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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Perturbations of existence families for abstract Cauchy problems


Authors: Ti-Jun Xiao and Jin Liang
Journal: Proc. Amer. Math. Soc. 130 (2002), 2275-2285
MSC (2000): Primary 47D06; Secondary 34G10
Published electronically: March 13, 2002
MathSciNet review: 1896409
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Abstract: In this paper, we establish Desch-Schappacher type multiplicative and additive perturbation theorems for existence families for arbitrary order abstract Cauchy problems in a Banach space: $u^{(n)}(t)=Au(t)$ $(t\geq 0)$; $u^{(j)}(0)=x_j (0\leq j\leq n-1)$. As a consequence, we obtain such perturbation results for regularized semigroups and regularized cosine operator functions. An example is also given to illustrate possible applications.


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Additional Information

Ti-Jun Xiao
Affiliation: Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China
Address at time of publication: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076, Tübingen, Germany
Email: xiaotj@ustc.edu.cn, tixi@fa.uni-tuebingen.de

Jin Liang
Affiliation: Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China
Address at time of publication: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076, Tübingen, Germany
Email: jliang@ustc.edu.cn, jili@fa.uni-tuebingen.de

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06627-3
PII: S 0002-9939(02)06627-3
Keywords: Existence family, abstract Cauchy problem, regularized semigroups, regularized cosine operator functions, perturbation
Received by editor(s): November 8, 2000
Published electronically: March 13, 2002
Additional Notes: This work was supported partly by the NSF of China, the Key-Project-Foundation of the Chinese Academy of Sciences, and the Ministry of Education of China
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2002 American Mathematical Society