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In this paper, we prove a new quantitative deformation lemma, and then gain a new mountain pass theorem in Hilbert spaces. By using the new mountain pass theorem, we obtain the new existence of two nontrivial periodic solutions for a class of nonlinear second-order discrete systems, which greatly improves the result in (Zhou et al. in Proc. R. Soc. Edinb. A 134:1013–1022, 2004).

On mountain pass theorem and its application to periodic solutions of some nonlinear discrete systems