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

Abstract

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.

Keywords:

• Second order evolution equation
• asymptotic behaviour
• quasi-nonexpansive operator
• quasiconvex function
• minimization

Mathematics Subject Classifications (2010):

• Primary: 35B40
• 47E05
• Secondary:37N40

Acknowledgments

This research is done while the third author was visiting the University of Texas at El Paso. This visit is funded partially by the Iran Ministry of Science, Research and Technology. The third author would like to thank Professor Djafari Rouhani and the Department of Mathematical Sciences for their hospitality at the University of Texas at El Paso. The authors are grateful to the referees for their valuable comments leading to the improvement of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research is done while the third author was visiting the University of Texas at El Paso. This visit is funded partially by the Iran Ministry of Science, Research and Technology.