Minimax Theory and its Applications 04 (2019), No. 2, 231--270
Copyright Heldermann Verlag 2019
From Convergence of Dynamical Equilibrium Systems to Bilevel Hierarchical Ky Fan Minimax Inequalities and Applications
Zaki Chbani
Cadi Ayyad University, Faculty of Sciences Semlalia, LIBMA Mathematics, 40000 Marrakech, Morocco
chbaniz@uca.ac.ma
Zakaria Mazgouri
Cadi Ayyad University, Faculty of Sciences Semlalia, LIBMA Mathematics, 40000 Marrakech, Morocco
zakariam511@gmail.com
Hassan Riahi
Cadi Ayyad University, Faculty of Sciences Semlalia, LIBMA Mathematics, 40000 Marrakech, Morocco
h-riahi@uca.ac.ma
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. In the process of doing so, we consider Fitzpatrick's transforms introduced in M. H. Alizadeh and N. Hadjisavvas [On the Fitzpatrick transform of a monotone bifunction, Optimization 62 (2013) 693--701] and their related Brézis-Haraux transforms for time-dependant equilibrium bifunctions, which prove to be a key tool in our convergence analysis. Afterwards, by means of a first-order linear sum approach of two real-valued bifunctions, where the penalization term is a positive measurable function, we present several results concerning the weak convergence in average, weak and strong convergence of trajectories to solutions to the bilevel equilibrium problem subject to our treatment.
Some applications, supported by numerical illustrations implemented in Scilab version 5.5.2, are thereafter discussed with respect to bilevel hierarchical minimization problem as well as dynamical systems for saddle convex-concave bifunctions. Our results generalize and extend some of those obtained in J. B. Baillon and H. Brézis [Une remarque sur le comportement asymptotique des semi-groupes non linéaires, Houston J. Math. 2 (1976) 5--7], and H. Attouch and M. O. Czarnecki [Asymptotic behavior of coupled dynamical systems with multiscale aspects, J. Differential Equations 248(6) (2010) 1315-1344]. We end the paper by concluding remarks and suggestions for research perspectives.
Keywords: Ky Fan minimax inequality, bilevel hierarchical Ky Fan minimax inequality, strongly monotone, asymptotic behavior, equilibrium Brézis-Haraux transform, equilibrium Fitzpatrick transform.
MSC: 34A12, 34A34, 34A40, 34D05
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