Fixed Point Theorem of $\{a,b,c\}$ Contraction and Nonexpansive Type Mappings in Weakly Cauchy Normed Spaces
Year: 2013
Analysis in Theory and Applications, Vol. 29 (2013), Iss. 3 : pp. 280–288
Abstract
Let $C$ be a closed convex weakly Cauchy subset of a normed space $X$. Then we define a new $\{a,b,c\}$ type nonexpansive and $\{a,b,c\}$ type contraction mapping $T$ from $C$ into $C$. These types of mappings will be denoted respectively by $\{a,b,c\}$-$n$type and $\{a,b,c\}$-$c$type. We proved the following:
1. If $T$ is $\{a,b,c\}$-$n$type mapping, then $\inf\{\|T(x)-x\|:x\in C\}=0$, accordingly $T$ has a unique fixed point. Moreover, any sequence $\{x_{n}\}_{n\in \mathcal{N}}$ in $C$ with $\lim_{n\to \infty}\|T(x_{n})-x_{n}\|=0$ has a subsequence strongly convergent to the unique fixed point of $T$.
2. If $T$ is $\{a,b,c\}$-$c$type mapping, then $T$ has a unique fixed point. Moreover, for any $x\in C$ the sequence of iterates $\{T^{n}(x)\}_{n\in \mathcal{N}}$ has subsequence strongly convergent to the unique fixed point of $T$.
This paper extends and generalizes some of the results given in [2,4,7] and [13].
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/ata.2013.v29.n3.8
Analysis in Theory and Applications, Vol. 29 (2013), Iss. 3 : pp. 280–288
Published online: 2013-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 9
Keywords: Fixed point generalized type of contaction and nonexpansive mappings normed space.
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