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$\ddot{你}\left(吨\right)+\gamma \dot{你}\left(吨\right)+(我-吨)你\left(吨\right)+\mathrm{\nabla }\phi \left(你\right(吨\left)\right)=0,$在哪里$吨：H\to H$在有界集上是准非扩张且 Lipschitz 连续的，并且$\phi ：H\to \mathbb{R}$是一个连续可微的拟凸函数，使得$\mathrm{\nabla }\phi$是有界集上的 Lipschitz 连续。我们研究这个方程解的渐近行为。假设算子有一些温和的条件，我们证明了解决方案在某个点上的弱收敛和强收敛$\mathrm{F}\mathrm{一世}\mathrm{X}\left(吨\right)\cap \left(\mathrm{\nabla }\phi {)}^{-1}\right(0)$. 对于上述方程的离散版本的解的渐近行为，我们也获得了类似的结果。最后，我们将我们的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。我们的工作受到 H. Attouch 和 PE Maingé 的论文的启发[二阶耗散演化方程的渐近行为结合了势能和非势能效应。ESAIM Control Optim 计算变量。2011;17:836–857.]，X. Goudou 和 J. Munier [具有摩擦动力系统的梯度和重球：准凸情况。数学程序 Ser B. 2009;116:173–191.]，以及 F. Alvarez 和 H. Attouch [通过对具有阻尼的非线性振荡器进行离散化来实现最大单调算子的惯性近似方法。在优化和相关主题方面表现得很好。设置有价值的肛门。2001;9:3–11.],