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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 $E$, $$u''(t)+ Bu'(t)+ Au(t)= 0,\quad t\ge 0,$$ 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:
34G20Nonlinear ODE in abstract spaces
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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) · Zbl 0493.34005
[3] Fattorini, H. O.: Second order linear differential equations in Banach spaces. (1985) · Zbl 0564.34063
[4] Goldstein, J. A.: Semigroups of linear operators and applications. (1985) · Zbl 0592.47034
[5] Krein, S. G.: Linear differential equations in Banach spaces. (1971) · Zbl 0236.47034
[6] Nagel, R.: One-parameter semigroups of positive operators. Lect. notes math. 1184 (1986)
[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. Proc. of amer. Math. soc. 113, 663-667 (1991) · 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, No. 10, 1-9 (1997) · Zbl 0972.34048
[19] Pazy, A.: Semigroups of linear operators and applications to partial differential equations. (1983) · 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. 948, 73-97 (1982)
[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