Periodic solutions for functional differential equations with periodic delay close to zero

This paper studies the existence of periodic solutions to the delay differential equation $$ dot{x}(t)=f(x(t-muau(t)),epsilon),. $$ The analysis is based on a perturbation method previously used for retarded differential equations with constant delay. By transforming the studied equation into a perturbed non-autonomous ordinary equation and using a bifurcation result and the Poincare procedure for this last equation, we prove the existence of a branch of periodic solutions, for the periodic delay equation, bifurcating from $mu=0$.