TY - JOUR
T1 - Fixed Point Theorem of $\{a,b,c\}$ Contraction and Nonexpansive Type Mappings in Weakly Cauchy Normed Spaces
JO - Analysis in Theory and Applications
VL - 3
SP - 280
EP - 288
PY - 2013
DA - 2013/07
SN - 29
DO - http://doi.org/10.4208/ata.2013.v29.n3.8
UR - https://global-sci.org/intro/article_detail/ata/5064.html
KW - Fixed point, generalized type of contaction and nonexpansive mappings, normed space.
AB - <p style="text-align: justify; margin-bottom: 10px;">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:<br/>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 $T$.<br/>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 $T$.<br/>This paper extends and generalizes some of the results given in [2,4,7] and [13].</p>