General auxiliary problem principle and solvability of a class of nonlinear mixed variational inequalities involving partially relaxed monotone mappings

Abstract

The approximation–solvability of the following class of nonlinear variational inequality (NVI) problems based on a new general auxiliary problem principle is presented: Find an element x∗ ∈ K such that 〈 T(x∗), x − x∗〉 + f (x) − f (x∗) 0 for all x ∈ K, where T : K → H is a partially relaxed monotone mapping from a nonempty closed convex subset K of a real Hilbert space H into H , and f : K → R is a continuous convex function on K . The general auxiliary problem principle is described as follows: for given iterate xk ∈ K and for a constant ρ > 0 , determine xk+1 such that (for k 0 ) 〈 ρT(xk) + ρL(xk+1) + h′(xk+1) − ρL(xk) − h′(xk), x − xk+1〉 + ρ[f (x) − f (xk+1)] 0 for all x ∈ K , where L : K → H is any mapping on K , h : K → R is a function on K and h′ is the derivative of h . Mathematics subject classification (2000): 49J40.