GÂTEAUX DIRECTIONAL DIFFERENTIABILITY OF THE GENERALIZED METRIC PROJECTION IN BANACH SPACES

doi:10.1007/s10473-025-0419-9

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (4): 1597-1618.doi: 10.1007/s10473-025-0419-9

GÂTEAUX DIRECTIONAL DIFFERENTIABILITY OF THE GENERALIZED METRIC PROJECTION IN BANACH SPACES

Jinlu LI

Department of Mathematics, Shawnee State University, 940 Second Street, Portsmouth, OH 45662, USA

收稿日期:2024-03-04 出版日期:2025-10-10 发布日期:2025-10-10

GÂTEAUX DIRECTIONAL DIFFERENTIABILITY OF THE GENERALIZED METRIC PROJECTION IN BANACH SPACES

Jinlu LI

Department of Mathematics, Shawnee State University, 940 Second Street, Portsmouth, OH 45662, USA

Received:2024-03-04 Online:2025-10-10 Published:2025-10-10
About author:Jinlu LI, E-mail: jli@shawnee.edu

摘要/Abstract

摘要： Let $X$ be a real uniformly convex and uniformly smooth Banach space and $C$ a nonempty closed and convex subset of $X$. Let $\Pi_C{:}~X\to C$ denote the generalized metric projection operator introduced by Alber in [1]. In this paper, we define the Gâteaux directional differentiability of $\Pi_{C.}$ We investigate some properties of the Gâteaux directional differentiability of $\Pi_C$. In particular, if $C$ is a closed ball, or a closed and convex cone (including proper closed subspaces), or a closed and convex cylinder, then, we give the exact representations of the directional derivatives of $\Pi_{C.}$ By comparing the results in [12] and this paper, we see the significant difference between the directional derivatives of the generalized metric projection operator $\Pi_{C}$ and the Gâteaux directional derivatives of the standard metric projection operator $P_{C.}$

关键词: uniformly convex and uniformly smooth Banach space, the generalized metric projection, directional differentiability of the generalized metric projection, directional derivative of the generalized metric projection

Abstract: Let $X$ be a real uniformly convex and uniformly smooth Banach space and $C$ a nonempty closed and convex subset of $X$. Let $\Pi_C{:}~X\to C$ denote the generalized metric projection operator introduced by Alber in [1]. In this paper, we define the Gâteaux directional differentiability of $\Pi_{C.}$ We investigate some properties of the Gâteaux directional differentiability of $\Pi_C$. In particular, if $C$ is a closed ball, or a closed and convex cone (including proper closed subspaces), or a closed and convex cylinder, then, we give the exact representations of the directional derivatives of $\Pi_{C.}$ By comparing the results in [12] and this paper, we see the significant difference between the directional derivatives of the generalized metric projection operator $\Pi_{C}$ and the Gâteaux directional derivatives of the standard metric projection operator $P_{C.}$

Key words: uniformly convex and uniformly smooth Banach space, the generalized metric projection, directional differentiability of the generalized metric projection, directional derivative of the generalized metric projection

Jinlu LI. GÂTEAUX DIRECTIONAL DIFFERENTIABILITY OF THE GENERALIZED METRIC PROJECTION IN BANACH SPACES[J]. 数学物理学报(英文版), 2025, 45(4): 1597-1618.

Jinlu LI. GÂTEAUX DIRECTIONAL DIFFERENTIABILITY OF THE GENERALIZED METRIC PROJECTION IN BANACH SPACES[J]. Acta mathematica scientia,Series B, 2025, 45(4): 1597-1618.

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GÂTEAUX DIRECTIONAL DIFFERENTIABILITY OF THE GENERALIZED METRIC PROJECTION IN BANACH SPACES

GÂTEAUX DIRECTIONAL DIFFERENTIABILITY OF THE GENERALIZED METRIC PROJECTION IN BANACH SPACES

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