Existence of solutions for second‐order impulsive differential inclusions

In this paper, we investigate the existence and multiplicity of solutions to the following second‐order impulsive Hamiltonian systems:−ρ ( x ) u ̇′ + A ( x ) u ∈ ∂F ( x , u ( x ) ) , a.e.x ∈ ( 0 , T ) ,Δ ρ ( x )u ̇i ( x j ) = ρx j +u ̇ix j +− ρx j −u ̇ix j −= I ij ( u i ( x j ) ) ,i = 1 , … , N , j = 1 , … , l ,α 1u ̇ ( 0 ) − α 2 u ( 0 ) = 0 ,β 1u ̇ ( T ) + β 2 u ( T ) = 0 ,where A : [ 0 , T ] → R N × Nis a continuousmap form the interval [0, T ] to the set of N‐order symmetric matrices. Our methods are based on critical point theory for nondifferentiable functionals. Copyright © 2014 John Wiley & Sons, Ltd.