Single-Step Algorithm for Variational Inequality Problems in Banach Spaces

Abstract

In this work, we propose a one-step algorithm for solving variational inequality problems in a 2-uniformly con vex Banach space. Weak convergence of the scheme to a solution of variational inequality is established under reasonable assumptions. More precisely, we proved the following theorem: Theorem Let E be a real 2-uniformly convex and uniformly smooth Banach space. Let C be nonempty closed convex subset of E. A : E → E∗ be monotone and Lipschitz with Lipschitz constant L. Let x0, x−1 ∈ E and defined the sequence {xn} by xn+1 = ΠC J−1 (Jxn − λnAxn − λn−1(Axn − Axn−1)), n ≥ 0; where {λn} ⊆ h , 1−2 2µL i for some > 0 and µ ≥ 1. Suppose Γ is nonempty and that the normalized duality mapping J is weakly sequentially continuous, then the sequence {xn} converges weakly to an element of Γ. Applications are also presented to show how our result can be applied to real life problems.