Metric Regularity and Lyusternik-Graves Theorem via Approximate Fixed Points of Set-Valued Maps in Noncomplete Metric Spaces

Abstract

This paper considers global metric regularity and approximate fixed points of set-valued mappings. We establish a very general Theorem extending to noncomplete metric spaces a recent result by A.L. Dontchev and R.T. Rockafellar on sharp estimates of the distance from a point to the set of exact fixed points of composition set-valued mappings. In this way, we find again the famous Nadler’s Theorem, and mainly, we accordingly come up with new conclusions in this research field concerning approximate versions of Lim’s Lemma as well as the celebrated global Lyusternik-Graves Theorem. The presented results are accompanied with examples and counter-examples when it is needed. Our approach follows up numerical procedures without recourse to convergence of Cauchy sequences. Moreover, we connect metric regularity to set-convergence in metric spaces such as Painleve-Kuratowski convergence and Pompeiu-Hausdorff convergence for sets of approximate fixed points of set-valued maps. In the same context, we analyse the possibilities of the passage of regularity estimates from approximate fixed points to exact ones under the motivation of some Beer’s observations related to Wijsman convergence. As a by-product, we obtain the approximative counterpart of a recent result by A. Arutyunov on coincidence points of set-valued maps besides a new characterization of globally metrically regular set-valued maps, wherein completeness and closedness conditions are not needed.