Abstract

The aim of this paper is to establish the following result: THEOREM 1. - Let XX be a finite-dimensional real Hilbert space, and let J:X→RJ:X\to {\bf R} be a C1C^1 function such that lim inf⁡∥x∥→+∞J(x)∥x∥2≥0.\liminf_{\|x\|\to +\infty}{{J(x)}\over {\|x\|^2}}\geq 0. Moreover, let x0∈Xx_0\in X and r,s∈Rr, s\in {\bf R}, with 0<r<s0<r<s, be such that inf⁡x∈XJ(x)<inf⁡∥x−x0∥≤sJ(x)≤J(x0)≤inf⁡r≤∥x−x0∥≤sJ(x).\inf_{x\in X}J(x)<\inf_{\|x-x_0\|\leq s}J(x)\leq J(x_0)\leq \inf_{r\leq\|x-x_0\|\leq s}J(x). Then, there exists λ^>0\hat\lambda> 0 such that the equation x+λ^J′(x)=x0x+\hat\lambda J'(x)=x_0 has at least three solutions. We will proceed as follows. We first give the proof of Theorem 1. Then, we discuss in detail the finite-dimensionality assumption on XX. 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 x0x_0 a local minimum of JJ) 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

Author information

Contact details are reproduced from the original publication and may be historical.

Biagio Ricceri

Department of Mathematics, University of Catania, Viale A. Doria 6, 95125 Catania, Italy

ricceri@dmi.unict.it

References

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Suggested citation

B. Ricceri. “A Multiplicity Theorem in 𝐑^n.” Journal of Convex Analysis 16 (2009), No. 3&4, 987–992. https://doi.org/10.68381/jca16060

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Published by Heldermann Verlag, 2009. Rights now held by Banach Press LLC.