Averaging, homotopy, and bounded solutions of ordinary differential equations (Q1174362)

The paper is concerned with the differential equation \[ \dot u=f(u,t)+ g(u,t) \] in \(n\) space variables where both \(f\) and \(g\) are almost periodic in time and satisfy the following additional hypothesis: There is a number \(p\) in the open unit interval such that \(f(\lambda u,t)=\lambda^ pf(u,t)\) for all nonnegative \(\lambda\) and such that \(\lim_{| u | \to \infty} g(u,t)/ | u |^ p=0\). Using the topological methods surrounding the Conley index, it is shown that if \(u \equiv 0\) is the only bounded solution of the averaged equation \(\dot u=\lim_{T \to \infty} {1 \over T} \int_ 0^ Tf(u,t)dt\), then the original differential equation has a bounded solution.