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Piskarev, S. I.

Almost-periodic solutions of second-order differential equations. (English. Russian original) Zbl 0658.34035

Sib. Math. J. 25, 451-460 (1984); translation from Sib. Mat. Zh. 25, No. 3(145), 137-147 (1984).
We give the fundamental properties of the infinitesimal generator A that corresponds to an almost-periodic cosine operator-function C(t). The cosine operator-function is related with an abstract linear second-order differential equation in the same way as the strongly continuous operator semigroups are related with an abstract first-order linear differential equation. Therefore, we may investigate the almost-periodicity property of the solution in terms of the infinitesimal generating operator.

Cited in 2 Documents

MSC:

34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations

Keywords:

linear second-order differential equation; infinitesimal generating operator
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References:

[1] Yu. L. Daletskii (Ju. L. Daleckii) and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, Amer. Math. Soc., Providence (1974).
[2] M. Amerio and G. Prouse, Almost-Periodic Functions and Functional Equations, Van Nostrand Reinhold, New York (1971). · Zbl 0215.15701
[3] H. Bart, ?Periodic strongly continuous semigroups,? Ann. Mat. Pura Appl.,115, 311-318 (1977). · Zbl 0376.47018 · doi:10.1007/BF02414722
[4] H. Bart and S. Goldberg, ?Characterizations of almost-periodic strongly continuous groups and semigroups,? Math. Ann.,236, 105-116 (1978). · doi:10.1007/BF01351384
[5] H. O. Fattorini, ?Uniformly bounded cosine functions in Hilbert space,? Indiana Univ. Math. J.,20, 411-425 (1970). · doi:10.1512/iumj.1970.20.20035
[6] E. Giusti, ?Funzione coseno periodiche,? Boll. Un. Mat. Ital.,22, No. 4, 478-485 (1967). · Zbl 0182.19304
[7] Yu. I. Lyubich, ?Almost-periodic functions in the spectral analysis of operators,? Dokl. Akad. Nauk SSSR,132, No. 3, 518-520 (1960).
[8] Yu. I. Lyubich, ?On completeness conditions for a system of eigenvectors of a correct operator,? Usp. Mat. Nauk,18, No. 1 (109), 165-171 (1963). · Zbl 0126.32001
[9] G. Vainikko, Analysis of Discretization Methods [in Russian], Tartu (1976). · Zbl 0323.65026
[10] G. M. Vainikko and S. I. Piskarev, ?Regularly consistent operators,? Izv. Vyssh. Uchebn. Zaved., Mat., No. 10, 25-36 (1977). · Zbl 0388.47007
[11] A. A. Samarskii and A. V. Gulin, ?On certain results and problems in the stability theory of difference schemes,? Mat. Sb.,99, (141), No. 3, 299-330 (1976).
[12] H. O. Fattorini, ?Ordinary differential equations in linear topological space. II,? J. Diff. Equations,6, No. 1, 50-70 (1969). · Zbl 0181.42801 · doi:10.1016/0022-0396(69)90117-X
[13] A. V. Gulin, ?Stability criteria for certain non-self-adjoint three-layer difference schemes,? Differents. Uravn.,16, No. 7, 1205-1210 (1980).
[14] M. Sova, ?Cosine operator functions,? Rozprawy Mat.,49, 3-46 (1966).
[15] B. Nagy, ?On cosine operator functions in Banach spaces,? Acta Sci. Math.,36, Nos. 3-4, 281-289 (1974). · Zbl 0273.47008
[16] S. I. Piskarev, ?The discretization of an abstract hyperbolic equation,? Uch. Zap. Tartusk. Univ.,500, 3-23.
[17] S. I. Piskarev, ?On the compact convergence of the resolvent and semigroups,? in: Materials of the Conference ?Methods of Algebra and Functional Analysis for the Study of Families of Operators,? Tartu (1978), pp. 71-74.
[18] A. A. Samarskii and E. S. Nikolaev, Methods of Solving Finite-Difference Equations [in Russian], Nauka, Moscow (1978).
[19] E. Hille and R. S. Phillips, Functional Analysis and Semigroups, Amer. Math. Soc., Providence (1957). · Zbl 0078.10004
[20] J. A. Goldstein, C. Radon, and R. E. Showalter, ?Convergence rate of ergodic limits for semigroups and cosine functions,? Semigroup Forum,12, 189-206 (1976). · Zbl 0326.47043 · doi:10.1007/BF02195925
[21] B. Nagy, ?On cosine operator functions,? Publ. Math.,26, Nos. 3-4, 133-138 (1979). · Zbl 0446.47030
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