An infinite dimensional 0-epi mapping with degree zero
Let \(X\) be a real Banach space and let \(D\subset X\) be an open and bounded set. A mapping \(T: \overline D\to X\) with \(0\not\in T(\partial D)\) is called 0-epi mapping on \(D\) if the equation \(Tx= g(x)\) is solvable in \(D\) for every compact continuous mapping \(g: \overline D\to X\) which vanishes identically on \(\partial D\). We show that there exists a mapping \(T: \ell^2\supset \overline D\to \ell^2\) such that the mapping \(I- T\) is 0-epi, but the degree \(\deg(I- T- g, D, 0)\) is well defined and equals zero, for any such function \(g\). This says that the degree theory cannot be applied directly to \(I- T- g\) in order to guarantee the solvability of \((I- T)x= g(x)\) in \(D\) for any mapping \(g\) as above. This fact provides a good justification for the study of 0-epi mappings by \textit{M. Furi}, \textit{M. Martelli} and \textit{A. Vignoli} [Ann. Mat. Pura Appl., IV. Ser. 124, 321-343 (1980; Zbl 0456.47051)] in infinite-dimensional spaces.
scientific article; zbMATH DE number 902552