Estimates for green's function of a vector differential equation with variable delays

We consider a vector functional - differential equation with variable delays. Estimates for various norms of the Green function are established. Applications of the derived estimates to equations with nonlinear causal mappings acting in space L² are discussed. Equations with causal mappings include differential-delay, integro-differential and other equations in a Euclidean space. Conditions that provide the absolute L²-stability of the considered equations are obtained. These conditions are explicitly formulated in terms of the entries of the characteristic matrices and Lipschitz constants of nonlinearities. The suggested approach is based on a combined use of the recent estimates for the Euclidean norm of matrix valued functions with some properties of the Green functions.