Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

BOUNDED AND PERIODIC SOLUTIONS OF INHOMOGENEOUS LINEAR EVOLUTION EQUATIONS

  • Duoc, Trinh Viet (Faculty of Mathematics, Mechanics, and Informatics VNU University of Science and Thang Long Institute of Mathematics and Applied Sciences)
  • Received : 2020.09.30
  • Accepted : 2021.08.23
  • Published : 2021.10.31
Download PDF
⟨ Previous Next ⟩

Abstract

The purpose of this paper is to prove unique existence of bounded solution and periodic solution to inhomogeneous linear evolution equations which trajectories of these solutions belong to given admissible Banach function space.

Keywords

Acknowledgement

The author would like to thank the anonymous referees for carefully reading the manuscript. Their comments and suggestions lead to the improvement of the paper.

References

  1. A. Beniani, N. Taouaf, and A. Benaissa, Well-posedness and exponential stability for coupled Lame system with viscoelastic term and strong damping, Comput. Math. Appl. 75 (2018), no. 12, 4397-4404. https://doi.org/10.1016/j.camwa.2018.03.037
  2. S. Boulaaras and N. Doudi, Global existence and exponential stability of coupled Lame system with distributed delay and source term without memory term, Bound. Value Probl. 2020 (2020), Paper No. 173, 21 pp. https://doi.org/10.1186/s13661-020-01471-9
  3. A. Choucha, S. Boulaaras, and D. Ouchenane, Exponential decay of solutions for a viscoelastic coupled Lame system with logarithmic source and distributed delay terms, Math. Methods Appl. Sci. 44 (2021), no. 6, 4858-4880. https://doi.org/10.1002/mma.7073
  4. A. Choucha, S. Boulaaras, D. Ouchenane, and S. Beloul, General decay of nonlinear viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping, logarithmic nonlinearity and distributed delay terms, Math. Methods Appl. Sci. 44 (2021), no. 7, 5436-5457. https://doi.org/10.1002/mma.7121
  5. Ju. L. Daleckii and M. G. Krein, Stability of solutions of differential equations in Banach space, translated from the Russian by S. Smith, Translations of Mathematical Monographs, Vol. 43, American Mathematical Society, Providence, RI, 1974.
  6. K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.
  7. N. T. Huy, Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line, J. Funct. Anal. 235 (2006), no. 1, 330-354. https://doi.org/10.1016/j.jfa.2005.11.002
  8. J. Kato, T. Naito, and J. S. Shin, Bounded solutions and periodic solutions to linear differential equations in Banach spaces, Vietnam J. Math. 30 (2002), suppl., 561-575.
  9. J. L. Massera and J. J. Schaffer, Linear Eifferential Equations and Function Spaces, Pure and Applied Mathematics, Vol. 21, Academic Press, New York, 1966.
  10. N. V. Minh, F. Rabiger, and R. Schnaubelt, Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line, Integral Equations Operator Theory 32 (1998), no. 3, 332-353. https://doi.org/10.1007/BF01203774
  11. R. Nagel and G. Nickel, Well-posedness for nonautonomous abstract Cauchy problems, in Evolution equations, semigroups and functional analysis (Milano, 2000), 279-293, Progr. Nonlinear Differential Equations Appl., 50, Birkhauser, Basel, 2002.
  12. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. https://doi.org/10.1007/978-1-4612-5561-1