Linear complementarity results for Z-matrices on Lorentz cone

Let K ⊃ ^Rn be the n-dimensional Lorentz cone. Given an n×n matrix M and q ε ^Rn, the Lorentz-cone linear complementarity problem LCLCP(M,q) is to find an x ε^Rn that satisfiesx ε K,y:=Mx+q ε Kand^yTx=0. We show that if M is a Z-matrix with respect to K, then M is positive stable if and only if LCLCP(M,q) has a non-empty finite solution set for all q ε ^Rn.