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We consider the following second order equationwhereis quasi-nonexpansive and Lipschitz continuous on bounded sets andis a continuously differentiable quasiconvex function such thatis 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. 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.

Asymptotic behaviour of a dynamical system governed by non-monotone potential and non-potential operators