An existence-uniqueness theorem and alternating contraction projection methods for inverse variational inequalities

Let C be a nonempty closed convex subset of a real Hilbert space [Formula: see text] with inner product [Formula: see text] , and let [Formula: see text] be a nonlinear operator. Consider the inverse variational inequality (in short, [Formula: see text] ) problem of finding a point [Formula: see text] such that [Formula: see text] In this paper, we prove that [Formula: see text] has a unique solution if f is Lipschitz continuous and strongly monotone, which essentially improves the relevant result in (Luo and Yang in Optim. Lett. 8:1261–1272, 2014). Based on this result, an iterative algorithm, named the alternating contraction projection method (ACPM), is proposed for solving Lipschitz continuous and strongly monotone inverse variational inequalities. The strong convergence of the ACPM is proved and the convergence rate estimate is obtained. Furthermore, for the case that the structure of C is very complex and the projection operator [Formula: see text] is difficult to calculate, we introduce the alternating contraction relaxation projection method (ACRPM) and prove its strong convergence. Some numerical experiments are provided to show the practicability and effectiveness of our algorithms. Our results in this paper extend and improve the related existing results.