A new projection and convergence theorems for the projections in Banach spaces

Abstract

In this paper, we introduce a new projection in a Banach space and show an example of the projections. Then, we study the Mosco convergence of a sequence of nonempty sets concerning the projections in a Banach space.

Shrinking projection algorithms for equilibrium problems with a bifunction defined on the dual space of a Banach space

“…Ibaraki and Takahashi [10] introduced a new resolvent of a maximal monotone operator in Banach spaces and the concept of the generalized nonexpansive mapping in Banach spaces. Honda et al [11], Kohsaka and Takahashi [12] also studied some properties for the generalized nonexpansive retractions in Banach spaces.…”

“…(see [10]) Let C be a nonempty closed subset of a smooth and strictly convex Banach space E such that there exists a sunny generalized nonexpansive retraction R from E onto C. Then, for any x E and z C, the following statements hold:…”

Hybrid method for the equilibrium problem and a family of generalized nonexpansive mappings in Banach spaces

“…Some methods have been proposed to solve the equilibrium problem in a Hilbert space, see for instance, Blum and Oettli [1], and Combettes and Hirstoaga [2]. On the other hand, Ibaraki and Takahashi [3] introduced a new resolvent of a maximal monotone operator in a Banach space and the concept of a generalized nonexpansive mapping in a Banach space. Ibaraki and Takahashi [3], and Kohsaka and Takahashi [5] also studied some properties for generalized nonexpansive retractions in Banach spaces.…”

“…On the other hand, Ibaraki and Takahashi [3] introduced a new resolvent of a maximal monotone operator in a Banach space and the concept of a generalized nonexpansive mapping in a Banach space. Ibaraki and Takahashi [3], and Kohsaka and Takahashi [5] also studied some properties for generalized nonexpansive retractions in Banach spaces. Recently, Takahashi and Zembayashi [12] considered the following equilibrium problem with a bifunction defined on the dual space of a Banach space.…”

Fixed Point Theorems for an Elastic Nonlinear Mapping in Banach Spaces

“…Let any ∈ be fixed, and then (⋅, ) is a convex function because of convexity of ‖⋅‖ 2 . Many nonlinear mappings which are defined by using (⋅, ⋅) are studied (see [2][3][4]). We also defined a nonlinear mapping which is called a -strongly nonexpansive mapping in [5] as follows.…”