Global continuation for bounded solutions of ordinary differential equations (Q1327687)

The author considers a parameter dependent family of differential equations of the form \((*)\) \(dx/dt= \mu F(x,t,\mu)\), where \(F\) is a continuous function of \((x,t,\mu)\in D\times \mathbb{R}\times [0,1]\), and gives some sufficient conditions which guarantee the existence of bounded solutions (i.e., \(\| x\|:= \sup_{t\in \mathbb{R}} | x(t)|<\infty\)) when \(F(x,t,\mu)\) is almost periodic in \(t\in \mathbb{R}\). Here, \(D\) stands for an open set in \(\mathbb{R}^ m\). The result is stated in the form of a theorem whose proof, which is rather long, is based upon applying Conley index theory to a family of skew-product flows associated with \((*)\). As a particular application he derives the sufficient conditions for the equation \(dx/dt= F(x,t)\) to have bounded solutions. Here \(F: \mathbb{R}^ m\times \mathbb{R}\to\mathbb{R}^ m\) is continuous and uniformly almost periodic in \(t\in \mathbb{R}\).