Almost convexity and continuous selections of the set-valued metric generalized inverse in Banach spaces
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Banach J. Math. Anal. https://doi.org/10.1007/s43037-020-00098-3 ORIGINAL PAPER
Almost convexity and continuous selections of the set‑valued metric generalized inverse in Banach spaces Shaoqiang Shang1 · Yunan Cui2 Received: 19 October 2019 / Accepted: 30 September 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract In this paper, we define almost convex space. Let T ∶ X → Y be a linear bounded operator. This paper shows that: (1) If X is almost convex and 2-strictly convex, Y is a Banach space, D(T) is closed, N(T) is an approximatively compact Chebyshev subspace of D(T) and R(T) is a 2-Chebyshev hyperplane of Y, then there exists a homogeneous selection T 𝜎 of T 𝜕 such that continuous points of T 𝜎 is dense on Y. (2) If X is locally uniformly convex, Y is reflexive, D(T) is closed, N(T) is a proximinal subspace of D(T) and R(T) is a closed hyperplane of Y, then T 𝜕 is single-valued, homogeneous and continuous on Y. The results are a perfect answer to the open problem posed by Nashed and Votruba (Bull Am Math Soc 80:831–835, 1974). Keywords Almost convex space · Continuous selection · Set-valued metric generalized inverse · 2-strictly convex Mathematics Subject Classification 46B20
Communicated by Mikhail Ostrovskii. * Shaoqiang Shang [email protected] Yunan Cui [email protected] 1
College of Mathematical Sciences, Harbin Engineering University, Harbin 150001, People’s Republic of China
2
Department of Mathematics, Harbin University of Science and Technology University, Harbin 150080, People’s Republic of China
Vol.:(0123456789)
S. Shang, Y. Cui
1 Introduction and preliminaries Let X denote a real Banach space with unit ball B(X) and unit sphere S(X). Let X* denote the dual space of Banach Ax∗ = A(x∗ ) = {x ∈ S(X) ∶ x∗ (x) = 1 = ‖x∗ ‖} space X. and ∗ ∗ ∗ A(x) = {x ∈ S(X ) ∶ x (x) = 1 = ‖x‖} . Let T denote a linear bounded operator from subspace of X into Banach space Y. Let D(T), N(T) and R(T) denote the domain, null space and range of T, respectively. Let M be a subspace of X. The set-valued mapping PM ∶ X → M � � PM (x) = z ∈ M ∶ ‖x − z‖ = dist(x, M) ∶= inf ‖x − y‖ y∈M
is said to be the metric projection operator. If PM (x) ≠ � for all x ∈ X , then we call X is proximinal. Let 𝜋M denote a selection for the set-valued mapping PM . Moreover, we know that if PM (x) is a singleton for all x ∈ X , then we call that M is a Chebyshev subspace of X. The theory of continuous selection is one of the important contents of set-valued analysis. In 1956, Michael [7] proposed the continuous selection theory when he studied continuous function expansion theory, fiber bundle theory and linear operators on closed sets. Continuous selection has developed into an interdisciplinary field of analysis and topology [5]. In addition, the theory of continuous selection have important applications in analysis, topology, differential equations, approximation theory and optimization [4, 9]. Mathematicians pay more and more attention to the existence of continuous selection with the development of the theory of c
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