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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
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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
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