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Theorems of Bohr-Neugebauer-type for almost-periodic differential equations. (English) Zbl 1068.34042
The paper deals with Bohr-Neugebauer-type theorems on the almost-periodicity of every bounded solution for linear nonhomogeneous equations and systems with constant coefficients. The right-hand side is assumed to be essentially bounded and almost-periodic in various metrics (in the sense of Bohr, Stepanov, Weyl, Besicovitch). Firstly, the properties of almost-periodicity in various metrics defined for measurable functions from $$\mathbb R$$ into $$\mathbb R^n$$ are investigated. The almost-periodicity results in the case of almost-periodic nonhomogenities in various metrics are proved on the basis of the integral representation of entirely bounded solutions.

##### MSC:
 34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations 34A30 Linear ordinary differential equations and systems
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##### References:
 [1] AMERIO L.-PROUSE G.: Almost Periodic Functions and Functional Equations. Van Nostrand-Reinhold, New York, 1971. · Zbl 0215.15701 [2] ANDRES J.: Almost periodic and bounded solutions of Carathéodory differential inclusions. Differential Integral Equations 12 (1999), 887-912. · Zbl 1017.34011 [3] ANDRES J.-BERSANI A. M.: Almost-periodicity problem as a fixed-point problem for evolution inclusions. Topol. Methods Nonlinear Anal. 18 (2001), 337-349. · Zbl 1013.34063 [4] ANDRES J.-BERSANI A. M.: Private communication. · Zbl 1238.68174 [5] ANDRES J.-BERSANI A. M.-GRANDE R. F.: Hierarchy of almost-periodic function spaces. Preprint, 2002. · Zbl 1133.42002 [6] ANDRES J.-BERSANI A. M-LEŚNIAK K.: On some almost-periodicity problems in various metrics. Acta Appl. Math. 65 (2001), 35-57. · Zbl 0997.34032 [7] BESICOVITCH A. S.: Almost Periodic Functions. Cambridge Univ. Press, Cambridge, 1932. · Zbl 0004.25303 [8] BOLES BASIT R.-TSEND L.: The generalized Bohr-Neugebauer theorem. Differ. Equ. 8 (1974), 1031-1035. · Zbl 0299.34081 [9] BOHR H.-NEUGEBAUER O.: Über lineare Differentialgleichungen mit konstanten Koeffizienten und fastperiodischer rechter Seite. Nachrichten Gottingen 1926 (1926), 8-22. · JFM 52.0464.01 [10] CORDUNEANU C.: Almost Periodic Functions. Interscience Publ., New York, 1968. · Zbl 0175.09101 [11] CORDUNEANU C.: Two qualitative inequalities. J. Differential Equations 64 (1986), 16-25. · Zbl 0598.34007 [12] DALETSKII, YU. L.-KREIN M. G.: Stability of Solution of Differential Equations in Banach Spaces. Nauka, Moscow, 1970. [13] DEMIDOVITCH B. P.: Lectures on the Mathematical Stability Theory. Nauka, Moscow, 1967. [14] FAVARD J.: Lecons sur les Fonctions Presque Periodiques. Gauthier-Villars, Paris, 1933. · Zbl 0007.34303 [15] FINK A. M.: Almost Periodic Differential Equations. Springer-Verlag, New York, 1974. · Zbl 0325.34039 [16] KRASNOSELSKII M. A.-BURD V. SH.-KOLESOV, YU. S.: Nonlinear Almost Periodic Oscillations. John Wiley, New York, 1971. [17] LEVITAN B. M.: Almost Periodic Functions. Gos. Izd. Tekh.-Teor. Lit., Moskow, 1953. · Zbl 1222.42002 [18] LEVITAN B. M.-ZHIKOV V. V.: Almost Periodic Functions and Differential Equations. Cambridge Univ. Press, London, 1982. · Zbl 0499.43005 [19] MASSERA J. L.-SCHAEFFER J. J.: Linear Differential Equations and Function Spaces. Academic Press, London, 1982. [20] N’GUEREKATA G. M.: Almost Automorphic and Almost Periodic Functions in Abstract Spaces. Kluwer Acad. Publ., New York, 2001. · Zbl 1001.43001 [21] STOINSKI S.: A connection between H-almost periodic functions and almost periodic functions of other types. Funct. Approx. Comment. Math. 3 (1976), 205-223. · Zbl 0344.42005 [22] YOSHIZAWA T.: Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions. Springer-Verlag, Berlin, 1975. · Zbl 0304.34051 [23] ZAIDMAN S.: Abstract Differential Equations. Pitman Publ. Ltd., San Francisco-London-Melbourne, 1979. · Zbl 0465.34002
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