Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A Sturm-Liouville theorem for some odd multivalued maps


Authors: Jo ao-Paulo Dias and Jesús Hernández
Journal: Proc. Amer. Math. Soc. 53 (1975), 72-74
MSC: Primary 47H99; Secondary 47A99
DOI: https://doi.org/10.1090/S0002-9939-1975-0377632-8
MathSciNet review: 0377632
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ T:H \to {2^H}$ be the subdifferential of a real l. s. c. convex function on an infinite dimensional, separable, real Hilbert space $ H$. Assuming that $ T$ is odd (i.e. $ T( - u) = - Tu,\;\forall u\;\epsilon H)$), $ 0\epsilon T(0),\;{(I + T)^{ - 1}}$ is compact and $ T(0)$ satisfies a geometrical condition, we prove that $ T$ has an infinite sequence $ \{ {\lambda _n}\} $ of eigenvalues such that $ 0 \leqslant {\lambda _{n\,\overrightarrow n }} + \infty $.


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

  • [1] H. Amann, Lusternik-Schnirelman theory and nonlinear eigenvalue problems, Math. Ann. 199 (1972), 55-72. MR 0350536 (50:3028)
  • [2] H. Brézis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, Contributions to Nonlinear Functional Analysis (E. H. Zarantonello, ed.), Academic Press, New York, 1971, pp. 101-156. MR 0394323 (52:15126)
  • [3] -, Opérateurs maximaux monotones et semi-groups de contractions dans les espaces de Hilbert, North-Holland, Amsterdam, 1973.
  • [4] J. P. Dias, Variational inequalities and eigenvalue problems for nonlinear maximal monotone operators in a Hilbert space, Amer. J. Math. (to appear). MR 0420354 (54:8368)
  • [5] -, Un théorème de Sturm-Liouville pour une classe d'opérateurs non linéaires maximaux monotones, J. Math. Anal. Appl. 47 (1974), 400-405. MR 0367737 (51:3979)
  • [6] M. A. Krasnosel'skii, Topological methods in the theory of nonlinear integral equations, GITTL, Moscow, 1956; English transl., Macmillan, New York, 1964. MR 20 #3464; 28 #2414. MR 0159197 (28:2414)
  • [7] P. H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math. 3 (1973), 161-202. MR 47 #9383. MR 0320850 (47:9383)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47H99, 47A99

Retrieve articles in all journals with MSC: 47H99, 47A99


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0377632-8
Keywords: Subdifferential, maximal monotone operator, genus of a closed symmetric subset, Sobolev spaces
Article copyright: © Copyright 1975 American Mathematical Society

American Mathematical Society