From Convergence of Dynamical Equilibrium Systems to Bilevel Hierarchical Ky Fan Minimax Inequalities and Applications
Keywords:
Ky Fan minimax inequality, bilevel hierarchical Ky Fan minimax inequality, strongly monotone, asymptotic behavior, equilibrium Brézis-Haraux transform, equilibrium Fitzpatrick transform.Abstract
Inspired by the variational formulation of continuous Cauchy-Lipschitz systems and forward (descent) or backward (proximal)methods, and motivated by the solvability of bilevel equilibrium problems, we introduce first-order continuous Evolution Dynamical Equilibrium Systems, (EDES) for short. Then, our primary goal is to study the existence and uniqueness of solutions to (EDES). Secondly, we study the asymptotic behaviour of trajectories of Dynamical Ky Fan Minimax Inequalities (NDEMI) with nonautonomous equilibrium bifunctions defined in Hilbert spaces under monotonicity conditions. In this way, we provide conditions guaranteeing the weak ergodic convergence, i.e., convergence in average, of trajectories to an equilibrium point of an appropriate limit monotone bifunction.
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