Multiplicity results for periodic solutions to delay differential equations via critical point theory

By the critical point theory, we study the existence and multiplicity of periodic solutions to the following system of delay differential equations:(*)x′(t)=-f(x(t-r)),where x∈Rn, f∈C(Rn,Rn), and r>0 is a given constant. A sufficient condition on the existence and multiplicity of periodic solutions for Eq. (*) is obtained. To the authors' knowledge, this is the first time, the existence of periodic solutions to systems of delay differential equations is dealt with by using variational approaches directly.