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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
DOI: https://doi.org/10.1090/S0002-9939-05-08218-3
Published electronically: June 20, 2005
MathSciNet review: 2161147
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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. P. Pucci and J. Serrin, A mountain pass theorem, J. Differential Equations, 60 (1985), 142-149. MR 0808262 (86m:58038)
  • 3. P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf. Ser. in Math., 65, Amer. Math. Soc., Providence, 1986. MR 0845785 (87j:58024)
  • 4. B. Ricceri, A further improvement of a minimax theorem of Borenshtein and Shul'man, J. Nonlinear Convex Anal., 2 (2001), 279-283. 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. Zalinescu, Convex analysis in general vector spaces, World Scientific, 2002. MR 1921556 (2003k:49003)
  • 7. E. Zeidler, Nonlinear functional analysis and its applications, vol. III, Springer-Verlag, 1985. MR 0768749 (90b:49005)

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Additional Information

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

DOI: https://doi.org/10.1090/S0002-9939-05-08218-3
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.

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