Linearized Stability for a New Class of Neutral Equations with State-Dependent Delay

Abstract

For neutral delay differential equations of the form

$$\begin{aligned} \dot{x}(t)=g(\partial x_t,x_t), \end{aligned}$$

with $g$ defined on an open subset of the space $C([-h,0],\mathbb {R}^n)\times C^1([-h,0],\mathbb {R}^n)$, we extend an earlier principle of linearized stability. The present result applies to a wider class of neutral differential equations

$$\begin{aligned} \dot{x}(t) = f(x(t),\dot{x}(t-\tau (x(t))), x(t-\sigma (x(t)))) \end{aligned}$$

with state-dependent delays which includes models for population dynamics with maturation delay.