Oscillation and comparison results in neutral differential equations and their applications to the delay logistic equation (Q912032)

This paper is a contribution to the Proceedings of the Conference on Population Dynamics held at the University of Mississippi, November 1986. It mainly addresses the question whether or not the characterization of oscillatory behavior in terms of the characteristic equation known for delay differential equations is valid for neutral differential equations too. This question has been positively answered since in a more general context than the one of this paper [see the reviewer and the author's paper, J. Differ. Equations 81, No.1, 98-105 (1989; Zbl 0691.34054)]. It is answered here (Theorem 3.1) in a case where the neutral part is dominated by the pure delay part. Other results presented in this paper relate oscillatory properties of equations with nonconstant coefficients or with nonlinear coefficients to those of an appropriate linear equation with constant coefficients. The main tool for that is a comparison theorem (Theorem 2.1) for solutions of differential inequalities. These results apply in particular to the logistic delay equation \[ x'(t)=rx(t)[1-K^{-1}\sum p_ ix(t-\tau_ i)], \] where \(p_ i>0\), \(\sum p_ i=1\), r and \(K>0\). This equation is oscillatory around the carrying capacity K if and only if its linearization around K is oscillatory.