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A critical points theorem and nonlinear differential problems. (English) Zbl 1087.58007
Both D. Averna and G. Bonanno [Topol. Methods Nonlinear Anal. 22, 93–103 (2003; Zbl 1048.58005)], and G. Bonanno [Nonlinear Anal., Theory Methods Appl. 54, 651–665 (2003; Zbl 1031.49006)] established a theorem on the existence of 3 critical points of a functional of the type ${\Phi }-\lambda J$ for each $\lambda$ in a suitable interval. In this paper, conditions are given ensuring that both theorems hold and hence the functional ${\Phi }-\lambda J$ has 3 critical points for each $\lambda$ in the union of those intervals. This result is then applied to the Dirichlet boundary value problem ${u}^{\text{'}\text{'}}+\lambda f\left(u\right)=0$, $u\left(0\right)=u\left(1\right)=0$.

##### MSC:
 58E05 Abstract critical point theory 34B15 Nonlinear boundary value problems for ODE