Abstract :
Let X be a real Banach space and let D;X be an open and bounded set. A
mapping T: DªX with 0fT D.is called 0-epi mapping on D if the equation
Txsg x. is solvable in D for every compact continuous mapping g: DªX
which vanishes identically on D. We show that there exists a mapping T:
l2>Dªl2such that the mapping IyT is 0-epi, but the degree deg IyTyg,
D, 0. is well defined and equals zero, for any such function g. This says that the
degree theory cannot be applied directly to IyTyg in order to guarantee the
solvability of IyT.xsg x.in D for any mapping g as above. This fact provides
a good justification for the study of 0-epi mappings by M. Furi, M. Martelli, and A.
Vignoli Ann. Mat. Pura Appl. 124, 1980, 321]343.in infinite dimensional spaces.
