?Mathematical formulae have been encoded as MathML and are displayed in this HTML version using MathJax in order to improve their display. Uncheck the box to turn MathJax off. This feature requires Javascript. Click on a formula to zoom.It is our great pleasure to devote this special issue to Equilibrium Problems and Minimax Inequalities which includes seven papers selected among those presented at the autumn school ASEM19, which was held during 25–26 September 2019, in El Jadida Morocco. ASEM19 has been supported by Chouaïb Doukkali University (UCD), Moroccan National Center for Scientific and Technical Research (CNRST), the faculty of sciences of UCD-El Jadida and its LMF-Laboratory of Fundamental Mathematics. We would like to thank Elisabeth Anna Sophia Köbis for her efficient management and very nice collaboration during the preparation of this issue.
The contributions presented in this issue incorporate relevant applied mathematical models ranging from mathematical programming, through game theory, mechanics, engineering to optimal control. The contents of this special edition are described as follows:
The paper ‘On the applications of a minimax theorem’ by B. Ricceri gives a nice overview of recent applications of a minimax theorem. The author discusses several problems such as uniquely remotal sets in normed spaces, multiple global minima for the integral functional of the calculus of variations, multiple periodic solutions for Lagrangian systems of relativistic oscillators and variational inequalities over balls in Hilbert spaces.
The article ‘Fast convex optimization via a third-order in time evolution equation’ by H. Attouch, Z. Chbani and H. Riahi develop fast convex optimization methods which are based on an evolution system of the third-order in time. The authors explore the minimization problem of a convex and continuously differentiable function over a Hilbert space, which enters the dynamic via its gradient. Using Lyapunov's analysis and temporal scaling techniques, they show a convergence rate of the values of the order and obtain the convergence of the trajectories towards optimal solutions. Then, they obtain similar convergence rates by analyzing the convergence of the proximal-based algorithms via temporal discretization.
A. Oussarhan and T. Amahroq in the paper ‘Existence of Lagrange multipliers for set optimization with application to vector equilibrium problems’ prove the existence of the Lagrange multipliers for constrained set-optimization problems. Then, as a consequence, they provide optimality conditions for weak vector equilibrium problems.
In the contribution, ‘Sharp estimates for approximate and exact solutions to quasi-optimization problems’, M. Ait Mansour, M. A Bahraoui and A. El Bekkali consider an implicit set-valued map representing solutions to a parametric quasi-optimization problem (QOpt). This equilibrium model is complex by its nature but constitutes a meaningful formulation of challenging problems such as quasi-convex programming and generalized Nash equilibria over constraints that depend on the solution. Firstly, the authors exploit a new recent variant of the celebrated Lim's Lemma considered in the context of metric regularity and approximate fixed points to establish quantitative stability for ε-approximate solutions to (QOpt) under parametric perturbations. Then, they derive sharp estimates for parametric exact solutions to (QOpt) by means of a qualitative stability analysis involving set-convergence ingredients. Finally, this work gives an application to a non-smooth mathematical program under polyhedral convex mappings.
The paper ‘Optimal control of a class of hemivariational inequalities for nonstationary Navier-Stokes equations: An equilibrium problem approach’ by O. Chadli and R. N. Mohapatra, deals with optimal control of nonstationary Navier–Stokes equations with nonlinear boundary conditions described by the subdifferential in the sense of Clarke. The authors' attention is devoted to the minimization of a general functional for a control problem whose state is a solution to a boundary value problem depending on the control itself. In particular, they underline the fact that the lower level of this problem is expressed by a hemivariational inequality associated with a nonconvex nonsmooth locally Lipschitz superpotential. The existence of solutions is proved via a convergence scheme based on mixed equilibria and a stability result with respect to variations of the control parameter.
The article ‘A dynamical approach for the quantitative stability of bilevel parametric equilibrium problems and applications’ by M. Ait Mansour, Z. Mazgouri and H. Riahi first presents Hölder and Lipschitz continuity of solutions to abstract dynamical mixed equilibrium problems together with an application to quantitative stability of parametric differential variational inclusions. Then, by using key conditions on Fitzpatrick transform of equilibrium bifunctions, the authors derive further quantitative stability for a parametric bilevel equilibrium problem as well as parametric mathematical programs with equilibrium constraints. They moreover discuss a specific applied model which interprets the role of the involved parameters in the context of optimal control problems whose state equation is defined by a variational inequality. Finally, a numerical illustration is reported to highlight the rate of convergence of the obtained stability results.
In the last contribution of this issue ‘An extended local principle of fixed points for weakly contractive set-valued mappings’, M. Ait Mansour, A. El Bekkali and J. Lahrache propose an extension of the well-known Contraction Mapping Principle for Set-Valued Mappings by considering the class of weakly contractive mappings, which allows to cover, extend and unify many assertions of fixed point theory. This extended fixed point principle is therefore applied to derive two quantitative stability results for fixed points of weakly contractive maps.
Finally, the editors would like to thank the authors who contributed to this issue and the reviewers for their numerous valuable suggestions.
Acknowledgments
On the behalf of local organizers of the ASEM19 school, we would like to warmly thank Professors Christiane Tammer and Juan Enrique Martínez-Legaz for their numerous valuable suggestions, without which this project would never have been achieved. We are equally grateful to Professor Biagio Ricceri and to all participants for their very nice contributions. Special thanks also go to Hedy Attouch and once more to Christiane Tammer for ensuring valuable mini-courses for Ph.D students in the framework of this event.
Mohamed Ait Mansour and Jaafar Lahrache
In the name of the participants of the school, we thank the local organizers for having arranged a very interesting meeting. We enjoyed the high quality of the organization and, very specially, their warm hospitality.
Christiane Tammer and Juan Enrique Martínez-Legaz
on the behalf of the external members of the Scientific Committee.
