Continuity of generalized metric projections in Banach spaces
Continuity of generalized metric projections in Banach spaces
复制标题
Banach 空间中广义度量投影的连续性
DOI：
10.1007/s13398-017-0453-0
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Liu Chunyan
中科院分区：
文献类型：
--
作者：
Zhang Zihou;Zhou Yu;Liu Chunyan
Let $$\mathcal {P}$$ P be the family of all proximinal subsets of a Banach spaceX. Let $$P:(X,\mathcal {P})\rightarrow 2^{X}$$ P : ( X , P ) → 2 X be the generalized metric projection defined as $$P(x,A)=P_{A}(x)=\{a\in A:\Vert x-a\Vert =d(x,A)\}$$ P ( x , A ) = P A ( x ) = { a ∈ A : ‖ x - a ‖ = d ( x , A ) } for any $$(x,A)\in (X,\mathcal {P})$$ ( x , A ) ∈ ( X , P ) , where $$P_{A}$$ P A is the usual metric projection onX. The mappingPis said to be (resp. weakly) upper semi-continuous at $$(x,A)\in (X,\mathcal {P})$$ ( x , A ) ∈ ( X , P ) in the Hausdorff sense if, for any (resp. weakly) open set $$W\supset P_A(x)$$ W ⊃ P A ( x ) , any $$\{x_n\}_{n=1}^{\infty }\subset X$$ { x n } n = 1 ∞ ⊂ X with $$x_n\rightarrow x$$ x n → x and any sequence $$\{A_n\}_{n=1}^{\infty }\subset \mathcal {P}$$ { A n } n = 1 ∞ ⊂ P with $$A_n\xrightarrow {H}A$$ A n → H A , there exists a $$N\in \mathbb {N}$$ N ∈ N such that $$P_{A_n}(x_n)\subset W …
Let $$\mathcal {P}$$ P be the family of all proximinal subsets of a Banach spaceX. Let $$P:(X,\mathcal {P})\rightarrow 2^{X}$$ P : ( X , P ) → 2 X be the generalized metric projection defined as $$P(x,A)=P_{A}(x)=\{a\in A:\Vert x-a\Vert =d(x,A)\}$$ P ( x , A ) = P A ( x ) = { a ∈ A : ‖ x - a ‖ = d ( x , A ) } for any $$(x,A)\in (X,\mathcal {P})$$ ( x , A ) ∈ ( X , P ) , where $$P_{A}$$ P A is the usual metric projection onX. The mappingPis said to be (resp. weakly) upper semi-continuous at $$(x,A)\in (X,\mathcal {P})$$ ( x , A ) ∈ ( X , P ) in the Hausdorff sense if, for any (resp. weakly) open set $$W\supset P_A(x)$$ W ⊃ P A ( x ) , any $$\{x_n\}_{n=1}^{\infty }\subset X$$ { x n } n = 1 ∞ ⊂ X with $$x_n\rightarrow x$$ x n → x and any sequence $$\{A_n\}_{n=1}^{\infty }\subset \mathcal {P}$$ { A n } n = 1 ∞ ⊂ P with $$A_n\xrightarrow {H}A$$ A n → H A , there exists a $$N\in \mathbb {N}$$ N ∈ N such that $$P_{A_n}(x_n)\subset W …