Farthest point problem and partial statistical continuity in normed linear spaces. Issue 4 (3rd April 2022)

• Record Type: Journal Article
• Authors: Basu, Sudeshna; Dey, Lakshmi Kanta; Som, Sumit
• Is Part Of: Quaestiones mathematicae. Volume 45:Issue 4(2022)
• Page Start: 595
• Page End: 604
• Publication Date: 2022-04-03
• Subjects: 46B20 -- 41A65 -- 41A50 -- 40A35; Continuity -- partial statistical continuity -- uniquely remotal -- farthest point map -- maximizing sequence -- approximate fixed point sequence; Mathematics -- Periodicals; 510.5
• DOI: 10.2989/16073606.2021.1886193
• Languages: English
• ISSNs: 1607-3606

Abstract

Abstract: In this paper, we prove that if E is an uniquely remotal subset of a real normed linear space such that E has a Chebyshev center c ∈ and the farthest point map F : → E restricted to [ c, F ( c )] is partially statistically continuous at c, then E is a singleton. We obtain a necessary and sufficient condition on uniquely remotal subsets of uniformly rotund Banach spaces to be a singleton. Moreover, we show that there exists a remotal set M having a Chebyshev center c such that the farthest point map F : ℝ → M is not continuous at c but is partially statistically continuous there in the multivalued sense.