On oscillation of solutions of scalar delay differential equation in critical case

Abstract

In this paper we study the oscillation problem for the known scalar delay differential equation. We assume that the coefficients of this equation have an oscillatory behaviour with an amplitude of oscillation tending to zero at infinity. The asymptotic formulae for the solutions of the considered equation in the so-called critical case are constructed. We give the conditions for existence of oscillatory or nonoscillatory solutions in terms of certain numerical quantities. The obtained results are illustrated by a number of examples.

Keywords

• asymptotic integration
• delay differential equation
• oscillation problem
• center manifold
• levinson's theorem