STRONG CONVERGENCE THEOREMS FOR GENERALIZED VARIATIONAL INEQUALITIES AND RELATIVELY WEAK NONEXPANSIVE MAPPINGS IN BANACH SPACES

Abstract

Abstract. In this paper, we introduce an iterative sequence by using ahybrid generalized f projection algorithm for nding a common elementof the set of xed points of a relatively weak nonexpansive mapping andthe set of solutions of a generalized variational inequality in a Banachspace. Our results extend and improve the recent ones announced byY. Liu [Strong convergence theorems for variational inequalities and rela-tively weak nonexpansive mappings, J. Glob. Optim. 46 (2010), 319{329],J. Fan, X. Liu and J. Li [Iterative schemes for approximating solutions ofgeneralized variational inequalities in Banach spaces, Nonlinear Analysis70 (2009), 3997{4007], and many others. 1. IntroductionLet B be a Banach space, B be the dual space of B. h;idenotes theduality pairing of B and B. We denote by J : B !2 B the normalizedduality mapping from Bto 2 B , de ned byJ(x) := fv2B : hv;xi= kvk 2 = kxk 2 g; 8x2B:The duality mapping Jhas the following properties:(i) if Bis smooth, then Jis single-valued;(ii) if Bis strictly convex, then Jis one-to-one;(iii) if Bis reexive, then Jis surjective;(iv) if Bis uniformly smooth, then Jis uniformly norm-to-norm continuouson each bounded subset of B.Let Bbe a reexive, strictly convex, smooth Banach space and Jthe dualitymapping from Binto B . Then J is also single-valued, one-to-one, surjective,and it is the duality mapping from B into B, i.e., J J = I. Let K be