Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
MathSciNet review: 1469403
Full-text PDF

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] E. B. Davies, ``One-parameter Semigroups,'' London Math. Soc. Monographs 15, Academic Press, 1980. MR 82i:47060
  • [2] R. deLaubenfels ``Existence Families, Functional Calculi and Evolution Equations,'' Lecture Notes in Mathematics 1570, Springer Verlag, Berlin 1994. MR 96b:47047
  • [3] R. deLaubenfels, G. Sun and S. Wang, Regularized semigroups, existence families and the abstract Cauchy problem, J. Diff. and Int. Eqns. 8 (1995), 1477-1496. MR 96j:47035
  • [4] R. deLaubenfels, Automatic extensions of functional calculi, Studia Math. 114 (1995), 237-259. MR 96f:47029
  • [5] J. A. Goldstein, ``Semigroups of Linear Operators and Applications,'' Oxford, New York, 1985. MR 87c:47056
  • [6] A. Pazy, ``Semigroups of Linear Operators and Applications to Partial Differential Equations,'' Springer, New York, 1983. MR 85g:47061
  • [7] J. A. van Casteren, ``Generators of Strongly Continuous Semigroups,'' Research Notes in Mathematics 115, Pitman, Boston, 1985. Zbl. 576:47023

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47D03, 34G10, 47D06, 47A60

Retrieve articles in all journals with MSC (1991): 47D03, 34G10, 47D06, 47A60

Additional Information

R. deLaubenfels
Affiliation: Scientia Research Institute, P. O. Box 988, Athens, Ohio 45701

S. Wang
Affiliation: Department of Mathematics, Nanjing University, Nanjing, Jiangsu 210008, People’s Republic of China

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