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

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

 

A general multiplicity theorem for certain nonlinear equations in Hilbert spaces


Author: Biagio Ricceri
Journal: Proc. Amer. Math. Soc. 133 (2005), 3255-3261
MSC (2000): Primary 47H50, 47J10, 47J30; Secondary 41A52, 41A65
Published electronically: June 20, 2005
MathSciNet review: 2161147
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we prove the following general result. Let $X$ be a real Hilbert space and $J:X\to {\bf R}$ a continuously Gâteaux differentiable, nonconstant functional, with compact derivative, such that

\begin{displaymath}\limsup_{\Vert x\Vert\to +\infty}{{J(x)}\over {\Vert x\Vert^2}}\leq 0 .\end{displaymath}

Then, for each $r\in ]\inf_{X}J,\sup_{X}J[$ for which the set $J^{-1}([r,+\infty[)$ is not convex and for each convex set $S\subseteq X$ dense in $X$, there exist $x_0\in S\cap J^{-1}(]-\infty,r[)$ and $\lambda>0$ such that the equation

\begin{displaymath}x=\lambda J'(x)+x_0\end{displaymath}

has at least three solutions.


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

  • 1. N. V. Efimov and S. B. Stechkin, Approximate compactness and Chebyshev sets, Soviet Math. Dokl., 2 (1961), 1226-1228.
  • 2. Patrizia Pucci and James Serrin, A mountain pass theorem, J. Differential Equations 60 (1985), no. 1, 142–149. MR 808262 (86m:58038), http://dx.doi.org/10.1016/0022-0396(85)90125-1
  • 3. Paul H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. MR 845785 (87j:58024)
  • 4. Biagio Ricceri, A further improvement of a minimax theorem of Borenshtein and Shul′man, J. Nonlinear Convex Anal. 2 (2001), no. 2, 279–283. Special issue for Professor Ky Fan. MR 1848707 (2002e:49011)
  • 5. I. G. Tsar'kov, Nonunique solvability of certain differential equations and their connection with geometric approximation theory, Math. Notes, 75 (2004), 259-271.
  • 6. C. Zălinescu, Convex analysis in general vector spaces, World Scientific Publishing Co., Inc., River Edge, NJ, 2002. MR 1921556 (2003k:49003)
  • 7. Eberhard Zeidler, Nonlinear functional analysis and its applications. III, Springer-Verlag, New York, 1985. Variational methods and optimization; Translated from the German by Leo F. Boron. MR 768749 (90b:49005)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47H50, 47J10, 47J30, 41A52, 41A65

Retrieve articles in all journals with MSC (2000): 47H50, 47J10, 47J30, 41A52, 41A65


Additional Information

Biagio Ricceri
Affiliation: Department of Mathematics, University of Catania, Viale A. Doria 6, 95125 Catania, Italy
Email: ricceri@dmi.unict.it

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08218-3
PII: S 0002-9939(05)08218-3
Keywords: Nonlinear equations, Hilbert spaces, local and global minima, critical points, level sets, minimax theory, Chebyshev sets
Received by editor(s): May 24, 2004
Published electronically: June 20, 2005
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.