×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: EuDML
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
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.