5] Jerome A. Goldstein, Semigroups of linear operators and applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. MR 790497
  • [6] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486
  • [7] J. A. van Casteren, ``Generators of Strongly Continuous Semigroups,'' Research Notes in Mathematics 115, Pitman, Boston, 1985. Zbl. 576:47023

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Additional Information

R. deLaubenfels
Affiliation: Scientia Research Institute, P. O. Box 988, Athens, Ohio 45701
Email: 72260.2403@compuserve.com

S. Wang
Affiliation: Department of Mathematics, Nanjing University, Nanjing, Jiangsu 210008, People’s Republic of China
Email: wang2598@netra.nju.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-98-04572-9
Received by editor(s): June 12, 1996
Received by editor(s) in revised form: March 20, 1997
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society