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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 Φ-λJ for each λ in a suitable interval. In this paper, conditions are given ensuring that both theorems hold and hence the functional Φ-λJ has 3 critical points for each λ in the union of those intervals. This result is then applied to the Dirichlet boundary value problem u '' +λf(u)=0, u(0)=u(1)=0.

58E05Abstract critical point theory
34B15Nonlinear boundary value problems for ODE