existence of three solutions for hemivariational inequalities driven with impulsive effects

In this paper we prove the existence of at least three solutions to the following second-order impulsive system: {−(ρ(x)u˙ )′ + a(x)u ∈ λ(∂j(x, u(x)) + µ∂k(x, u(x))), a.e. t ∈ (0, t ),∆(ρ(x)u˙ i(xj )) = ρ(x+)u˙ i(x+) − ρ(x−)u˙ i(x−) = iij (ui(xj )),i = 1, . . . , n, j = 1, . . . , l,α1u˙ (0) − α2u(0) = 0,} β1u˙ (t ) + β2u(t ) = 0, where a : [0; t] → r ^n˟n is a continuous map from the interval [0; t] to the set of n-order symmetric matrixes. the approach is fully based on a recent three critical points theorem of teng [k. teng, two nontrivial solutions for hemivariational inequalities driven by nonlocal elliptic operators, nonlinear anal. (rwa) 14 (2013) 867-874].