TY - JOUR
T1 - More on Fixed Point Theorem of $\{a,b,c\}$-Generalized Nonexpansive Mappings in Normed Spaces
JO - Analysis in Theory and Applications
VL - 1
SP - 1
EP - 11
PY - 2013
DA - 2013/03
SN - 29
DO - http://doi.org/10.4208/ata.2013.v29.n1.1
UR - https://global-sci.org/intro/article_detail/ata/4509.html
KW - Fixed point theorem, $\{a,b,c\}$-generalized-nonexpansive mapping, asymptotic center, Browder's strong convergence Theorem.
AB - <p style="text-align: justify;">Let $X$ be a weakly Cauchy normed space in which the parallelogram law holds, $C$ be a bounded closed convex subset of $X$ with one contracting point and $T$ be an $\{a,b,c\}$-generalized-nonexpansive mapping from $C$ into $C$. We prove that the infimum of the set $\{\| x-T(x) \|\}$ on $C$ is zero, study some facts concerning the $\{a,b,c\}$-generalized-nonexpansive mapping and prove that the asymptotic center of any bounded sequence with respect to $C$ is singleton. Depending on the fact that the $\{a,b,0\}$-generalized-nonexpansive mapping from $C$ into $C$ has fixed points, accordingly, another version of the Browder&#39;s strong convergence theorem for mappings is given.</p>