Contractive and optimal sets in modular spaces

Abstract

We study contractive, existence and optimal sets in modular spaces. We define the analogous classes of sets with respect to a modular and generalize a number of results due to Beauzamy and Maurey [3] and Bruck [9] to modular spaces. We then apply them to show that in certain Köthe sequence spaces all of these notions are equivalent, extending a result of Davis and Enflo [11] for lp spaces, 1 < p < ∞. We also show that in class of Orlicz sequence spaces the theorem that any optimal set in lp spaces, 1 < p < ∞, is an intersection of half-spaces determined by one-complemented hyperplanes in lp, is no longer true. We then find a class of optimal sets, called strongly ρ-optimal, such that this characterization holds true in Musielak-Orlicz sequence spaces. We provide examples which show that all families of sets discussed in the paper like optimal, existence or contractive with respect to a norm or a modular or their strong counterparts, are not equivalent in general.

Contractive and optimal sets in Musielak-Orlicz spaces with a smoothness condition

“…Actually, in this article we will compare contractive and optimal sets defined in a more general way using the modular, since our study is devoted to a wide class of Musielak-Orlicz sequence spaces. This will be done precisely using the main result of [6] and some arguments of Kamińska and Lewicki from [12]. Our results complete in some sense the results from [12] providing a characterization of strongly contractive sets in the sense of the modular ρ Φ in a Musielak-Orlicz space (n) Φ when Φ satisfies condition (M ).…”

“…This permitted Kamińska and Lewicki to characterize in [12] the contractive sets in Musielak-Orlicz spaces with the condition (S) and to compare these sets with optimal sets. Condition (S), though not really restrictive, excludes such regular functions as t p for p ∈ [1, 2) (thus the Jamison-Kamińska-Lewicki Theorem does not not work for p ).…”

“…Then we turn to the counterparts of the Kamińska-Lewicki results from [12] concerning contractive and optimal sets in Musielak-Orlicz spaces.…”

“…However, Remark 3.3. Theorem 1.5 from [12] implies that in case Φ = h Φ and Φ satisfies (s), the space Φ is smooth (cf. Theorem 2.7).…”

Nonexpansive retracts in Banach spaces

“…This is indeed known to be the case for the ℓ p ( [DE,E2]) and E2]) spaces, 1 < p < ∞, as well as for strictly convex, smooth and reflexive Köthe sequence spaces ( [KL,Theorem 3.8]). …”