On the boundedness of the solutions of the nonlinear oscillating equation (Q1192129)

The differential equation under consideration has the form (1) \(x''+f(x)h(x')x'+g(x)k(x')=0\) in \(\mathbb{R}\), where \(f,g,h,k\) are continuous and \(h,k\) are positive in \(\mathbb{R}\). Conditions are given which are necessary and sufficient for the boundedness of all solutions of (1). The theorems do not require the usual conditions \(f(x)\geq 0\) for all \(x\) or \(xg(x)>0\) for all \(x\neq 0\).