Weak and strong convergence of an inertial proximal method for solving Ky Fan minimax inequalities

In this paper we present the relaxed inertial proximal algorithm for Ky Fan minimax inequalities. Based on Opial lemma, we propose a weak convergence result to a solution of the problem by eliminating in the algorithm (RIPAFAN) the Browder–Halpern’s factor of contraction. We present after, a first result of strong convergence by adding a strong monotonicity condition. Secondly, we eliminate the strong monotonicity and add a Browder–Halpern’s contraction factor in the algorithm (RIPAFAN) and then ensure the strong convergence to a selected solution with respect to the contraction factor. Some examples are proposed. The first one concerns the convex minimization where the objective function is only controlled with a provided well conditioning. In the second one, we propose monotone set-valued variational inequalities. The last example deals with the problem of fixed point for a nonexpansive set-valued operator.