In this paper, we consider the second-order Hamiltonian system $$ \ddot{x}+V^{\prime}(x)=0,\quad x\in \mathbb{R}^{N}. $$ We use the monotonicity assumption introduced by Bartsch and Mederski (Arch. Ration. Mech. Anal. 215:283–306, 2015). When V is even, we can release the strict convexity hypothesis, which is used by Bartsch and Mederski combined with the monotonicity assumption. When V is noneven, we weaken the strict convexity assumption and introduce another hypothesis (see (V10)). Then in both cases, we can build the homomorphism between the Nehari manifold and the unit sphere of some suitable space. Using the Nehari manifold method introduced by Szulkin (J. Funct. Anal. 257:3802–3822 2009), we prove the existence of T-periodic solutions with minimal period T.

中文翻译：

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二阶偶数和非偶数哈密顿系统的周期解

在本文中，我们考虑二阶哈密顿系统$$ \ddot{x}+V^{\prime}(x)=0,\quad x\in \mathbb{R}^{N}。$$ 我们使用 Bartsch 和 Mederski 引入的单调性假设 (Arch. Ration. Mech. Anal. 215:283–306, 2015)。当V为偶数时，我们可以释放严格的凸性假设，这是Bartsch和Mederski结合单调性假设使用的。当V非偶数时，我们削弱严格的凸性假设并引入另一个假设（参见（V10））。那么在这两种情况下，我们都可以建立 Nehari 流形和某个合适空间的单位球面之间的同态。使用 Szulkin 引入的 Nehari 流形方法 (J. Funct. Anal. 257:3802–3822 2009)，我们证明了具有最小周期 T 的 T 周期解的存在性。