Remarks on \(u_{0}\)-positive operators (Q1031976)

The notion of a \(u_0\)-positive operator introduced by \textit{M.\,A.\thinspace Krasnosel'skij} in [``Positive solutions of operator equations'' (Groningen: P.\,Noordhoff) (1964; Zbl 0121.10604)] is used to prove the existence and nonexistence of fixed points of some nonlinear maps in cones. The spectral radius of a \(u_0\)-positive operator, which appears to be a unique positive eigenvalue with eigenfunction in the cone, plays the crucial role in proofs. The paper ends with one result on nonexistence of nontrivial positive solutions to Hammerstein integral equations.