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

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