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Metric Regularity and Lyusternik-Graves Theorem via Approximate Fixed Points of Set-Valued Maps in Noncomplete Metric Spaces
Set-Valued and Variational Analysis ( IF 1.6 ) Pub Date : 2020-10-29 , DOI: 10.1007/s11228-020-00553-1
Mohamed Ait Mansour , Mohamed Amin Bahraoui , Adham El Bekkali

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 Painlevé-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.



中文翻译:

非完全度量空间中通过集值映射的近似不动点的度量正则和Lyusternik-Graves定理

本文考虑了全局度量规则性和集值映射的近似不动点。我们建立了一个非常通用的定理,该定理扩展到AL Dontchev和RT Rockafellar的最新结果,该结果是对从点到构图集值映射的确切固定点集的距离的清晰估计。这样,我们再次找到了著名的纳德勒定理,并且主要地,因此,我们在该研究领域中得出了有关林氏引理的近似形式以及著名的全球Lyusternik-Graves定理的新结论。提出的结果在需要时附带示例和反示例。我们的方法遵循数值过程,而无需求助于柯西序列的收敛性。此外,我们将度量规则性与度量空间(例如Painlevé-Kuratowski收敛和Pompeiu-Hausdorff收敛)中的集合收敛性联系起来,以获取集值映射的近似固定点集。在相同的上下文中,我们分析了在比尔关于Wijsman收敛的观察结果的推动下,将正则性估计从近似固定点传递到精确点的可能性。作为副产品,我们获得了A. Arutyunov在集值地图的重合点上的最新结果的近似对应物,此外,它不需要全局度量规则集值地图的新特征,其中不需要完整性和封闭性条件。我们分析了在比尔关于Wijsman收敛的观察结果的推动下,将正则性估计从近似固定点传递到精确点的可能性。作为副产品,我们获得了A. Arutyunov在集值地图的重合点上的最新结果的近似对应物,此外,它不需要全局度量规则集值地图的新特征,其中不需要完整性和封闭性条件。我们分析了在比尔关于Wijsman收敛的观察结果的推动下,将正则性估计从近似固定点传递到精确点的可能性。作为副产品,我们获得了A. Arutyunov在集值地图的重合点上的最新结果的近似对应物,此外,它不需要全局度量规则集值地图的新特征,其中不需要完整性和封闭性条件。

更新日期:2020-10-30
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