## Abstract and Applied Analysis

### Seminormal Structure and Fixed Points of Cyclic Relatively Nonexpansive Mappings

#### Abstract

Let A and B be two nonempty subsets of a Banach space X. A mapping T : $A\cup B\to A\cup B$ is said to be cyclic relatively nonexpansive if T(A) $\subseteq B$ and T(B) $\subseteq A$ and $∥Tx-Ty∥\le ∥x-y∥$ for all ($x\text{,}y$) $\in A{\times}B$. In this paper, we introduce a geometric notion of seminormal structure on a nonempty, bounded, closed, and convex pair of subsets of a Banach space X. It is shown that if (A, B) is a nonempty, weakly compact, and convex pair and (A, B) has seminormal structure, then a cyclic relatively nonexpansive mapping T : $A\cup B\to A\cup B$ has a fixed point. We also discuss stability of fixed points by using the geometric notion of seminormal structure. In the last section, we discuss sufficient conditions which ensure the existence of best proximity points for cyclic contractive type mappings.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 123613, 8 pages.

Dates
First available in Project Euclid: 26 March 2014

https://projecteuclid.org/euclid.aaa/1395858408

Digital Object Identifier
doi:10.1155/2014/123613

Mathematical Reviews number (MathSciNet)
MR3166561

Zentralblatt MATH identifier
07021756

#### Citation

Gabeleh, Moosa; Shahzad, Naseer. Seminormal Structure and Fixed Points of Cyclic Relatively Nonexpansive Mappings. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 123613, 8 pages. doi:10.1155/2014/123613. https://projecteuclid.org/euclid.aaa/1395858408

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