Let X be an infinite dimensional F-normed space and r a positive number such that the closed ball B_r(X) of radius r is properly contained in X. For a bounded subset A of X, the Hausdorff measure of noncompactness gamma(A) of A is the infimum of all $\eps >0$ such that A has a finite $\eps$-net in X. A retraction R of B_r(X) onto its boundary is called k-ball contractive if $\gamma(RA) \le k \gamma(A)$ for each subset A of B_r(X). The main aim of this talk is to give examples of regular F-normed ideal spaces in which there is a 1-ball contractive retraction or, for any $\eps>0$, a $(1+ \eps)$-ball contractive retraction with positive lower Hausdorff measure of noncompactness.
Caponetti, D. (2010). On k-ball contractive retractions in F-normed ideal spaces.
On k-ball contractive retractions in F-normed ideal spaces
CAPONETTI, Diana
2010-01-01
Abstract
Let X be an infinite dimensional F-normed space and r a positive number such that the closed ball B_r(X) of radius r is properly contained in X. For a bounded subset A of X, the Hausdorff measure of noncompactness gamma(A) of A is the infimum of all $\eps >0$ such that A has a finite $\eps$-net in X. A retraction R of B_r(X) onto its boundary is called k-ball contractive if $\gamma(RA) \le k \gamma(A)$ for each subset A of B_r(X). The main aim of this talk is to give examples of regular F-normed ideal spaces in which there is a 1-ball contractive retraction or, for any $\eps>0$, a $(1+ \eps)$-ball contractive retraction with positive lower Hausdorff measure of noncompactness.
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