Linearization of solution operators for state-dependent delay equations: A simple example

Abstract

For state-dependent delay equations, it may easily happen that the equation is not differentiable. This hampers the formulation and the pr∞f of the Principle of Linearized Stability.

The fact that an equation is not dif-ferentiable does not, by itself, imply that the solution operators are not dif-ferentiable. And indeed, the aim of this paper is to present a simple example with differentiable solution operators despite of lack of differentiability of the equation.

The example takes the form of a renewal equation and is motivated by a population dynamical model.

Keywords

• Consumer-resource
• Differentiability of solution operators
• Linearized stability
• Maturation delay
• Size structure
• State-dependent delay