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

Shaoqiang Shang,Yunan Cui

doi:10.1155/2014/384639

Abstract

Authors investigate the metric generalized inverses of linear operators in Banach spaces. Authors prove by the methods of geometry of Banach spaces that, ifXis approximately compact andXis 2-strictly convex, then metric generalized inverses of bounded linear operators inXare 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 mappingT∂to be continuous mapping is given.

Highlights

Let (X, ‖ ⋅ ‖) be a real Banach space
Authors prove by the methods of geometry of Banach spaces that, if X is approximatively compact and X is 2-strictly convex space, metric generalized inverse of a bounded linear operator is upper semicontinuous
(1) X1 is a 2-strictly convex Banach space; (2) for any y ∈ Y, there exist x1 ∈ D(T) and x2 ∈ D(T) such that the set-valued mapping satisfies the equality T∂(y) = [x1, x2]; (3) the set-valued mapping T∂ is upper semicontinuous; (4) for any y ∈ Y, the set-valued mapping T∂ is lower semicontinuous at y if and only if the function g(y) = sup{‖z1 − z2‖ : z1, z2 ∈ T∂(y)} is lower semicontinuous at y

Summary

Introduction

Let (X, ‖ ⋅ ‖) be a real Banach space. Let S(X) and B(X) denote the unit sphere and the unit ball, respectively. Authors investigate the metric generalized inverses of linear operators in Banach spaces. Authors prove by the methods of geometry of Banach spaces that, if X is approximatively compact and X is 2-strictly convex space, metric generalized inverse of a bounded linear operator is upper semicontinuous.

Results

Conclusion