## Abstract and Applied Analysis

### Fixed Point Theorems for an Elastic Nonlinear Mapping in Banach Spaces

Hiroko Manaka

#### Abstract

Let $E$ be a smooth Banach space with a norm $‖·‖$. Let $V(x,y)={‖x‖}^{2}+{‖y‖}^{2}-2{\langle}x,Jy{\rangle}$ for any $x,y\in E$, where ${\langle}·,·{\rangle}$ stands for the duality pair and $J$ is the normalized duality mapping. We define a $V$-strongly nonexpansive mapping by $V(·,·)$. This nonlinear mapping is nonexpansive in a Hilbert space. However, we show that there exists a $V$-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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2015, Special Issue (2015), Article ID 760671, 9 pages.

Dates
First available in Project Euclid: 17 August 2015

https://projecteuclid.org/euclid.aaa/1439816297

Digital Object Identifier
doi:10.1155/2015/760671

Mathematical Reviews number (MathSciNet)
MR3378563

Zentralblatt MATH identifier
06929077

#### Citation

Manaka, Hiroko. Fixed Point Theorems for an Elastic Nonlinear Mapping in Banach Spaces. Abstr. Appl. Anal. 2015, Special Issue (2015), Article ID 760671, 9 pages. doi:10.1155/2015/760671. https://projecteuclid.org/euclid.aaa/1439816297

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