Two-step Systems for G-h-relaxed Pseudococoercive Nonlinear Variational Problems Based on Projection Methods

The approximation-solvability of a generalized system of nonlinear variational inequalities (SNVI) involving relaxed pseudococoercive mappings, based on the convergence of a system of projection methods, is presented. The class of relaxed pseudococoercive mappings is more general than classes of strongly monotone and relaxed cocoercive mappings. Let K1 and K2 be nonempty closed convex subsets of real Hilbert spaces H1 and H2, respectively. The two-step SNVI problem considered here is as follows: find an element (x∗, y∗) ∈ H1 ×H2 such that (g(x∗), g(y∗)) ∈ K1 ×K2 and 〈S(x∗, y∗), g(x)− g(x∗)〉 ≥ 0 ∀ g(x) ∈ K1, 〈T (x∗, y∗), h(y)− h(y∗)〉 ≥ 0 ∀ h(y) ∈ K2, where S : H1 ×H2 → H1, T : H1 ×H2 → H2, g : H1 → H1 and h : H2 → H2 are nonlinear mappings. 2000 Mathematics Subject Classification: 49J40, 65B05.