We consider a second-order differential equation governed by a quasiconvex function with a nonempty set of minimizers. Assuming that the gradient of this function is Lipschitz continuous, the existence of solutions to the gradient system is guaranteed. We study the asymptotic behavior of these solutions in continuous and discrete times. More precisely, we show that, if a solution is bounded, then it converges weakly to a critical point of the function; otherwise, it goes to infinity (in norm). We also provide several sufficient conditions for obtaining strong convergence in both continuous and discrete cases. Our work is motivated by an open problem proposed by Khatibzadeh and Moroşanu (J Convex Anal 26:1175–1186, 2019), and we solve this problem in the case, where the gradient of the function is Lipschitz continuous on bounded sets.

中文翻译：

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由拟凸函数控制的梯度系统的长期行为

我们考虑由具有非空极小值集的拟凸函数控制的二阶微分方程。假设该函数的梯度是 Lipschitz 连续的，则保证梯度系统的解存在。我们研究了这些解在连续和离散时间内的渐近行为。更准确地说，我们证明，如果一个解是有界的，那么它会弱收敛到函数的一个临界点；否则，它将趋于无穷大（在规范中）。我们还提供了在连续和离散情况下获得强收敛的几个充分条件。我们的工作受到 Khatibzadeh 和 Moroşanu (J Convex Anal 26:1175–1186, 2019) 提出的一个开放问题的启发，我们在函数梯度在有界集上是 Lipschitz 连续的情况下解决了这个问题。