Exponential stability of linear functional equations with delay

We study the invertibility of the operator I— S , where I is the identity operator and S is an internal superposition operator, that acts in Lebesgue spaces of integrable functions on R + . We prove the relationship between the invertibility of the operator and the exponential stability of the corresponding functional equation. Key results are the solution representation formula, which allows the stability analysis to be reduced to obtaining estimates for the solution of a single functional equation with a unit external perturbation, and a number of statements equivalent to the invertibility of the operator I— S .