N ov 2 00 1 Mackey - Glass type delay differential equations near the boundary of absolute stability

For equations x(t) = −x(t) + ζf(x(t − h)), x ∈ R, f (0) = −1, ζ > 0, with C-nonlinearity f which has negative Schwarzian derivative and satisfies xf(x) < 0 for x 6= 0, we prove 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.