Directional differentiability of the metric projection in Bochner spaces
Applicable Nonlinear Analysis, Volume 1, Issue 1, June 2024, Pages: 79–109
JINLU LI
Department of Mathematics, Shawnee State University, Portsmouth Ohio 45662, USA
Abstract
In this paper, we consider the Gâteaux directional differentiability of metric projection operator and its properties in uniformly convex and uniformly smooth Bochner space \(L_{p}(S ; X)\), in which \((S, \mathcal{A}, \mu)\) is a positive measure space and \(X\) is a uniformly convex and uniformly smooth Banach space. Let (arbitrary) \(A \in \mathcal{A}\) with \(\mu(A)>0\) and define a subspace \(L_{p}(A ; X)\) of \(L_{p}(S ; X)\), which is considered as a closed and convex subset of \(L_{p}(S ; X)\). We first study the properties of the normalized duality mapping in \(L_{p}(S ; X)\) and in \(L_{p}(A ; X)\). For any \(c \in L_{p}(A ; X)\) and \(r>0\), we define a closed ball \(B_{A}(c ; r)\) in \(L_{p}(A ; X)\) and a cylinder \(C_{A}(c ; r)\) in \(L_{p}(S ; X)\) with base \(B_{A}(c ; r)\). Then, we investigate some optimal properties of the corresponding metric projections \(P_{L_{p}(A ; X)}, P_{B_{A}(c ; r)}\) and \(P_{C_{A}(c ; r)}\) that include the inverse images, the Gâteaux directional differentiability and the precise solutions of their Gâteaux directional derivatives.
Cite this Article as
Jinlu Li, Directional differentiability of the metric projection in Bochner spaces, Applicable Nonlinear Analysis, 1 (2024), 79–109