Relative Stability for Ascending and Positively Homogeneous Operators on Banach Spaces

For a closed convex cone K in a real Banach space with a norm increasing on K, and a selfmapping T of K which is continuous, positively homogeneous, and ascending it is shown that the nonlinear eigenvalue problem Tx* = λ*x* has a unique solution x* ∈ K − {0} (up to a scalar), λ* ∈ R+ which is relatively stable in the sense that for a suitable function c of K into R+[formula] Moreover, an estimate for the speed of convergence is given.