### A three critical points theorem and its applications to the ordinary Dirichlet problem

DOI: http://dx.doi.org/10.12775/TMNA.2003.029

#### Abstract

The aim of this paper is twofold. On one hand we establish a three

critical points theorem for functionals depending on a real parameter

$\lambda \in \Lambda$, which is different from the one proved by B. Ricceri

in [Arch. Math. {\bf 75} (2000), 220-226]

and gives an

estimate of where $\Lambda$ can be located. On the other hand, as an

application of the previous result, we prove an existence theorem of three

classical solutions for a two-point boundary value problem which is

independent from the one by J. Henderson and H. B. Thompson

[J. Differential Equations {\bf 166} (2000), 443-454].

Specifically, an example is given where the key assumption of

[J. Differential Equations {\bf 166} (2000), 443-454] fails. Nevertheless, the existence of three

solutions can still be deduced using our theorem.

critical points theorem for functionals depending on a real parameter

$\lambda \in \Lambda$, which is different from the one proved by B. Ricceri

in [Arch. Math. {\bf 75} (2000), 220-226]

and gives an

estimate of where $\Lambda$ can be located. On the other hand, as an

application of the previous result, we prove an existence theorem of three

classical solutions for a two-point boundary value problem which is

independent from the one by J. Henderson and H. B. Thompson

[J. Differential Equations {\bf 166} (2000), 443-454].

Specifically, an example is given where the key assumption of

[J. Differential Equations {\bf 166} (2000), 443-454] fails. Nevertheless, the existence of three

solutions can still be deduced using our theorem.

#### Keywords

Critical points; three solutions; two point boundary value problem

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