Focal points of half-linear second order differential equations (Q5942912)

Here, the author studies the distribution of focal points of solutions to the half-linear second-order differential equation \[ (r(t)\Phi(x'))'+c(t)\Phi(x)=0,\quad \Phi(x)=|x|^{p-2}x,\;p>1. \tag{*} \] Recall that a point \(b\) is said to be the focal point to the point \(a<b\) if there exists a nontrivial solution \(x\) to (*) such that \(x'(a)=0=x(b)\). Based on the Riccati technique and the variational principle, various comparison theorems for focal points of two equations of the form (*) are obtained. The situation when on the right-hand-side of (*) a function \(f\) appears instead of \(0\) (nonhomogeneous half-linear equations) is also discussed. The obtained results extend and complete the results given in the papers by \textit{J. Jaroš} and \textit{T. Kusano} [Acta Math. Univ. Comen., New Ser. 68, No. 1, 137-151 (1999; Zbl 0926.34023)] and \textit{R. Navarro} and \textit{J. Sarabia} [Appl. Math. Comput. 79, No. 2-3, 203-206 (1996; Zbl 0867.34019)].