Advertisement

Asymptotic Behavior of Linear Almost Periodic Differential Equations

  • Bui Xuan Dieu
  • Luu Hoang Duc
  • Stefan Siegmund
  • Nguyen Van Minh
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 157)

Abstract

The present paper is concerned with strong stability of solutions of non-autonomous equations of the form \(\dot{u}(t) = A(t)u(t)\), where A(t) is an unbounded operator in a Banach space depending almost periodically on t. A general condition on strong stability is given in terms of Perron conditions on the solvability of the associated inhomogeneous equation.

Keywords

Strong stability Non-autonomous equation Almost periodicity Evolution semigroup Perron type conditions 

Notes

Acknowledgements

The first author was supported by the Vietnamese Ministry of Education and Training (MOET) Scholarship Scheme (Project 322) and the Graduate Academy (GA) of the TU Dresden (PSPElement: F-00361-553-52A-2330000) in accordance with the funding regulations of the German Research Foundation (DFG). The second author was supported by DFG under grant number Si801/6-1 and NAFOSTED under grant number 101.02-2011.47.

References

  1. 1.
    Arendt, W., Batty, C.J.K.: Asymptotically almost periodic solutions of inhomogeneous Cauchy problems on the half-line. Bull. Lond. Math. Soc. 31, 291–304 (1999)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Arendt, W., Rabiger, F., Sourour, A.: Spectral properties of the operator equation AX + XB = Y. Q. J. Math. Oxford Ser. (2) 45 (178), 133–149 (1994)Google Scholar
  3. 3.
    Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F.: Vector-Valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics, vol. 96. Birkhäuser Verlag, Basel (2001)Google Scholar
  4. 4.
    Ballotti, M.E., Goldstein, J.A., Parrott, M.E.: Almost periodic solutions of evolution equations. J. Math. Anal. Appl. 138, 522–536 (1989)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Basit, B.: Harmonic analysis and asymptotic behavior of solutions to the abstract Cauchy problem. Semigroup Forum 54, 58–74 (1997)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Batty, C.J.K., Van Neerven, J., Räbiger, F.: Local spectra and individual stability of uniform bounded C 0-semigroups. Trans. Am. Math. Soc. 350, 2071–2085 (1998)CrossRefMATHGoogle Scholar
  7. 7.
    Boumenir, A., Van Minh, N., Kim Tuan, V.: Frequency modules and nonexistence of quasi-periodic solutions of nonlinear evolution equations. Semigroup Forum 76, 58–70 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chicone, C., Latushkin, Y.: Evolution Semigroups in Dynamical Systems and Differential Equations. Mathematical Surveys and Monographs, vol. 70. American Mathematical Society, Providence, RI (1999)Google Scholar
  9. 9.
    Chill, R., Tomilov, Y.: Stability of operators semigroups: ideas and results. In: Perspectives in Operator Theory, vol. 75, pp. 71–109. Banach Center Publications, Polish Academy Science, Warszawa (2007)Google Scholar
  10. 10.
    Ellis, R., Johnson, R.A.: Topological dynamics and linear differential systems. J. Differ. Equ. 44, 21–39 (1982)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Fink, A.M.: Almost Periodic Differential Equations. Springer, Berlin/Heidelberg/New York (1974)CrossRefMATHGoogle Scholar
  12. 12.
    Hino, Y., Naito, T., Van Minh, N., Shin, J.S.: Almost Periodic Solutions of Differential Equations in Banach Spaces. Taylor & Francis, London/New York (2002)MATHGoogle Scholar
  13. 13.
    Johnson, R.A., Sell, G.R.: Smoothness of spectral subbundles and reducibility of quasiperiodic linear differential systems. J. Differ. Equ. 41, 262–288 (1981)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Levitan, B.M., Zhikov, V.V.: Almost Periodic Functions and Differential Equations. Moscow University Publishing House, Moscow (1978). English translation by Cambridge University Press, Cambridge, UK (1982)Google Scholar
  15. 15.
    Murakami, S., Naito, T., Van Minh, N.: Evolution semigroups and sums of commuting operators: a new approach to the admissibility theory of function spaces. J. Differ. Equ. 164, 240–285 (2000)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Naito, T., Van Minh, N.: Evolutions semigroups and spectral criteria for almost periodic solutions of periodic evolution equations. J. Differ. Equ. 152, 358–376 (1999)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)CrossRefMATHGoogle Scholar
  18. 18.
    Preda, C.: \((L^{p}(\mathbb{R}_{+},X),L^{q}(\mathbb{R}_{+},X))\)-admissibility and exponential dichotomies of cocycles. J. Differ. Equ. 249, 578–598 (2010)Google Scholar
  19. 19.
    Preda, P., Pogan, A., Preda, C.: Schäffer spaces and exponential dichotomy for evolutionary processes. J. Differ. Equ. 230, 378–391 (2006)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Preda, C., Preda, P., Craciunescu, A.: Criterions for detecting the existence of the exponential dichotomies in the asymptotic behavior of the solutions of variational equations. J. Funct. Anal. 258, 729–757 (2010)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Quoc Phong, V.: Stability and almost periodicity of trajectories of periodic processes. J. Differ. Equ. 115, 402–415 (1995)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Sacker, R.J., Sell, G.R.: A spectral theory for linear differential systems. J. Differ. Equ. 27, 320–358 (1978)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Thieu Huy, N.: Exponentially dichotomous operators and exponential dichotomy of evolution equations on the half-line. Int. Equ. Oper. Theory 48, 497–510 (2004)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Thieu Huy, N.: Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line. J. Funct. Anal. 235, 330–354 (2006)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Van Minh, N.: Asymptotic behavior of individual orbits of discrete systems. Proc. Am. Math. Soc. 137, 3025–3035 (2009)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Van Minh, N.: A spectral theory of continuous functions and the Loomis-Arendt-Batty-Vu theory on the asymptotic behavior of solutions of evolution equations. J. Differ. Equ. 247, 1249–1274 (2009)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Van Minh, N., Räbiger, F., Schnaubelt, R.: On the exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half line. Int. Equ. Oper. Theory 32, 332–353 (1998)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    van Neerven, J.M.A.M.: The asymptotic behaviour of semigroups of linear operator. In: Operator Theory, Advances and Applications, vol. 88. Birkhäuser Verlag, Basel /Boston/Berlin (1996)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Bui Xuan Dieu
    • 1
    • 2
  • Luu Hoang Duc
    • 1
    • 3
  • Stefan Siegmund
    • 1
  • Nguyen Van Minh
    • 4
    • 5
  1. 1.Department of MathematicsDresden University of TechnologyDresdenGermany
  2. 2.School of Applied Mathematics & Informatics, Hanoi University of Science and TechnologyHa NoiViet Nam
  3. 3.Institute of MathematicsVietnam Academy of Science and TechnologyHa NoiViet Nam
  4. 4.Department of MathematicsColumbus State UniversityColumbusUSA
  5. 5.Department of Mathematics & StatisticsUniversity of Arkansas at Little RockLittle RockUSA

Personalised recommendations