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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


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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