International Journal of Differential Equations

On the Almost Periodic Solutions of Differential Equations on Hilbert Spaces

Nguyen Thanh Lan

Full-text: Open access

Abstract

For the differential equation u ( t ) = A u ( t ) + f ( t ) , t 0 on a Hilbert space H , we find the necessary and sufficient conditions that the above-mentioned equation has a unique almost periodic solution. Some applications are also given.

Article information

Source
Int. J. Differ. Equ., Volume 2009 (2009), Article ID 575939, 11 pages.

Dates
Received: 18 January 2009
Accepted: 13 April 2009
First available in Project Euclid: 26 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1485399814

Digital Object Identifier
doi:10.1155/2009/575939

Mathematical Reviews number (MathSciNet)
MR2525716

Zentralblatt MATH identifier
1207.34054

Citation

Lan, Nguyen Thanh. On the Almost Periodic Solutions of Differential Equations on Hilbert Spaces. Int. J. Differ. Equ. 2009 (2009), Article ID 575939, 11 pages. doi:10.1155/2009/575939. https://projecteuclid.org/euclid.ijde/1485399814


Export citation

References

  • W. Arendt and C. J. K. Batty, “Almost periodic solutions of first- and second-order Cauchy problems,” Journal of Differential Equations, vol. 137, no. 2, pp. 363–383, 1997.MR1456602
  • B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, Cambridge, UK, 1982.MR690064
  • S. Murakami, T. Naito, and N. Van Minh, “Evolution semigroups and sums of commuting operators: a new approach to the admissibility theory of function spaces,” Journal of Differential Equations, vol. 164, no. 2, pp. 240–285, 2000.MR1765556
  • V. Q. Phong, “A new proof and generalizations of Gearhart's theorem,” Proceedings of the American Mathematical Society, vol. 135, no. 7, pp. 2065–2072, 2007.MR2299482
  • W. M. Ruess and V. Q. Phong, “Asymptotically almost periodic solutions of evolution equations in Banach spaces,” Journal of Differential Equations, vol. 122, no. 2, pp. 282–301, 1995.MR1355893
  • H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, vol. 18 of North-Holland Mathematical Library, North-Holland, Amsterdam, The Netherlands, 1978.MR503903
  • W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, vol. 96 of Monographs in Mathematics, Birkhäuser, Basel, Switzerland, 2001.MR1886588
  • J. Prüss, “On the spectrum of ${C}_{0}$-semigroups,” Transactions of the American Mathematical Society, vol. 284, no. 2, pp. 847–857, 1984.MR743749
  • R. Nagel and E. Sinestrari, “Inhomogeneous Volterra integrodifferential equations for Hille-Yosida operators,” in Functional Analysis (Essen, 1991), K. D. Bierstedt, A. Pietsch, W. M. Ruess, and D. Vogt, Eds., vol. 150 of Lecture Notes in Pure and Applied Mathematics, pp. 51–70, Marcel Dekker, New York, NY, USA, 1994.MR1241671