Multiplicity results of periodic solutions for a class of first order delay differential equations

This paper is concerned with the periodic boundary value problemequation(1){u′(t)=−Λu(t+r)−f(t,u(t−r)),u(0)=−u(2r),u(0)=u(4r) where r>0r>0 is a given constant, −π2r<Λ<3π2r is a parameter, and f∈C(R1×Rn,Rn)f∈C(R1×Rn,Rn) satisfies f(t+r,z)=f(t,z)f(t+r,z)=f(t,z) for all z∈Rnz∈Rn. The variational principle is given and some multiplicity results of periodic solutions of (1) are obtained via variational methods.