An abstract result on asymptotically positively homogeneous differential equations (Q1359573)

This article presents a new result on the nondegeneracy of a vector field \(I-P-M\) in a real Banach space \(X\); here \(P\) is a linear and \(M\) a positive 1-homogeneous operator. The simple abstract result allows the author to prove some new existence theorems of \(\omega\)-periodic solutions for asymptotically positively homogeneous differential equations of the form \[ \dot x=g(t,x)+ h(t,x) \] with a continuous function \(g(t,x): [0,\omega]\times \mathbb{R}^n\to\mathbb{R}^n\), positively homogeneous with respect to \(x\), and a Carathéodory function \(h(t,x): [0,\omega]\times \mathbb{R}^n\to \mathbb{R}^n\) satisfying the condition \[ \limsup_{|x|\to\infty} \frac{|h(t,x)|}{|x|}. \]