대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제38권4호
  • /
  • Pages.1153-1162
  • /
  • 2023
  • /
  • 1225-1763(pISSN)
  • /
  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

CIRCULAR SPECTRUM AND ASYMPTOTIC PERIODIC SOLUTIONS TO A CLASS OF NON-DENSELY DEFINED EVOLUTION EQUATIONS

  • Le Anh Minh (Department of Mathematical Analysis Hong Duc University) ;
  • Nguyen Ngoc Vien (Faculty of Foundations Hai Duong University)
  • 투고 : 2023.01.29
  • 심사 : 2023.04.26
  • 발행 : 2023.10.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, for the bounded solution of the non-densely defined non-autonomous evolution equation, we present the condition for asymptotic periodicity by using the circular spectral theory of functions on the half line and the extrapolation theory of non-densely defined evolution equation.

키워드

과제정보

The authors would like to thank the anonymous referees who provided useful and detailed comments on the earlier versions of the manuscript.

참고문헌

  1. T. A. Burton, Stability and periodic solutions of ordinary and functional-differential equations, Mathematics in Science and Engineering, 178, Academic Press, Inc., Orlando, FL, 1985.
  2. C. Cuevas and M. Pinto, Existence and uniqueness of pseudo almost periodic solutions of semilinear Cauchy problems with non dense domain, Nonlinear Anal. 45 (2001), no. 1, Ser. A: Theory Methods, 73-83. https://doi.org/10.1016/S0362-546X(99)00330-2
  3. G. Da Prato and E. Sinestrari, Differential operators with nondense domain, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), no. 2, 285-344 (1988).
  4. B. de Andrade and C. Cuevas, S-asymptotically ω-periodic and asymptotically ω-periodic solutions to semi-linear Cauchy problems with non-dense domain, Nonlinear Anal. 72 (2010), no. 6, 3190-3208. https://doi.org/10.1016/j.na.2009.12.016
  5. K. Ezzinbi and M. Jazar, New criteria for the existence of periodic and almost periodic solutions for some evolution equations in Banach spaces, Electron. J. Qual. Theory Differ. Equ. 2004 (2004) , No. 6, 12 pp. https://doi.org/10.14232/ejqtde.2004.1.6
  6. K. Ezzinbi and J. H. Liu, Periodic solutions of non-densely defined delay evolution equations, J. Appl. Math. Stoch. Anal. 15 (2002), no. 2, 113-123. https://doi.org/10.1155/S1048953302000114
  7. G. Guhring and F. Rabiger, Asymptotic properties of mild solutions of nonautonomous evolution equations with applications to retarded differential equations, Abstr. Appl. Anal. 4 (1999), no. 3, 169-194. https://doi.org/10.1155/S1085337599000214
  8. H. Henriquez, M. Pierri, and P. Taboas, On S-asymptotically ω-periodic functions on Banach spaces and applications, J. Math. Anal. Appl. 343 (2008), no. 2, 1119-1130. https://doi.org/10.1016/j.jmaa.2008.02.023
  9. D. B. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, 840, Springer, Berlin, 1981.
  10. N. T. Huy and Ngo Quy Dang, Existence, uniqueness and conditional stability of periodic solutions to evolution equations, J. Math. Anal. Appl. 433 (2016), no. 2, 1190-1203. https://doi.org/10.1016/j.jmaa.2015.07.059
  11. J. H. Liu, G. M. N'Guerekata, and N. V. Minh, Topics on stability and periodicity in abstract differential equations, Series on Concrete and Applicable Mathematics, 6, World Sci. Publ., Hackensack, NJ, 2008. https://doi.org/10.1142/9789812818249
  12. V. T. Luong, D. V. Loi, N. V. Minh, and H. Matsunaga, A Massera theorem for asymptotic periodic solutions of periodic evolution equations, J. Differential Equations 329 (2022), 371-394. https://doi.org/10.1016/j.jde.2022.05.010
  13. V. T. Luong, N. H. Tri, and N. V. Minh, Asymptotic behavior of solutions of periodic linear partial functional differential equations on the half line, J. Differential Equations 296 (2021), 1-14. https://doi.org/10.1016/j.jde.2021.05.053
  14. N. V. Minh, H. Matsunaga, N. D. Huy, and V. T. Luong, A Katznelson-Tzafriri type theorem for difference equations and applications, Proc. Amer. Math. Soc. 150 (2022), no. 3, 1105-1114. https://doi.org/10.1090/proc/15722
  15. N. V. Minh, G. M. N'Guerekata, and S. Siegmund, Circular spectrum and bounded solutions of periodic evolution equations, J. Differential Equations 246 (2009), no. 8, 3089-3108. https://doi.org/10.1016/j.jde.2009.02.014
  16. J. W. Pruss, Periodic solutions of the thermostat problem, in Differential equations in Banach spaces (Bologna, 1985), 216-226, Lecture Notes in Math., 1223, Springer, Berlin, 1986. https://doi.org/10.1007/BFb0099195
  17. W. M. Ruess and V. Q. Ph'ong, Asymptotically almost periodic solutions of evolution equations in Banach spaces, J. Differential Equations 122 (1995), no. 2, 282-301. https://doi.org/10.1006/jdeq.1995.1149
  18. J. B. Serrin Jr., A note on the existence of periodic solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 3 (1959), 120-122. https://doi.org/10.1007/BF00284169
  19. C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc. 200 (1974), 395-418. https://doi.org/10.2307/1997265
  20. W. Zhang, D. Zhu, and P. Bi, Existence of periodic solutions of a scalar functional differential equation via a fixed point theorem, Math. Comput. Modelling 46 (2007), no. 5-6, 718-729. https://doi.org/10.1016/j.mcm.2006.12.026