Weak and strong convergence for quasi-nonexpansive mappings in Banach spaces
Bull. Korean Math. Soc. 2012 Vol. 49, No. 4, 799-813
https://doi.org/10.4134/BKMS.2012.49.4.799
Published online July 1, 2012
Gang Eun Kim
Pukyong National University
Abstract : In this paper, we first show that the iteration $\{x_{n}\}$ defined by $x_{n+1}=P((1-\alpha_{n})x_{n}+\alpha_{n}TP[\beta_{n}Tx_{n}+(1-\beta_{n})x_{n}])$ converges strongly to some fixed point of $T$ when $E$ is a real uniformly convex Banach space and $T$ is a quasi-nonexpansive non-self mapping satisfying Condition ${\mathbf A}$, which generalizes the result due to Shahzad \cite{Sha}. Next, we show the strong convergence of the Mann iteration process with errors when $E$ is a real uniformly convex Banach space and $T$ is a quasi-nonexpansive self-mapping satisfying Condition ${\mathbf A}$, which generalizes the result due to Senter-Dotson \cite{Sen}. Finally, we show that the iteration $\{x_{n}\}$ defined by $x_{n+1}=\alpha_n S x_{n}+\beta_{n}T[\alpha_{n}^\prime S x_{n}+\beta_n ^{\prime} Tx_n+\gamma_{n}^{\prime}v_{n}]+\gamma_{n}u_{n}$ converges strongly to a common fixed point of $T$ and $S$ when $E$ is a real uniformly convex Banach space and $T, S$ are two quasi-nonexpansive self-mappings satisfying Condition ${\mathbf D}$, which generalizes the result due to Ghosh-Debnath \cite{Gho}.
Keywords : weak and strong convergence, fixed point, Opial's condition, Condition ${\mathbf A}$, Condition ${\mathbf D}$, quasi-nonexpansive mapping
MSC numbers : 47H05, 47H10
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