Proper $k$-ball-contractive mappings in $C_b^m[0, + \infty)$

Abstract

In this paper we deal with the Banach space $C_b^m[0, + \infty)$ of all $m$-times continuously derivable, bounded with all derivatives up to the order $m$, real functions defined on $[0,+ \infty)$. We prove, for any $\epsilon > 0$, the existence of a new proper $k$-ball-contractive retraction with $k < 1+ \epsilon$ of the closed unit ball of the space onto its boundary, so that the Wośko constant $W_\gamma (C_b^m[0, + \infty))$ is equal to $1$.