GENERALIZED AUXILIARY PROBLEM PRINCIPLE AND SOLVABILITY OF A CLASS OF NONLINEAR VARIATIONAL INEQUALITIES INVOLVING COCOERCIVE AND CO-LIPSCHITZIAN MAPPINGS

ABSTRACT. The approximation-solvability of the following class of nonlinear variational in-equality (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 xk ∈ K and, for constants ρ> 0 and σ> 0), find xk+1 such that〈 ρ(S − T) (yk)+ h ′ (xk+1) − h ′ (yk) , x − xk+1〉+ρ(f(x)−f(xk+1)) ≥ 0 for all x ∈ K, where〈 σ(S − T) (xk)+ h ′ (yk) − h ′ (xk) , x − yk 〉 + σ(f(x) − f(yk)) ≥ 0 for all x ∈ K, where h is a functional on K and h ′ the derivative of h. Key words and phrases: Generalized auxiliary variational inequality problem, Cocoercive mappings, Approximation-solvability, Approximate solutions, Partially relaxed monotone mappings. 2000 Mathematics Subject Classification. 49J40. 1