Let X be a Banach space of differentiable functions and A: X
|| Ax1 - Ax2 \mathord/ |
\vphantom Ax1 - Ax2 X X || \leqslant k|| x1 - x2 \mathord | / |
\vphantom x1 - x2 X X ||( x1 ,x2 Î X ),\left\| {{{Ax_1 - Ax_2 } \mathord{\left/ {\vphantom {{Ax_1 - Ax_2 } X}} \right. \kern-\nulldelimiterspace} X}} \right\| \leqslant k\left\| {{{x_1 - x_2 } \mathord{\left/ {\vphantom {{x_1 - x_2 } X}} \right. \kern-\nulldelimiterspace} X}} \right\|\left( {x_1 ,x_2 \in X} \right),
| ((1)) |
a(AM \mathord | / |
\vphantom AM X X) \leqslant Ka(M \mathord | / |
\vphantom M X X)(M Ì Xbounded),\alpha ({{AM} \mathord{\left/ {\vphantom {{AM} X}} \right. \kern-\nulldelimiterspace} X}) \leqslant K\alpha ({M \mathord{\left/ {\vphantom {M X}} \right. \kern-\nulldelimiterspace} X})(M \subset Xbounded),
| ((2)) |
where
(M/X) denotes the Hausdorff measure of noncompactness. We caracterize (1) e (2) in some nonideal spaces X which occur frequently in applications.
Work supported by MPI — Italy.