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<p>Let <i>X</i> be a real uniformly convex and uniformly smooth Banach space and <i>C</i> a nonempty closed and convex subset of <i>X</i>. Let Π<sub><i>C</i></sub>: <i>X</i> → <i>C</i> denote the generalized metric projection operator introduced by Alber in [1]. In this paper, we define the Gâteaux directional differentiability of Π<sub><i>C</i></sub>. We investigate some properties of the Gâteaux directional differentiability of Π<sub><i>C</i></sub>. In particular, if <i>C</i> 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 Π<sub><i>C</i></sub>. By comparing the results in [12] and this paper, we see the significant difference between the directional derivatives of the generalized metric projection operator Π<sub><i>C</i></sub> and the Gâteaux directional derivatives of the standard metric projection operator <i>P</i><sub><i>C</i></sub>.</p>

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Gâteaux directional differentiability of the generalized metric projection in Banach spaces

  • 期刊论文

  • 发表日期：

    2025年4月8日

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  • Jinlu Li

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 ΠC: X → C denote the generalized metric projection operator introduced by Alber in [1]. In this paper, we define the Gâteaux directional differentiability of ΠC. We investigate some properties of the Gâteaux directional differentiability of ΠC. 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 ΠC. By comparing the results in [12] and this paper, we see the significant difference between the directional derivatives of the generalized metric projection operator ΠC and the Gâteaux directional derivatives of the standard metric projection operator PC.