We consider the following second order equation $\ddot{u}\left(t\right)+\gamma \dot{u}\left(t\right)+(I-T)u\left(t\right)+\mathrm{\nabla }\varphi \left(u\right(t\left)\right)=0,$where $T:H\to H$ is quasi-nonexpansive and Lipschitz continuous on bounded sets and $\varphi :H\to \mathbb{R}$ is a continuously differentiable quasiconvex function such that $\mathrm{\nabla }\varphi$ is Lipschitz continuous on bounded sets. We study the asymptotic behaviour of solutions to this equation. Assuming some mild conditions on the operators, we prove weak and strong convergence of solutions to some point in $\mathrm{F}\mathrm{i}\mathrm{x}\left(T\right)\cap \left(\mathrm{\nabla }\varphi {)}^{-1}\right(0)$. We also obtain similar results for the asymptotic behaviour of solutions to the discrete version of the above equation. Finally, we apply our results to solving a minimization problem and approximating a common fixed point of two mappings. Our work is motivated by the papers of H. Attouch and P. E. Maingé [Asymptotic behaviour of second-order dissipative evolution equations combining potential with non-potential effects. ESAIM Control Optim Calc Var. 2011;17:836–857.], X. Goudou and J. Munier [The gradient and heavy ball with friction dynamical systems: the quasiconvex case. Math Program Ser B. 2009;116:173–191.], and F. Alvarez and H. Attouch [An inertial proximal method for maximal monotone operators via dicretization of a nonlinear oscillator with damping. Well-posedness in optimization and related topics. Set Valued Anal. 2001;9:3–11.], and extends some of their results.

我们考虑以下二阶方程 $\ddot{ü}（Ť）+\gamma \dot{ü}（Ť）+（\mathrm{一世}-Ť）ü（Ť）+\mathrm{\nabla }\varphi （ü（Ť））=0，$哪里 $Ť：H\to H$ 在有界集上是拟非扩张和Lipschitz连续的， $\varphi ：H\to \mathbb{[R}$ 是连续可微的拟凸函数，使得 $\mathrm{\nabla }\varphi$Lipschitz在有界集合上是连续的。我们研究了该方程解的渐近行为。假设运营商有一些温和的条件，我们证明解决方案的弱点和强点收敛到$\mathrm{F}\mathrm{一世}\mathrm{X}（Ť）\cap （\mathrm{\nabla }\varphi {）}^{-\mathrm{1个}}（0）$。对于上述方程的离散版本的解的渐近性，我们也获得了相似的结果。最后，我们将结果应用于解决最小化问题并逼近两个映射的公共不动点。我们的工作是由H. Attouch和PEMaingé[二阶耗散发展方程的渐近行为，结合势和非势效应的结果]推动的。ESAIM控制Optim Calc Var。2011; 17：836–857。]，X。Goudou和J. Munier [具有摩擦动力学系统的梯度球和重球：拟凸情况。Math Program Ser B. 2009; 116：173–191。]，以及F. Alvarez和H. Attouch [一种通过阻尼阻尼的非线性振荡器的离散化来实现最大单调算子的惯性近端方法。在优化和相关主题方面具有良好的定位。设定值的肛门。2001; 9：3-11。]，