Attractivity properties of infinite delay Mackey-Glass type equations

Abstract

In this paper, several sufficient conditions are established for the global stability of the positive steady state of a scalar functional differential equation $x'=-Lx_t+f(x_t), \ x\geq 0 \; (1)$. The basic idea of the paper is to reduce an infinite dimensional system generated by $(1)$ in some "friendly" spaces to the study of associated one-dimensional maps. In this way, we improve earlier results concerning not only the scalar Lasota-Wazewska and Mackey-Glass equations with infinite distributed delay but also the multidimensional Goodwin oscillator with infinite delay.