Linearized instability for differential equations with dependence on the past derivative

Abstract

We provide a criterion for instability of equilibria of equations in the form x ˙ ( t ) = g ( x t ′ , x t ) , which includes neutral delay equations with state-dependent delay. The criterion is based on a lower bound Δ > 0 for the delay in the neutral terms, on regularity assumptions of the functions in the equation, and on spectral assumptions on a semigroup used for approximation. The spectral conditions can be verified studying the associated characteristic equation. Estimates in the C 1 -norm, a manifold containing the state space X 2 of the equation and another manifold contained in X 2 , and an invariant cone method are used for the proof. We also give mostly self-contained proofs for the necessary prerequisites from the constant delay case, and conclude with an application to a mechanical example.