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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 904164, 10 pages
http://dx.doi.org/10.1155/2013/904164
Research Article

Existence and Approximation of Attractive Points of the Widely More Generalized Hybrid Mappings in Hilbert Spaces

1Graduate Institute of Business and Management, College of Management, Chang-Gung University, Kwei-Shan, Taoyuan Hsien 330, Taiwan
2Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo 152-8552, Japan

Received 30 January 2013; Accepted 5 June 2013

Academic Editor: Mohamed Amine Khamsi

Copyright © 2013 Sy-Ming Guu and Wataru Takahashi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the widely more generalized hybrid mappings which have been proposed to unify several well-known nonlinear mappings including the nonexpansive mappings, nonspreading mappings, hybrid mappings, and generalized hybrid mappings. Without the convexity assumption, we will establish the existence theorem and mean convergence theorem for attractive point of the widely more generalized hybrid mappings in a Hilbert space. Moreover, we prove a weak convergence theorem of Mann’s type and a strong convergence theorem of Shimizu and Takahashi’s type for such a wide class of nonlinear mappings in a Hilbert space. Our results can be viewed as a generalization of Kocourek, Takahashi and Yao, and Hojo and Takahashi where they studied the generalized hybrid mappings.