Oscillation properties of 𝑦ⁿ+𝑝𝑦=0

Abstract

The purpose of this paper is to give necessary and sufficient conditions for ( k , n − k ) (k,n - k) disconjugacy of y n + p y = 0 {y^n} + py = 0 . The results are then applied to give counterexamples to the following theorems of Nehari. If n is even and p is positive either all solutions of y n + p y = 0 {y^n} + py = 0 oscillate or none do. If n is even and p is negative and y n + p y = 0 {y^n} + py = 0 has an oscillatory solution then all positive nonoscillatory solutions are either strongly increasing or strongly decreasing.