On a problem posed by Vladimir Ivanovich Zubov (Q2002683)

Consider the nonautonomous system \[\frac{dx}{dt}= f(x)+ h(t,x),\tag{\(*\)}\] where \(h:\mathbb{R}\times \mathbb{R}^n\to \mathbb{R}^n\) is \(\omega\)-periodic in the first variables under the assumption that the autonomous system \(\frac{dx}{dt}= f(x)\) has the origin \(x=0\) as a globally asymptotically stable equilibrium point. The authors derive conditions on \(f\) and \(h\) such that \((*)\) has for any \(\omega\) at least one \(\omega\)-periodic solution. The proof is based on the mapping degree. The conditions on \(f\) and \(h\) include that \(f\) is coercive and satisfies some extendability conditions and that \(h\) is uniformly bounded.