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Let C be a nonempty closed convex subset of a Banach space E with the dual E*, let T : C → E* be a continuous mapping, and let S : C → C be a relatively nonexpansive mapping. In this paper, by employing the notion of generalized projection operator we study the variational inequality (for short, VI(T - f,C)): find x ϵ C such that ⩾ 0 for all y ϵ C, where f ϵ E* is a given element. By combining the approximate proximal point scheme both with the modified Ishikawa iteration and with the modified Halpern iteration for relatively nonexpansive mappings, respectively, we propose two modified versions of the approximate proximal point scheme L. C. Ceng and J. C. Yao (2008) for finding approximate solutions of the VI(T - f,C). Moreover, it is proven that these iterative algorithms converge strongly to the same solution of the VI(T - f,C), which is also a fixed point of S. [ABSTRACT FROM AUTHOR]

The article presents a discussion that looks into the wide class of inexact proximal-type methods for handling the generalized variational inequality problem with maximal monotone operator in a reflexive Banach space. An extension of Kamimura and Takahashi's algorithm is also provided with the aim of developing a generic inexact proximal point algorithm. A convergence analysis which unifies a broad class of proximal-type methods associated with finding Hilbert spaces or Banach spaces is also included.