A proof of the Dancer's conjecture on fixed point indices of mappings in cones.
Let \(C\) be a total cone in a real Banach space \(E\), \(A:C\to C\) be a compact and \(C^1\)-operator, \(y\in C\), \(C_y=\{v\in E:y+tv\in C \text{ for some } t>0\}\) and \(S_y=\{v:\pm v\in \overline{C_y} \}\). Denote by \(\widetilde{S_y}\) the quotient space \(E/S_y\) and by \(\widetilde{A}\) a quotient mapping. Moreover, let \(r(\widetilde{A})\) denote the spectral radius of \(\widetilde{A}\). \textit{E. N. Dancer} [J. Reine Angew. Math. 371, 46--66 (1986; Zbl 0597.47034)] proposed the following conjecture: If \(y\) is an isolated fixed point of \(A\) in \(C\) and \(r(\widetilde{A})>1\), then \(i_C(A,y)=0\). The above conjecture is proved in section 4 of the present paper.
scientific article; zbMATH DE number 2056387