Uniform Gateaux differentiability and weak uniform rotundity in Musielak-Orlicz function spaces of Bochner type equipped with the Luxemburg norm
type: Item
instance of: scholarly article
title: Uniform Gateaux differentiability and weak uniform rotundity in Musielak-Orlicz function spaces of Bochner type equipped with the Luxemburg norm
zbMATH Open document ID: 1252.46012
author: Shaoqiang Shang
author: Henryk Hudzik
author: Yunan Cui
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2012-03-19T00:00:00Z
review text: Let \(\Phi\) and \(\Psi\) be a Musielak-Orlicz function and its complementary function in the sense of Young, respectively. The main result of this paper concerns weak uniform rotundity in Musielak-Orlicz function spaces of Bochner type equipped with the Luxemburg norm (see Theorem 2 and Lemmas 3-6). Namely, it is shown that if the dual space \(X^{\ast}\) of a Banach space \(X\) has the Radon-Nykodym property, then the space \(L_{M}(X)\) is weakly uniformly rotund if and only if \(M\) and \(N\) satisfy the \(\Delta_{2}\) condition, \(M(t,u)\) is strictly convex with respect to \(u\) for \(\mu\)-a.e.\ \(t\in T\) and \(X\) is a weakly uniformly rotund space. On the basis of this result criteria for uniform Gateâux differentiability of Musielak-Orlicz spaces equipped with the Orlicz norm are deduced.
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46E40
zbMATH DE Number: 6015710
zbMATH Keywords: weak uniform rotundity
zbMATH Keywords: uniform Gâteaux differentiability
zbMATH Keywords: reflexivity
zbMATH Keywords: Musielak-Orlicz function space of Bochner type
zbMATH Keywords: \(\Delta_2\) condition
reviewed by: Pawel Foralewski
MaRDI profile type: Publication
full work available at URL: j.na.2011.11.012
OpenAlex ID: W2044049218
cites work: Q5777588
cites work: Q4338822
cites work: Q5315206
cites work: Uniform Gateaux differentiability and weak uniform rotundity of Musielak-Orlicz function spaces
cites work: Q4299104
cites work: Uniformly non-l n (1) Musielak-Orlicz Spaces of Bochner Type
cites work: Local uniform rotundity and weak local uniform rotundity of Musielak--Orlicz sequence spaces endowed with the Orlicz norm
cites work: Smooth, very smooth and strongly smooth points in Musielak-Orlicz sequence spaces
cites work: Criteria for complex strongly extreme points of Musielak--Orlicz function spaces
cites work: On some properties of Musielak-Orlicz spaces and their subspaces of order continuous elements
cites work: Q4879588
cites work: Maluta’s coefficient and Opial’s properties in Musielak–Orlicz sequence spaces equipped with the Luxemburg norm
cites work: On fully rotundity properties and approximative compactness in some Banach sequence spaces
cites work: Extreme points and strong U-points in Musiełak-Orlicz sequence spaces equipped with the Orlicz norm
cites work: Criteria for monotonicity properties of Musielak-Orlicz spaces equipped with the Amemiya norm
cites work: On some uniform convexities and smoothness in certain sequence spaces
cites work: Q3416979
cites work: Maluta’s coefficient in Musielak-Orlicz sequence spaces equipped with the Orlicz norm
cites work: Q5455801
cites work: Uniformly non-\(\ell^{(1)}_n\) Musielak-Orlicz sequence spaces
cites work: Lebesgue and Sobolev spaces with variable exponents
cites work: Q3989724
cites work: On weighted critical imbeddings of Sobolev spaces
cites work: Q5325674
cites work: Q4027017
cites work: Q3943455
cites work: Q3938038
cites work: Q3701961
cites work: Q3678186
cites work: Q3679731
cites work: Q3678187
cites work: Convexity in Musielak-Orlicz spaces
cites work: Q3715645
cites work: Q3779236
cites work: Q5313953
cites work: On some local geometric properties in Musielak-Orlicz function spaces
cites work: Approximative compactness and full rotundity in Musielak--Orlicz spaces and Lorentz--Orlicz spaces
cites work: On non-effective weights in Orlicz spaces
cites work: Monotonicity properties of Musielak-Orlicz spaces and dominated best approximation in Banach lattices
cites work: Q3137385
cites work: Q4780281
cites work: Points of monotonicity in Musielak--Orlicz function spaces endowed with the Luxemburg norm
cites work: Q4508968
cites work: Q4298813
cites work: Q5753025
cites work: Smooth points of Musielak-Orlicz sequence spaces equipped with the Luxemburg norm
cites work: Smoothness in Musielak-Orlicz spaces equipped with the Orlicz norm
cites work: Q3920139
cites work: Q3658544
cites work: Strict convexity of sequence Orlicz-Musielak spaces with Orlicz norm
cites work: Q3692278
cites work: Q5185513
cites work: Q3749578
cites work: Uniform rotundity of Musielak-Orlicz sequence spaces
cites work: Q3975361
cites work: Q4386403
cites work: \(P\)-convexity of Musielak-Orlicz sequence spaces of Bochner type
cites work: \(P\)-convexity of Musielak-Orlicz function spaces of Bochner type

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cites work: Q3200054

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cites work: Strictly and uniformly monotone Musielak-Orlicz spaces and applications to best approximation

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cites work: Strictly and uniformly monotone sequential Musielak-Orlicz spaces

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cites work: Orlicz spaces and modular spaces

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cites work: On the points of local uniform rotundity and weak local uniform rotundity in Musielak-Orlicz sequence spaces equipped with the Orlicz norm

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cites work: Q3998720

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cites work: Handbook of metric fixed point theory

Uniform Gateaux differentiability and weak uniform rotundity in Musielak-Orlicz function spaces of Bochner type equipped with the Luxemburg norm (DOI)
DOI: 10.1016/J.NA.2011.11.012