Integral averages and the asymptotic behavior of solutions of second order ordinary differential equations (Q1193165)

Consider the second order differential equation \((*)\;x''+a(t)f(x)=0\), where \(a\in C[t_ 0,\infty)\) and \(f\in C^ 1(\mathbb{R}\backslash\{0\})\). Moreover let \(xf(x)>0\) and \(f'(x)\geq 0\) for \(x\in\mathbb{R}\backslash\{0\}\). Under the hypothesis \(\limsup_{t\to\infty}t^{-1}\int^ t\int^ sa(r)dr ds>-\infty\), that is weaker as the hypothesis used in the previous papers by the author, he proves when \((*)\) has solutions being asymptotically linear.