Stability and existence of multiple periodic solutions for a quasilinear differential equation with maxima

We study the stability of periodic solutions of the scalar delay differential equation x' = -deltax(t) + p max(u is an element of [t-h,t]) x(u) + f(t), (*) where f(t) is a periodic forcing term and delta, p is an element of R. We study stability in the first approximation showing that the non-smooth equation (*) can be linearized along some 'non-singular' periodic solutions. Then the corresponding variational equation together with the Krasnosel'skij index are used to prove the existence of multiple periodic solutions to (*). Finally, we apply a generalization of Halanay's inequality to establish conditions for global attractivity in equations with maxima.