Perturbations of existence families for abstract Cauchy problems
HTML articles powered by AMS MathViewer
- by Ti-Jun Xiao and Jin Liang PDF
- Proc. Amer. Math. Soc. 130 (2002), 2275-2285 Request permission
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.References
- Proceedings of the Second World Congress of Nonlinear Analysts. Part 6, Elsevier Ltd, Oxford, 1997. Held in Athens, July 10–17, 1996; Nonlinear Anal. 30 (1997), no. 6. MR 1603027
- G. Da Prato, Semigruppi di crescenza $n$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 20 (1966), 753–782 (Italian). MR 222710
- G. Da Prato, Semigruppi regolarizzabili, Ricerche Mat. 15 (1966), 223–248 (Italian). MR 225199
- E. B. Davies and M. M. H. Pang, The Cauchy problem and a generalization of the Hille-Yosida theorem, Proc. London Math. Soc. (3) 55 (1987), no. 1, 181–208. MR 887288, DOI 10.1112/plms/s3-55.1.181
- Ralph deLaubenfels, Existence and uniqueness families for the abstract Cauchy problem, J. London Math. Soc. (2) 44 (1991), no. 2, 310–338. MR 1136443, DOI 10.1112/jlms/s2-44.2.310
- Ralph deLaubenfels, Existence families, functional calculi and evolution equations, Lecture Notes in Mathematics, vol. 1570, Springer-Verlag, Berlin, 1994. MR 1290783, DOI 10.1007/BFb0073401
- Ralph deLaubenfels and Fuyuan Yao, Regularized semigroups of bounded semivariation, Semigroup Forum 53 (1996), no. 3, 369–383. MR 1406782, DOI 10.1007/BF02574151
- Wolfgang Desch and Wilhelm Schappacher, Some generation results for perturbed semigroups, Semigroup theory and applications (Trieste, 1987) Lecture Notes in Pure and Appl. Math., vol. 116, Dekker, New York, 1989, pp. 125–152. MR 1009392
- Odo Diekmann, Mats Gyllenberg, and Horst R. Thieme, Perturbing semigroups by solving Stieltjes renewal equations, Differential Integral Equations 6 (1993), no. 1, 155–181. MR 1190171
- Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. MR 1721989
- Jerome A. Goldstein, Ralph deLaubenfels, and James T. Sandefur Jr., Regularized semigroups, iterated Cauchy problems and equipartition of energy, Monatsh. Math. 115 (1993), no. 1-2, 47–66. MR 1223244, DOI 10.1007/BF01311210
- M. Hieber, A. Holderrieth, and F. Neubrander, Regularized semigroups and systems of linear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992), no. 3, 363–379. MR 1205405
- M. Jung, Multiplicative perturbations in semigroup theory with the $(Z)$-condition, Semigroup Forum 52 (1996), no. 2, 197–211. MR 1371803, DOI 10.1007/BF02574096
- Hermann Kellerman and Matthias Hieber, Integrated semigroups, J. Funct. Anal. 84 (1989), no. 1, 160–180. MR 999494, DOI 10.1016/0022-1236(89)90116-X
- J.-L. Lions, Les semi groupes distributions, Portugal. Math. 19 (1960), 141–164 (French). MR 143045
- Chung-Cheng Kuo and Sen-Yen Shaw, $C$-cosine functions and the abstract Cauchy problem. I, II, J. Math. Anal. Appl. 210 (1997), no. 2, 632–646, 647–666. MR 1453196, DOI 10.1006/jmaa.1997.5420
- Isao Miyadera, $C$-semigroups and semigroups of linear operators, Differential equations (Plovdiv, 1991) World Sci. Publ., River Edge, NJ, 1992, pp. 133–143. MR 1195545
- Hirokazu Oka, Linear Volterra equations and integrated solution families, Semigroup Forum 53 (1996), no. 3, 278–297. MR 1406775, DOI 10.1007/BF02574144
- S. Piskarëv and S.-Y. Shaw, Multiplicative perturbations of $C_0$-semigroups and some applications to step responses and cumulative outputs, J. Funct. Anal. 128 (1995), no. 2, 315–340. MR 1319959, DOI 10.1006/jfan.1995.1034
- S. Piskarëv and S.-Y. Shaw, Perturbation and comparison of cosine operator functions, Semigroup Forum 51 (1995), no. 2, 225–246. MR 1345113, DOI 10.1007/BF02573631
- N. Tanaka, $C$-semigroups of linear operators in Banach spaces - a generalization of the Hille-Yosida theorem, thesis, Waseda University, 1992.
- Ti-Jun Xiao and Jin Liang, The Cauchy problem for higher-order abstract differential equations, Lecture Notes in Mathematics, vol. 1701, Springer-Verlag, Berlin, 1998. MR 1725643, DOI 10.1007/978-3-540-49479-9
- T. J. Xiao and J. Liang, Higher order abstract Cauchy problems and their existence, uniqueness families, J. London Math. Soc., to appear.
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
- MR Author ID: 269685
- 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
- MR Author ID: 238393
- Email: jliang@ustc.edu.cn, jili@fa.uni-tuebingen.de
- 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
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 2275-2285
- MSC (2000): Primary 47D06; Secondary 34G10
- DOI: https://doi.org/10.1090/S0002-9939-02-06627-3
- MathSciNet review: 1896409