Absolute value equations

We investigate existence and nonexistence of solutions for NP-hard equations involving absolute values of variables: Ax − x = b, where A is an arbitrary n × n real matrix. By utilizing an equivalence relation to the linear complementarity problem (LCP) we give existence results for this class of absolute value equations (AVEs) as well as a method of solution for special cases. We also give nonexistence results for our AVE using theorems of the alternative and other arguments.