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].