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Volume 29, Issue 1
More on Fixed Point Theorem of $\{a,b,c\}$-Generalized Nonexpansive Mappings in Normed Spaces

S. M. Ali

Anal. Theory Appl., 29 (2013), pp. 1-11.

Published online: 2013-03

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  • Abstract

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's strong convergence theorem for mappings is given.

  • AMS Subject Headings

42B25, 42B20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{ATA-29-1, author = {}, title = {More on Fixed Point Theorem of $\{a,b,c\}$-Generalized Nonexpansive Mappings in Normed Spaces}, journal = {Analysis in Theory and Applications}, year = {2013}, volume = {29}, number = {1}, pages = {1--11}, abstract = {

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's strong convergence theorem for mappings is given.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2013.v29.n1.1}, url = {http://global-sci.org/intro/article_detail/ata/4509.html} }
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 -

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's strong convergence theorem for mappings is given.

S. M. Ali. (1970). More on Fixed Point Theorem of $\{a,b,c\}$-Generalized Nonexpansive Mappings in Normed Spaces. Analysis in Theory and Applications. 29 (1). 1-11. doi:10.4208/ata.2013.v29.n1.1
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