Global Attractivity of the Zero Solution for Wright's Equation

Abstract

In 1955 E.M. Wright proved that all solutions of the delay differential equation $\dot x(t) = -\alpha (e^{x(t-1)}-1)$ converge to zero as $t\to\infty$ for $\alpha\in(0,3/2]$ and conjectured that this is even true for $\alpha\in(0,\pi/2)$. The present paper proves the conjecture for $\alpha\in[1.5,1.5706]$ (compare with $\pi/2=1.570796\ldots$). The first part of the proof verifies that it is sufficient to guarantee the nonexistence of slowly oscillating periodic solutions, and it shows that slowly oscillating periodic solutions with small amplitudes cannot exist. In the second part a computer-assisted proof is given to exclude slowly oscillating periodic solutions with large amplitudes.