Generalized Auxiliary Problem Principle and Solvability of a Class of Nonlinear Variational Inequalities Involving Cocoercive and Co-lipschitzian Mappings

The approximation-solvability of the following class of nonlinear variational inequality (NVI) problems, based on a new generalized auxiliary problem principle, is discussed. Find an element x∗ ∈K such that 〈(S − T ) (x∗) , x− x∗〉+ f(x)− f (x∗) ≥ 0 for all x ∈ K, where S, T : K → H are mappings from a nonempty closed convex subset K of a real Hilbert space H into H , and f : K → R is a continuous convex functional on K. The generalized auxiliary problem principle is described as follows: for given iterate x ∈ K and, for constants ρ > 0 and σ > 0), find x such that 〈 ρ(S − T ) ( y ) + h′ ( x ) − h′ ( y ) , x− x 〉 +ρ(f(x)−f(x)) ≥ 0 for all x ∈ K, where 〈 σ(S − T ) ( x ) + h′ ( y ) − h′ ( x ) , x− y 〉 + σ(f(x) − f(y)) ≥ 0 for all x ∈ K, where h is a functional on K and h′ the derivative of h.