×

zbMATH — the first resource for mathematics

A characterization of norm continuity of propagators for second order abstract differential equations. (English) Zbl 0932.34064
Summary: The authors obtain a concise characterization of the norm continuity for \(t>0\) of propagators for the complete second-order abstract differential equation on a Banach space \(E\), \[ u''(t)+ Bu'(t)+ Au(t)= 0,\quad t\geq 0, \] with \(B\in L(E)\). As a consequence, they discover that a strongly continuous cosine operator function or operator group is norm continuous for \(t>0\) if and only if its generator is bounded.

MSC:
34G20 Nonlinear differential equations in abstract spaces
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Lions, J.L., Un remarque sur LES applications du théorème de Hille-Yosida, J. math. soc. Japan, 9, 62-70, (1957) · Zbl 0078.08304
[2] Fattorini, H.O., The Cauchy problem, (1983), Addison-Wesley Reading, MA · Zbl 0493.34005
[3] Fattorini, H.O., Second order linear differential equations in Banach spaces, (1985), Elsevier Science Amsterdam · Zbl 0564.34063
[4] Goldstein, J.A., Semigroups of linear operators and applications, (1985), Oxford New York · Zbl 0592.47034
[5] Krein, S.G., Linear differential equations in Banach spaces, (1971), Amer. Math. Soc Providence · Zbl 0636.34056
[6] Nagel, R., One-parameter semigroups of positive operators, () · Zbl 0585.47030
[7] Favini, A.; Obrecht, E., Conditions for parabolicity of second order abstract differential equations, Differential integral equations, 4, 1005-1020, (1991) · Zbl 0735.34043
[8] Huang, F.L., On the mathematical model for linear elastic systems with analytic damping, SIAM control and optimization, 26, 714-724, (1988) · Zbl 0644.93048
[9] Liang, J.; Xiao, T.J., Norm continuity (for t > 0) of propagators of arbitrary order abstract differential equations in Hilbert spaces, J. math. anal. appl., 204, 124-137, (1996) · Zbl 0879.34058
[10] J. Liang and T.J. Xiao, Wellposedness results for certain classes of higher order abstract Cauchy problems connected with integrated semigroups, Semigroup Forum (to appear). · Zbl 0892.34054
[11] Mel’nikova, I.V.; Filinkov, A.I., The connection between well-posedness of the Cauchy problem for an equation and for a system in a Banach space, Soviet math. dokl., 37, 647-651, (1988) · Zbl 0669.34061
[12] Neubrander, F., Integrated semigroups and their applications to complete second order problems, Semigroup forum, 38, 233-251, (1989) · Zbl 0686.47038
[13] Xio (Xiao), T.J.; Liang, J., On complete second order linear differential equations in Banach spaces, Pacific J. math., 142, 175-195, (1990) · Zbl 0663.34053
[14] Xio (Xiao), T.J.; Liang, J., A note on the propagators of second order linear differential equations in Hilbert spaces, (), 663-667 · Zbl 0746.47024
[15] Xio (Xiao), T.J.; Liang, J., Complete second order linear differential equations with almost periodic solutions, J. math. anal. appl., 163, 136-146, (1992) · Zbl 0754.34062
[16] Xio (Xiao), T.J.; Liang, J., Analyticity of the propagators of second order linear differential equations in Banach spaces, Semigroup forum, 44, 356-363, (1992) · Zbl 0782.47041
[17] Xiao, T.J.; Liang, J., Entire solutions of higher order abstract Cauchy problems, J. math. anal. appl., 208, 298-310, (1997) · Zbl 0879.34062
[18] Xiao, T.J.; Liang, J., Semigroups arising from elastic systems with dissipation, Computers math. applic., 33, 10, 1-9, (1997) · Zbl 0972.34048
[19] Pazy, A., Semigroups of linear operators and applications to partial differential equations, (1983), Springer-Verlag New York · Zbl 0516.47023
[20] Kurepa, S., A cosine functional equation in Hilbert space, Can. J. math., 12, 45-50, (1960) · Zbl 0090.10001
[21] Lutz, D., Strongly continuous operator cosine functions, (), 73-97
[22] Fattorini, H.O., Extension and behavior at infinity of solutions of certain linear operational differential equations, Pacific J. math, 33, 583-615, (1970) · Zbl 0181.42601
[23] Goldstein, J.A., Semigroups and second-order differential equations, J. funct. anal., 4, 50-70, (1969) · Zbl 0179.14605
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.