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By applying the combination of discrete variational method and approximation, developed in a previous study [J. Kuang, W. Chen, and Z. Guo,  Periodic solutions with prescribed minimal period for second-order even Hamiltonian systems , Commun. Pure Appl. Anal.  21  (2022), no. 1, 47–59], we overcome some difficulties in the absence of Ambrosetti-Rabinowitz condition and obtain new sufficient conditions for the existence of periodic solutions with prescribed minimal period for second-order Hamiltonian systems with asymptotically linear nonlinearities.

Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities