Hybrid Approximate Proximal Point Algorithms for Variational Inequalities in Banach Spaces

Lc Ceng,Jc Yao,Sm Guu

doi:10.1155/2009/275208

Abstract

Let be a nonempty closed convex subset of a Banach space with the dual , let be a continuous mapping, and let be a relatively nonexpansive mapping. In this paper, by employing the notion of generalized projection operator we study the variational inequality (for short, VI( ): find such that for all , where 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( ). Moreover, it is proven that these iterative algorithms converge strongly to the same solution of the VI( ), which is also a fixed point of .

Highlights

Let E be a real Banach space with the dual E∗
2 Banach space E: Journal of Inequalities and Applications xn 1 ΠC J−1 Jxn − λ T xn − f, n 1, 2, . . . , 1.2 where J : E → E∗ is the normalized duality mapping on E and ΠC : E → C is the generalized projection operator which assigns to an arbitrary point x ∈ E the minimum point of the functional φ y, x x 2 − 2 x, Jy y 2 with respect to y ∈ C
Strong convergence results on these two iterative algorithms are established; that is, under appropriate conditions, both the sequence {xn} generated by algorithm 1.5 and the sequence {xn} generated by algorithm 1.6 converge strongly to the same point ΠF S x0, which is a solution of the VI T − f, C

Summary

Introduction

Let E be a real Banach space with the dual E∗. As usually, ·, · denotes the duality pairing between E and E∗. If E is a real Hilbert space, ·, · denotes its inner product. Let C be a nonempty closed convex subset of E and T : C → E∗ be a mapping. Given f ∈ E∗, let us consider the following variational inequality problem for short, VI T − f, C : find an element x ∈ C such that y − x, T x − f ≥ 0 ∀y ∈ C. Suppose that the VI T − f, C 1.1 has a unique solution x∗ ∈ C. For any x0 ∈ C, define the following successive sequence in a uniformly convex and uniformly smooth

Banach space E

Preliminaries

Main Results