Can a solution of a linear delay differential equation have an infinite number of isolated zeros on a finite interval?

Abstract

A solution of a linear delay differential equation can have an infinite number of isolated zeros on a finite interval. As is well known, for ordinary differential equations this is impossible.As is well known, there exist upper and lower bounds for the distance between two adjacent zeros for a solution of the linear ordinary differential equationThe simplest upper bound is the following: b − a ≤ π µ , where q(t) ≥ µ 2 . Lower bounds can be obtained by Lyapunov's theorem (see Cor. 5In particular, this inequality implies the number of zeros for any solution of (1) is finite on any finite interval (see [1], Chap. IV).The situation for delay differential equations (both first and second order) is absolutely different. For these equations the phenomenon of "sticking together" for two solutions on some interval is possible.

Upper Bounds for the Distance between Adjacent Zeros of First-Order Linear Differential Equations with Several Delays

“…where υ > 0, q ∈ C([t 0 , ∞), [0, ∞)), has attracted the interest of many mathematicians; for example, [4][5][6][7][11][12][13][14][20][21][22][24][25][26][27][28][29][30]. The purpose of most of these works was to obtain new estimations for the upper bound (UB) between successive zeros of all solutions.…”

“…On the other hand, some studies have obtained some fundamental results for the lower bound (LB) between successive zeros of Equation (2); see [5][6][7]13,20,21]. Barr [5] and El-Morshedy [13] proved that the zeros of a solution of Equation ( 2) with an initial function that has a finite number of zeros do not accumulate.…”