On Weak Well-posedness of the Nearest Point and Mutually Nearest Point Problems in Banach Spaces

Let G be a nonempty closed subset of a Banach space X. Let ℬ(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal B}(X)$$\end{document} be the family of nonempty bounded closed subsets of X endowed with the Hausdorff distance and ℬG(X)={A∈ℬ(X):A∩G∅}¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\cal B}_G}(X) = \overline {\{ A \in {\cal B}(X):A \cap G\emptyset \} } $$\end{document}, where the closure is taken in the metric space (ℬ(X),H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\cal B}(X),H)$$\end{document}. For x ∈ X and F∈ℬG(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F \in {{\cal B}_G}(X)$$\end{document}, we denote the nearest point problem inf{∥x − g∥: g ∈ G} by min(x, G) and the mutually nearest point problem inf{∥f − g∥: f ∈ F,g ∈ G} by min(F, G). In this paper, parallel to well-posedness of the problems min(x, G) and min(F, G) which are defined by De Blasi et al., we further introduce the weak well-posedness of the problems min(x, G) and min(F, G). Under the assumption that the Banach space X has some geometric properties, we prove a series of results on weak well-posedness of min(x, G) and min(F, G). We also give two sufficient conditions such that two classes of subsets of X are almost Chebyshev sets.