Attractivity properties of infinite delay Mackey-Glass type equations
The authors study attractivity properties of the unique positive steady state of the scalar functional-differential equation \(x'(t)=-\delta x(t)+ f(x_t),\;\;x\geq 0\), where \(f\) is a nonlinear functional defined in the cone \(C_I^+\) of nonegative continuous functions \(\varphi:I\to\mathbb{R}^+\), \(\mathbb{R}^+=[0,\infty)\), \(I\subseteq (-\infty,0]\). New sufficient conditions for the global stability and some persistence results are established. The main idea of the authors is to use the properties of the one-dimensional map \(h:\mathbb{R}^+\to\mathbb{R}^+\), \(h(c):=f(\varphi_c(\cdot))\), with \(\varphi_c(s)\equiv c\), \(s\in I\), \(c\in\mathbb{R}^+\). By this way, the authors improve some earlier results concerning the scalar Lasota-Wazewska and Mackey-Glass equation, and the multidimensional Goodwin oscillator with finite and infinite delay. The last section of the paper presents a generalization of the attractivity results to the equation \(x'(t)=\int_0^\tau x(t-s) dp(s)+f(x_t)\) with some nonconstant nondecreasing function \(p\), \(p(0)=0\).
scientific article; zbMATH DE number 1839921