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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
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.

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Additional Information

Biagio Ricceri
Affiliation: Department of Mathematics, University of Catania, Viale A. Doria 6, 95125 Catania, Italy

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.