Exact regions of oscillation for a neutral differential equation

Abstract

This paper is concerned with a neutral differential equation with four constant coefficients, one delay and one advancement. By means of the theory of envelopes, we consider all possible values of the parameters involved in the equation and obtain a complete set of necessary and sufficient conditions for all solutions to be oscillatory.

The exact region of oscillation for a first order neutral differential equation with delays

“…The envelope C is composed of four curves C (σ − τ ), λ 1 ), the third to where λ ∈ (λ 1 , λ 2 ), and the fourth to where λ ∈ (λ 2 , +∞). Furthermore, the function y = C (1) IV (x) is strictly negative, strictly decreasing and strictly concave on (0, +∞), the function y = C (2) IV (x) is strictly decreasing and strictly convex on (−∞, p * 1 ), the function y = C (3) IV (x) is strictly decreasing and strictly concave on (p * 2 , p * 1 ), and the function y = C (4) IV (x) is strictly decreasing and strictly convex on (p * 2 , +∞). As in the previous Case I, let Ω IV be the set of all points in the plane which are located strictly above the curve C (2) IV and strictly above the curve C (4) IV :…”

“…As we will see, however, the 'geometric' method of envelopes in [3] can be applied and give rise to convex 'regions of oscillation' in the (p, q)-plane. The nice convexity property will then yield sharp and explicit oscillation criteria for (3), while conditions in [1] are 'algebraic' in nature, and it seems difficult to derive similar oscillation criteria from them. Although the method of envelopes [4] has been employed by the authors in several studies, for the sake of completeness we briefly go through the essentials of the theory of envelopes which are needed in the sequel.…”