Relative stability for ascending and positively homogeneous operators on Banach spaces (Q1342481)

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 the author shows that the nonlinear eigenvalue problem \(Tx^*= \lambda^* x^*\) has a unique solution \(x^*\in K- \{0\}\) (up to a scalar), where \(\lambda^*\in \mathbb{R}_ +\) is relatively stable in the sense that for a suitable function \(c\) of \(K\) into \(\mathbb{R}_ +\), \[ \lim_{n\to\infty} {T^ n x\over \lambda^{*n}}= c(x)x^*\qquad\text{for all }x\in K. \] Moreover, he gives an estimate for the speed of convergence.