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- Title
On Weak Well-posedness of the Nearest Point and Mutually Nearest Point Problems in Banach Spaces.
- Authors
Zhang, Zi Hou; Liu, Chun Yan; Zhou, Yu; Zhou, Jing
- Abstract
Let G be a nonempty closed subset of a Banach space X. Let ℬ (X) be the family of nonempty bounded closed subsets of X endowed with the Hausdorff distance and ℬ G (X) = { A ∈ ℬ (X) : A ∩ G ∅ } ¯ , where the closure is taken in the metric space (ℬ (X) , H) . For x ∈ X and F ∈ 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.
- Subjects
BANACH spaces
- Publication
Acta Mathematica Sinica, 2021, Vol 37, Issue 8, p1303
- ISSN
1439-8516
- Publication type
Article
- DOI
10.1007/s10114-021-0121-3