Hybrid Steepest Descent Method for Variational Inequalities

Let (E, ρ) be a metric space and K be a nonempty subset of E. For every x∊E, the distance between the point x and K is denoted by ρ(x, K) and is defined by the following minimum problem: $\rho {\rm{(}}x{\rm{,}}K{\rm{) : = }}\mathop {{\rm{inf}}}\limits_{y \in K} {\rm{ }}\rho {\rm{(}}x{\rm{,}}y{\rm{)}}{\rm{.}}$ The metric projection operator (also called the nearest point mapping) Pk defined on E is a mapping from E to 2K such that $P_K {\rm{(}}x{\rm{) : = \{ }}z{\rm{ }} \in {\rm{ }}K{\rm{ : }}\rho {\rm{(}}x{\rm{, }}z{\rm{) = }}\rho {\rm{(}}x{\rm{,}}K{\rm{)\} }}\forall {\rm{ }}x{\rm{ }} \in {\rm{ }}E{\rm{.}}$.