×

zbMATH — the first resource for mathematics

On generators of integrated \(C\)-semigroups and \(C\)-cosine functions. (English) Zbl 0804.47044
Summary: The following two theorems are proved:
(1) the generator of an exponentially equicontinuous \(n\)-times integrated \(C\)-cosine function also generates an exponentially equicontinuous \([(n+1)/2]\)-times integrated \(C\)-semigroup;
(2) If \(A\) and \(-A\) are generators of exponentially equicontinuous \(n\)- times integrated \(C\)-semigroups, then \(A^ 2\) generates an exponentially equicontinuous \(n\)-times integrated \(C\)-cosine function.

MSC:
47D06 One-parameter semigroups and linear evolution equations
47D09 Operator sine and cosine functions and higher-order Cauchy problems
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Arendt, W.,Vector valued Laplace transforms and Cauchy problems, Israel J. Math.59 (1987), 327–352. · Zbl 0637.44001 · doi:10.1007/BF02774144
[2] Davies, E. B., and M. M. Pang,The Cauchy problem and a generalization of the Hille-Yosida theorem, Proc. London Math. Soc.55 (1987), 181–208. · Zbl 0651.47026 · doi:10.1112/plms/s3-55.1.181
[3] deLaubenfels, R.,C-semigroups and the Cauchy problem, Journal of Functional Analysis, to appear. · Zbl 0717.47014
[4] deLaubenfels, R.,Existence and uniqueness families for the abstract Cauchy problem, J. London Math. Soc., to appear. · Zbl 0766.47011
[5] Fattorini, H. O.,Ordinary differential equations in linear topological spaces, I., J. Differential Eq.5 (1968), 72–105. · Zbl 0175.15101 · doi:10.1016/0022-0396(69)90105-3
[6] Fattorini, H. O., ”The Second order linear differential equations in Banach spaces,” North-Holland Mathematics Studies 108, North-Holland, 1985. · Zbl 0564.34063
[7] Goldstein, J. A.,The universal addability problem for generators of cosine functions and operator groups, Houston J. Math.6 (1980), 365–373. · Zbl 0454.47019
[8] Kellermann, H., and M. Hieber,Integrated semigroups, J. Funct. Anal.84 (1989), 160–180. · Zbl 0689.47014 · doi:10.1016/0022-1236(89)90116-X
[9] Li, Y.-C., and S.-Y. Shaw,Integrated C-semigroups and the abstract Cauchy problem, 1991, preprint. · Zbl 0892.47042
[10] Li, Y.-C., and S.-Y. Shaw,Integrated C-cosine functions and the abstract Cauchy problem, 1991, preprint.
[11] Neubrander, F.,Integrated semigroups and their application to the abstract Cauchy problem, Pacific J. Math.135 (1988), 111–155. · Zbl 0675.47030
[12] Tanaka, N., and I. Miyadera,Exponentially bounded C-semigroups and integrated semigroups, Tokyo J. Math.12 (1989), 99–115. · Zbl 0702.47028 · doi:10.3836/tjm/1270133551
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.