Existence of Fixed Points of Firmly Nonexpansive-Like Mappings in Banach Spaces

Koji Aoyama,Fumiaki Kohsaka

doi:10.1155/2010/512751

Abstract

AbstractThe aim of this paper is to obtain some existence theorems related to a hybrid projection method and a hybrid shrinking projection method for firmly nonexpansive-like mappings (mappings of type (P)) in a Banach space. The class of mappings of type (P) contains the classes of resolvents of maximal monotone operators in Banach spaces and firmly nonexpansive mappings in Hilbert spaces.

Highlights

Many problems in optimization, such as convex minimization problems, variational inequality problems, minimax problems, and equilibrium problems, can be formulated as the problem of solving the inclusion 0 ∈ Au1.1 for a maximal monotone operator A : E → 2E∗ defined in a Banach space E; see, for example, 1–5 for convex minimization problems, 3, 5, 6 for variational inequality problems, 3, 5, 7 for minimax problems, and 8 for equilibrium problems
In 2000, Solodov and Svaiter 9 proved the following strong convergence theorem for maximal monotone operators in Hilbert spaces
Let H be a Hilbert space, A : H → 2H a maximal monotone operator such that A−10 is nonempty, and Jr the resolvent of A defined by Jr I rA −1 for all r > 0

Summary

Introduction

Many problems in optimization, such as convex minimization problems, variational inequality problems, minimax problems, and equilibrium problems, can be formulated as the problem of solving the inclusion. 1.1 for a maximal monotone operator A : E → 2E∗ defined in a Banach space E; see, for example, 1–5 for convex minimization problems, 3, 5, 6 for variational inequality problems, 3, 5, 7 for minimax problems, and 8 for equilibrium problems. This method is sometimes called a hybrid projection method; see Bauschke and Combettes 10 on more general results for a class of nonlinear operators including that of resolvents of maximal monotone operators in Hilbert spaces. Ohsawa and Takahashi 11 obtained a generalization of Theorem 1.1 for maximal monotone operators in Banach spaces. Kamimura and Takahashi 17 obtained another generalization of Theorem 1.1 for maximal monotone operators in Banach spaces. We show that the boundedness of the generated sequences is equivalent to the existence of fixed points of mappings of type P

Preliminaries

If xn 1

Existence Theorems

Ck and

Deduced Results