Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Periodic solutions for nonlinear evolution equations in a Banach space

Author: Ioan I. Vrabie
Journal: Proc. Amer. Math. Soc. 109 (1990), 653-661
MSC: Primary 34G20; Secondary 34C25, 47H15, 58D25
MathSciNet review: 1015686
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove an existence result for $ T$-periodic mild solutions to nonlinear evolution equations of the form

$\displaystyle u'(t) + Au(t) \mathrel\backepsilon F(t,u(t)),\quad t \in {R_ + }.$

Here $ (X,\vert\vert \cdot \vert\vert)$ is a real Banach space, $ A:D(A) \subset X \to {2^X}$ is an operator with $ A - aI$ $ m$-accretive for some $ a > 0$ and such that $ - A$. generates a compact semigroup, while $ F:{R_ + } \times \overline {D(A)} \to X$ is a Carathéodory mapping which is $ T$-periodic with respect to its first argument and satisfies

$\displaystyle \mathop {\lim }\limits_{r \to + \infty } \tfrac{1}{r}\sup \left\{... {R_ + },v \in \overline {D(A)} ,\vert\vert v\vert\vert \leq r} \right\} < a.$

. As a consequence, we obtain an existence theorem for $ T$-periodic solutions to the porous medium equation.

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

  • [1] P. Baras, Compacité de l'opérateur $ f \mapsto u$ solution d'une équation non-linéaire $ (du/dt) + Au \mathrel\backepsilon f$, C. R. Acad. Sci. Paris 286 (1978), 1113-1116. MR 0493554 (58:12548)
  • [2] V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei Bucureşti, Noordhoff, 1976. MR 0390843 (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 626739 (84a:34068)
  • [4] T. D. Benavides, Generic existence of periodic solutions of differential equations in Banach spaces, Bull. Pol. Acad. Sci. (Math.) 33 (1985), 129-135. MR 805026 (87a:34072)
  • [5] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans un espace de Hilbert, North-Holland, 1973.
  • [6] F. E. Browder, Existence of periodic solutions for nonlinear equations of evolution, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1100-1103. MR 0177295 (31:1558)
  • [7] M. G. Crandall, Nonlinear semigroups and evolution governed by accretive operators, Proc. Sympos. Pure Math., vol. 45, Part 1, Amer. Math. Soc., Providence, RI, 1986, pp. 305-337. MR 843569 (87h:47140)
  • [8] K. Deimling, Periodic solutions of differential equations in Banach spaces, Manuscripta Math. 24 (1978), 31-44. MR 0499551 (58:17384)
  • [9] J. H. Lightbourne III, Periodic solutions for a differential equation in Banach space, Trans. Amer. Math. Soc. 238 (1978), 285-299. MR 0481337 (58:1456)
  • [10] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non-linéaires, Gauthier-Villars, Paris, 1969. MR 0259693 (41:4326)
  • [11] N. H. Pavel, Nonlinear evolution operators and semigroups. Applications to partial differential equations, Lecture Notes in Math., vol. 1260, Springer-Verlag, Berlin and New York, 1987. MR 900380 (88j:47087)
  • [12] J. Prüss, Periodic solutions of semilinear evolution equations, Nonlinear Anal. T.M.A. 3 (1979), 601-612.
  • [13] I. I. Vrabie, The nonlinear version of Pazy's local existence theorem, Israel J. Math. 32 (1979), 221-235. MR 531265 (82a:47064)
  • [14] -, Compactness methods and flow-invariance for perturbed nonlinear semigroups, An. Stiint. Univ. "A1. I. Cuza" Iaşi Secţ. I a Mat. 27 (1981), 119-125. MR 618716 (82i:47101)
  • [15] -, Compactness methods for nonlinear evolutions, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 32, Longman Scientific & Technical, 1987. MR 932730 (90f:47101)
  • [16] -, Periodic solutions of nonlinear evolution equations in a Hilbert space, in preparation.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 34G20, 34C25, 47H15, 58D25

Retrieve articles in all journals with MSC: 34G20, 34C25, 47H15, 58D25

Additional Information

PII: S 0002-9939(1990)1015686-4
Keywords: Accretive operator, compact semigroups, periodic solution, porous medium equation
Article copyright: © Copyright 1990 American Mathematical Society

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia