Multivalued pseudo-contractive mappings defined on unbounded sets in Banach spaces

Claudio H. Morales

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In this paper we prove that for each $1 < p, \tilde{p} < \infty$ , the Banach space $(l^{\tilde{p}}, {∥\cdot∥}_{\tilde{p}})$ can be equivalently renormed in such a way that the Banach space $(l^{\tilde{p}}, {∥\cdot∥}_{L, α, β, p, \tilde{p}})$ is LUR and has a diametrically complete set with empty interior. This result extends the Maluta theorem about existence of such a set in $l^{2}$ with the Day norm. We also show that the Banach space $(l^{\tilde{p}}, {∥\cdot∥}_{L, α, β, p, \tilde{p}})$ has the weak fixed point property for nonexpansive mappings.

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A Banach space X is reflexive if and only if every bounded sequence xₙ in X contains a norm attaining subsequence. This means that it contains a subsequence $x_{n_{k}}$ for which $s u p_{f \in S_{X *}} l i m s u p_{k \to \infty} f (x_{n_{k}})$ is attained at some f in the dual unit sphere $S_{X *}$ . A Banach space X is not reflexive if and only if it contains a normalized sequence xₙ with the property that for every $f \in S_{X *}$ , there exists $g \in S_{X *}$ such that $l i m s u p_{n \to \infty} f (x ₙ) < l i m i n f_{n \to \infty} g (x ₙ)$ . Combining this with a result of Shafrir, we conclude that every infinite-dimensional Banach space contains an unbounded...

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We present an example of finite mappings of algebraic varieties f:V → W, where V ⊂ kⁿ, $W \subset k^{n + 1}$ , and $F : k ⁿ \to k^{n + 1}$ such that ${F |}_{V} = f$ and gdeg F = 1 < gdeg f (gdeg h means the number of points in the generic fiber of h). Thus, in some sense, the result of this note improves our result in J. Pure Appl. Algebra 148 (2000) where it was shown that this phenomenon can occur when V ⊂ kⁿ, $W \subset k^{m}$ with m ≥ n+2. In the case V,W ⊂ kⁿ a similar example does not exist.

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