A saddle point theorem for functional state-dependent delay differential equations

The aim of the work is to study the stability of equilibrium points for some functional differential equations with state-dependent delay. As a preliminary step, existence, continuation, uniqueness and smoothness results have been shown for solutions. From a given functional differential equation with state-dependent delay, a linearized equation is constructed which gives sufficient conditions for asymptotic stability of equilibrium solutions. Also, a saddle point theorem is shown in the case where the equilibrium point is a hyperbolic equilibrium point for the linearized equation.