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Existence of periodic solutions
for nonlinear evolution equations
with pseudo monotone operators


Author: Naoki Shioji
Journal: Proc. Amer. Math. Soc. 125 (1997), 2921-2929
MSC (1991): Primary 47H05, 47H17, 47H30, 35B10, 35F25
DOI: https://doi.org/10.1090/S0002-9939-97-03984-1
MathSciNet review: 1402888
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we study the existence of $T$-periodic solutions for the problem

\begin{displaymath}u' (t) +A(t)u(t) =0, \quad t\in \mathbb R, \end{displaymath}

where $A(t)$ is a $T$-periodic, pseudo monotone mapping from a reflexive Banach space into its dual.


References [Enhancements On Off] (What's this?)

  • 1. H. Amann, Periodic solutions of semilinear parabolic equations, in Nonlinear analysis (A collection of papers in honor of Erich H. Rothe, Ed. L. Cesari, R. Kannan and H. F. Weinberger), 1-29, Academic Press, New York, 1978. MR 80a:35009
  • 2. V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Noordhoff, Leyden, 1976. MR 52:11666
  • 3. R. I. Becker, Periodic solutions of semilinear equations of evolution of compact type, J. Math. Anal. Appl. 82 (1981), 33-48. MR 84a:34068
  • 4. F. E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. Symp. Pure Math. 18 part 2, 1976. MR 53:8982
  • 5. J. P. Cascaval and I. I. Vrabie, Existence of periodic solutions for a class of nonlinear evolution equations, Rev. Mate. Univ. Complutense Madrid 7 (1994), 325-338. MR 95i:34113
  • 6. R. E. Edwards, Functional analysis, Holt, Rinehart and Winston, New York, 1965. MR 36:4308
  • 7. J. P. Gossez, Existence of periodic solutions for some first order ordinary differential equations, in Equadiff 78 (Convegno Internaz. Equaz. Diff. Ordin. ed Funzionali, R. Conti, G. Sestini and G. Villari ed.), Firenze, 1978, 361-379. MR 84m:34051
  • 8. N. Hirano, Abstract nonlinear Volterra equations with positive kernels, SIAM J. Math. Anal. 17 (1986), 403-414. MR 87d:45028
  • 9. N. Hirano, Nonlinear Volterra equations with positive kernels, in Nonlinear and convex analysis (Santa Barbara, Calif., 1985), Lecture Notes in Pure and Appl. Math., 107, 83-98, 1987. MR 88k:45004
  • 10. N. Hirano, Nonlinear evolution equations with nonmonotonic perturbations, Nonlinear Anal. 13 (1989), 599-609. MR 90e:34114
  • 11. N. Hirano, Existence of periodic solutions for nonlinear evolution equations in Hilbert space, Proc. Amer. Math. Soc. 120 (1994), 185-192. MR 94b:34087
  • 12. J. Mawhin, Topological degree methods in nonlinear boundary value problems, CBMS Regional Conf. Ser. in Math. 40, Amer. Math. Soc., Providence, 1979. MR 80c:47055
  • 13. J. Prüss, Periodic solutions of semilinear evolution equations, Nonlinear Anal. 3 (1979), 221-235. MR 81a:34061
  • 14. M. Renardy and R. C. Rogers, An introduction to partial differential equations, Springer-Verlag, New York, 1993. MR 94c:35001
  • 15. I. I. Vrabie, Periodic solutions for nonlinear evolution equations in a Banach space, Proc. Amer. Math. Soc. 109 (1990), 653-661. MR 90k:34080
  • 16. E. Zeidler, Nonlinear functional analysis and its applications II/A, Linear monotone operators, Springer-Verlag, New York, 1990. MR 91b:47001
  • 17. E. Zeidler, Nonlinear functional analysis and its applications II/B, Nonlinear monotone operators, Springer-Verlag, New York, 1990. MR 91b:47002

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Additional Information

Naoki Shioji
Affiliation: Faculty of Engineering, Tamagawa University, Tamagawa Gakuen, Machida, Tokyo 194, Japan
Email: shioji@eng.tamagawa.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-97-03984-1
Keywords: Evolution equations, pseudo monotone operators, periodic solutions
Received by editor(s): December 19, 1995
Received by editor(s) in revised form: April 30, 1996
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 American Mathematical Society

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