On a nonlinear second order differential equation with oscillating coefficient (Q1802861)

The nonlinear equation (1) \((p(t)x'(t))'+g(t,x(t))=0\) is considered, where \(p:[0,\infty)\to R\), \(g:[0,\infty)\times R\to R\) are continuous and \(p(t)>0\). Sufficient conditions and necessary conditions are established which ensure that (1) has at least a positive decreasing solution which tends to zero as \(t\to\infty\) and at least a positive decreasing solution which has a nonzero limit as \(t\to\infty\).