ABSTRACT
The Vallee-Poussin theorem about differential inequality plays an important role in the analysis of oscillation properties of solutions and solvability and positivity of solutions to boundary value problems for ordinary differential equations. The Vallee-Poussin theorem generally speaking is not valid for delay and neutral differential equations. In this chapter, we obtain assertions about validity of this theorem for delay and neutral equations. We propose theorem about six equivalences, connecting the assertion on differential inequality, an assertion about integral inequality, estimate of the spectral radius of a corresponding operator, nonoscillation of solution on a corresponding interval and sign-constancy of the Cauchy function and sign-constancy of Green's function of two-point problem. Results of this Chapter is a basis for estimates of distance between adjacent zeros of a solution and new conditions of validity of the Sturm theorem in which conditions on smallness of delay are essentially weaker.
