Directional differentiability of the metric projection in Bochner spaces

In this paper, we consider the G\^ateaux directional differentiability of metric projection operator and its properties in uniformly convex and uniformly smooth Bochner space Lₚ(S ; X), in which (S, A, μ) is a positive measure space and X is a uniformly convex and uniformly smooth Banach space. Let (arbitrary) A ∈ A with μ(A)>0 and define a subspace Lₚ(A ; X) of Lₚ(S ; X), which is considered as a closed and convex subset of Lₚ(S ; X). We first study the properties of the normalized duality mapping in Lₚ(S ; X) and in Lₚ(A ; X). For any c ∈ Lₚ(A ; X) and $r>0$, we define a closed ball BA(c ; r) in Lₚ(A ; X) and a cylinder CA(c ; r) in Lₚ(S ; X) with base BA(c ; r). Then, we investigate some optimal properties of the corresponding metric projections P_Lₚ(A ; X), P_BA(c ; r) and P_CA(c ; r) that include the inverse images, the G\^ateaux directional differentiability and the precise solutions of their G\^ateaux directional derivatives.