Proper 1-ball contractive retractions in Banach spaces of measurable functions

Abstract

In this paper we consider the Wos' ko problem of evaluating, in an infinite-dimensional Banach space X, the infimum of all k ^ 1 for which there exists a fc-ball contractive retraction of the unit ball onto its boundary. We prove that in some classical Banach spaces the best possible value 1 is attained. Moreover we give estimates of the lower H-measure of noncompactness of the retractions we construct.

Examples of proper k-ball contractive retractions in F-normed spaces

“…We observe that in the particular case α( f ) = 2 1+ f the mapping Q coincides with that introduced in [18] (see also [5,6,12]). …”

“…In fact, since the Δ 2 -condition holds we have E ϕ = L ϕ . Then we consider the mapping P ϕ : B(L ϕ ) → L ϕ defined, as in [6], by…”

“…Briefly we recall that W (X) 6 [17] for any infinite dimensional Banach space X, reaching the value 4 or 3 depending on the geometry of the space. Moreover W (X) = 1 in some Banach spaces of continuous functions [5,15,18], in some classical Banach spaces of measurable functions [6] and in Banach spaces whose norm is monotone with respect to some basis [1].…”

Hausdorff Norms of Retractions in Banach Spaces of Continuous Functions

“…The constant W (X) was introduced by Wośko in [11], where it is proved that W (C[0, 1]) = 1. The same result has been extended in [3] and [9] to other Banach spaces of real continuous functions. On the other hand we observe that there is not a unified method to evaluate W (X), most of the evaluations have required individual constructions in each space X (see, for example, [1,4,10]).…”

Optimal retraction problem for proper $k$-ball-contractive mappings in $C^m [0,1]$

“…Concerning general results in the setting of Banach spaces, in [27] it was proved that W γ (X) ≤ 6 for any Banach space X, reaching the value 4 or 3 depending on the geometry of the space. Moreover it has been proved that W γ (X) = 1 in some spaces of continuous functions ( [7], [15]), in some classical Banach spaces of measurable functions ( [12]) and in Banach spaces whose norm is monotone with respect to some basis ( [3]). In [10] the problem of evaluating the Wośko constant has been considered in the setting of F -normed spaces.…”