2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/174678We succeeded to isolate a special class of concave Young-functions enjoying the so-called \emph{density-level property}. In this class there is a proper subset whose members have each the so-called degree of contraction denoted by $c^{\ast}$, and map bijectively the interval $[ c^{\ast}, \infty) $ onto itself. We constructed the fixed point of each of these functions. Later we proved that every positive number $b$ is the fixed point of a concave Young-function having $b$ as degree of contraction. We showed that every concave Young-function is square integrable with respect to a specific Lebesgue measure. We also proved that the concave Young-functions possessing the density-level property constitute a dense set in the space of concave Young-functions with respect to the distance induced by the $L^{2}$-norm.Analysis of PDEsPrimary 47H10, 47H11,26A18; Secondary 26A06, 26A09,33B30, 37C25text