Oscillation and Block-Diagonalization

Asymptotic methods are used for the two-term differential equation y n.sp x.y, with p x. a real analytic function at infinity. The differential equation is written as a system Y9sA x.Y, and block diagonalization is utilized to study the asymptotic solutions in the case of coalescing eigenvalues. It is shown that the system may be reduced to one standard system of dimension 2=2. We provide a necessary and sufficient condition for various types of oscillation of the two-term differential equation near infinity. This complements Hille]Kneser theory which gives a sufficient condition and, separately, a necessary condition for oscillation, while it leaves some cases unresolved