A characterization of norm continuity of propagators for second order abstract differential equations. (English) Zbl 0932.34064

Liang, Jin; Xiao, Tijun

Comput. Math. Appl. 36, No. 2, 87-94 (1998).

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

Keywords:

norm continuity; propagators; complete second-order abstract differential equation; operator group

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: Addison-Wesley Reading, MA · Zbl 0493.34005
[3]	Fattorini, H. O., Second Order Linear Differential Equations in Banach Spaces (1985), Elsevier Science: Elsevier Science Amsterdam · Zbl 0564.34063
[4]	Goldstein, J. A., Semigroups of Linear Operators and Applications (1985), Oxford: Oxford New York · Zbl 0592.47034
[5]	Krein, S. G., Linear Differential Equations in Banach Spaces (1971), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 0636.34056
[6]	Nagel, R., One-parameter semigroups of positive operators, (Nagel, R., Lect. Notes Math., Volume 1184 (1986), Springer-Verlag: Springer-Verlag New York) · 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
[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, (Proc. of Amer. Math. Soc., 113 (1991)), 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: 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, (Lecture Notes in Math., Volume 948 (1982), Springer-Verlag: Springer-Verlag New York), 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