WEAK AND STRONG CONVERGENCE FOR QUASI-NONEXPANSIVE MAPPINGS IN BANACH SPACES

Title & Authors
WEAK AND STRONG CONVERGENCE FOR QUASI-NONEXPANSIVE MAPPINGS IN BANACH SPACES
Kim, Gang-Eun;

Abstract
In this paper, we first show that the iteration {$\small{x_n}$} defined by $\small{x_{n+1}=P((1-{\alpha}_n)x_n +{\alpha}_nTP[{\beta}_nTx_n+(1-{\beta}_n)x_n])}$ 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 A, which generalizes the result due to Shahzad [11]. 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 A, which generalizes the result due to Senter-Dotson [10]. Finally, we show that the iteration {$\small{x_n}$} defined by $\small{x_{n+1}={\alpha}_nSx_n+{\beta}_nT[{\alpha}^{\prime}_nSx_n+{\beta}^{\prime}_nTx_n+{\gamma}^{\prime}_n{\upsilon}_n]+{\gamma}_nu_n}$ 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 D, which generalizes the result due to Ghosh-Debnath [3].
Keywords
weak and strong convergence;fixed point;Opial's condition;Condition A;Condition D;quasi-nonexpansive mapping;
Language
English
Cited by
1.
An algorithm for finding common solutions of various problems in nonlinear operator theory, Fixed Point Theory and Applications, 2014, 2014, 1, 9
2.
Convergence theorems of a new iteration for two nonexpansive mappings, Journal of Inequalities and Applications, 2014, 2014, 1, 82
3.
Modified Halpern-type iterative methods for relatively nonexpansive mappings and maximal monotone operators in Banach spaces, Fixed Point Theory and Applications, 2014, 2014, 1, 237
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