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

Abstract

For equations $x^{\prime }(t)=-x(t)+\zeta f(x(t-h)),x\in \mathbb{R},f^{\prime }(0)=-1,\zeta >0,$ with $C^3$-nonlinearity $f$ which has negative Schwarzian derivative and satisfies $xf(x)<0$ for $x\ne 0$, we prove convergence of all solutions to zero when both $\zeta -1>0$ and $h(\zeta -1{)}^{1/8}$ are less than some constant (independent on $h,\zeta$). 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.

16 pages, 1 figure, accepted for publication in the Journal of Mathematical Analysis and Applications