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

We consider the following second order equation u(t) + gamma(u) over dot(t) + (I - T)u (t) + del phi(u(t)) = 0, where T : H -> H is quasi-nonexpansive and Lipschitz continuous on bounded sets and phi : H -> R is a continuously differentiable quasiconvex function such that del(phi) 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 Fix(T) boolean AND (del phi)(-1)(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. Mainge [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. Wellposedness in optimization and related topics. Set Valued Anal. 2001;9:3-11.], and extends some of their results.

Automated summary

The paper investigates the asymptotic behavior of solutions to a second-order equation involving quasi-nonexpansive operators and continuously differentiable quasiconvex functions, proving weak and strong convergence of solutions and obtaining similar results for the discrete version of the equation. The results are applied to solving a minimization problem and approximating common fixed points of two mappings, extending some previous works in the field.