Global Stability for Price Models with Delay

Abstract

Consider the delay differential equationẋ(t) = a r 0 x(t − s) dη(s) − g(x(t)) and the neutral type differential equationẏ(t) = a r 0ẏ (t − s) dµ(s) − g(y(t)) where a > 0, g : R → R is smooth, ug(u) > 0 for u = 0, s 0 g(u) du → ∞ as |s| → ∞, r > 0, η and µ are nonnegative functions of bounded variation on [0, r], η(0) = η(r) = 0, r 0 η(s) ds = 1, µ is nondecreasing, µ does not have a singular part, r 0 dµ = 1. Both equations can be interpreted as price models. Global asymptotic stability of y = 0 is obtained, in case a ∈ (0, 1), for the neutral equation by using a Lyapunov functional. Then this result is applied to get global asymptotic stability of x = 0 for the (non-neutral) delay differential equation provided a ∈ (0, 1). As particular cases, two related global stability conjectures are solved, with an affirmative answer.

A new stability test for linear neutral differential equations

“…and its particular case with a constant delay lim sup Note that the inequalities 2r 0 |ρ|(2 − r 0 |ρ|) < 3 2 and r 0 ρ < 1 imply r 0 |ρ| < 0.5.…”

“…Many applied problems lead to neutral differential equations as their mathematical models, for example, a model of a controlled motion of a rigid body, a distributed network (a long line with tunnel diodes), models of infection diseases, a price model in economic dynamics, see, for example, [3,14,15]. Though neutral delay differential equations describe important applied models, from mechanics to disease spread in epidemiology, compared to other classes of equations, stability theory for neutral equations with variable coefficients and delays is not sufficiently developed.…”