2-Strict Convexity and Continuity of Set-Valued Metric Generalized Inverse in Banach Spaces

Authors investigate the metric generalized inverses of linear operators in Banach spaces. Authors prove by the methods of geometry of Banach spaces that, if is approximately compact and is 2-strictly convex, then metric generalized inverses of bounded linear operators in are upper semicontinuous. Moreover, authors also give criteria for metric generalized inverses of bounded linear operators to be lower semicontinuous. Finally, a sufficient condition for set-valued mapping to be continuous mapping is given. 1. Introduction Let be a real Banach space. Let and denote the unit sphere and the unit ball, respectively. By we denote the dual space of . Let , , and denote the set of natural numbers, reals, and nonnegative reals, respectively. Let and . By we denote that is weakly convergent to . denotes closed hull of (weak closed hull) and dist denotes the distance of and . Let be a nonempty subset of . Then the set-valued mapping is called the metric projection operator from onto . A subset of is said to be proximal if for all (see [1]). is said to be semi-Chebyshev if is at most a singleton for all . is said to be Chebyshev if it is proximal and semi-Chebyshev. It is well known that (see [1]) is reflexive if and only if each closed convex subset of is proximal and that is strictly convex if and only if each convex subset of is semi-Chebyshev. Definition 1 (see [2]). A nonempty subset of is said to be approximatively compact if, for any and any satisfying , the sequence has a subsequence converging to an element in . is called approximatively compact if every nonempty closed convex subset of is approximatively compact. Definition 2 (see [3]). Set-valued mapping is called upper semicontinuous at , if, for each norm open set with , there exists a norm neighborhood of such that for all in . is called lower continuous at , if, for any and any in with , there exists such that as . is called continuous at , if is upper semicontinuous and is lower continuous at . Let us present the history of the approximative compactness and related notions. This notion has been introduced by Jefimow and Stechkin in [2] as a property of Banach spaces, which guarantees the existence of the best approximation element in a nonempty closed convex set for any . In 2007, Chen et al. (see [4]) proved that a nonempty closed convex of a midpoint locally uniformly rotund space is approximately compact if and only if is Chebyshev set and the metric projection operator is continuous. In 1972, Oshman (see [5]) proved that the metric projection operator is upper semicontinuous. Definition 3 (see