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 $${\cal B}(X)$$ be the family of nonempty bounded closed subsets of X endowed with the Hausdorff distance and $${{\cal B}_G}(X) = \overline {\{ A \in {\cal B}(X):A \cap G\emptyset \} } $$ , where the closure is taken in the metric space $$({\cal B}(X),H)$$ . For x ∈ X and $$F \in {{\cal B}_G}(X)$$ , 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.