Oscillation and Nonoscillation Criteria for Second-order Linear Differential Equations

Sufficient conditions for oscillation and nonoscillation of second-order linear equations are established. 1. Statement of the Problem and Formulation of Basic Results Consider the differential equation u′′ + p(t)u = 0, (1) where p : [0, +∞[→ [0, +∞[ is an integrable function. By a solution of equation (1) is understood a function u : [0,+∞[→] − ∞, +∞[ which is locally absolutely continuous together with its first derivative and satisfies this equation almost everywhere. Equation (1) is said to be oscillatory if it has a nontrivial solution with an infinite number of zeros, and nonoscillatory otherwise. It is known (see [1]) that if for some λ < 1 the integral ∫ +∞ sλp(s)ds diverges, then equation (1) is oscillatory. Therefore, we shall always assume below that +∞ ∫ sλp(s)ds < +∞ for λ < 1. 1991 Mathematics Subject Classification. 34C10.