Existence and multiplicity of periodic solutions for second-order systems at resonance

Abstract

The purpose of this paper is to prove the existence and multiplicity of periodic solutions for the following second-order system at resonance: { u ̈ + m 2 w 2 u = ∇ F ( t , u ) , a.e. t ∈ [ 0 , T ] , u ̇ ( 0 ) − u ̇ ( T ) = u ( 0 ) − u ( T ) = 0 , where m > 0 . Some new existence theorems are obtained by the variational methods.