Mackey–Glass type delay differential equations near the boundary of absolute stability

For an equation x (t )= −x(t) + ζf (x(t − h)), x ∈ R, f (0) = −1, ζ > 0, with C3- nonlinearity f which has a negative Schwarzian derivative and satisfies xf (x) < 0 for x = 0, we prove the convergence of all solutions to zero when both ζ − 1 > 0 and h(ζ −1)1/8 are less than some constant (independent on h, ζ ). This result gives additional insight to the conjecture about the equivalence between local and global asymptotical stabilities in the Mackey–Glass type delay differential equations.  2002 Elsevier Science (USA). All rights reserved.