Mohamed Ait Mansour, “On the global contraction mapping principle: Improvement and perspectives“

In this talk we first stress the original motivation of the stability of equilibrium and fixed points theories. Then, we go into approximate fixed points of set-valued maps by listing the fundamental properties they enjoy. We, thereafter, present our global approximate contraction mapping principle, which ensures the existence of approximate fixed points for a set-valued contracting mapping in metric spaces and, at the meantime, estimates the distance between the set of those approximate points and any other point of the metric space under consideration. We show that our principle, being a key result, has actually immediate and interesting consequences such as the improvement of the famous Lim’s Lemma and Global Lyusternik-Graves Theorem for which we obtain a meaningful extended approximate format, constituting henceforth the key quantitative stability property of approximate fixed points of set-valued maps defined in general metric spaces. Finally, we
discuss the extension of our parametric stability analysis to dynamic equilibrium problems.