An infinite sequence of nonoscillation theorems for second-order nonlinear differential equations of Euler type
The purpose of this paper is to improve nonoscillation criteria for the nonlinear differential equation \[ t^2 x''+ g(x)= 0,\quad t> 0,\tag{\(*\)} \] where \(g(x)\) satisfies a suitable smoothness condition for the uniqueness of solutions to the initial value problem and the signum condition \[ xg(x)> 0\quad\text{if }x\neq 0. \] Equation \((*)\) is said to be oscillatory (resp. nonoscillatory) in case all nontrivial solutions are oscillatory (resp. nonoscillatory). A nontrivial solution \(x(t)\) to \((*)\) is said to be oscillatory if there exists a sequence \(\{t_n\}\) tending to \(\infty\) such that \(x(t_n)= 0\). Otherwise, it is said to be nonoscillatory. The Riemann-Weber version of the Euler differential equation and positive orbits to a Liénard system are considered.
scientific article; zbMATH DE number 1788018