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Journal of Convex Analysis 16 (2009), No. 3, 987--992
Copyright Heldermann Verlag 2009
A Multiplicity Theorem in Rn
Biagio Ricceri
Department of Mathematics, University of Catania, Viale A. Doria 6, 95125 Catania, Italy
ricceri@dmi.unict.it
[Abstract-pdf]
The aim of this paper is to establish the following result:\par
\medskip
THEOREM 1. - {\it Let $X$ be a finite-dimensional real Hilbert space,
and let $J:X\to {\bf R}$ be a $C^1$ function such that
$$\liminf_{\|x\|\to +\infty}{{J(x)}\over {\|x\|^2}}\geq 0\ .$$
Moreover, let $x_0\in X$ and $r, s\in {\bf R}$, with
$0 0$ such that the equation
$$x+\hat\lambda J'(x)=x_0$$
has at least three solutions.}\par
\medskip
We will proceed as follows. We first give the proof of Theorem 1.
Then, we discuss in detail the finite-dimensionality assumption on
$X$. More precisely, we will show not only that it can not be dropped,
but also that it is very hard to imagine some additional condition
(different from being $x_0$ a local minimum of $J$) under which one
could adapt the given proof to the infinite-dimensional case. We
finally conclude presenting an application of Theorem 1 to a discrete
boundary value problem.
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