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Abstract and Applied Analysis
Volume 2015, Article ID 760671, 9 pages
Research Article

Fixed Point Theorems for an Elastic Nonlinear Mapping in Banach Spaces

Department of Mathematics, Graduate School of Environment and Information Sciences, Yokohama National University, Tokiwadai, Hodogayaku, Yokohama 240-8501, Japan

Received 17 October 2014; Accepted 8 December 2014

Academic Editor: Kyung Soo Kim

Copyright © 2015 Hiroko Manaka. 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.


Let be a smooth Banach space with a norm . Let for any , where stands for the duality pair and is the normalized duality mapping. We define a -strongly nonexpansive mapping by . This nonlinear mapping is nonexpansive in a Hilbert space. However, we show that there exists a -strongly nonexpansive mapping with fixed points which is not nonexpansive in a Banach space. In this paper, we show a weak convergence theorem and strong convergence theorems for fixed points of this elastic nonlinear mapping and give the existence theorem.