On unique solvability and Picard’s iterative method for absolute value equations

Abstract

In this paper, we deal with unique solvability and numerical solution of absolute value equations (AVE), Ax - B |x| = b, (A, B ∈ ℝ^nxn, b ∈ ℝⁿ). Under some weaker conditions, a simple proof is given for the unique solvability of AVE. Furthermore, we demonstrate with an example that these results are reliable to detect the unique solvability of AVE. These results are also extended to the unique solvability of standard and horizontal linear complementarity problems. Finally, we suggest a Picard iterative method to compute an approximated solution of some uniquely solvable AVE problems where its globally linear convergence is guaranteed via one of our weaker sufficient conditions.