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Strong convergence of a hybrid viscosity approximation method with perturbed mappings for nonexpansive and accretive operators. (English) Zbl 1219.47102
Summary: Recently, H.-K. Xu [J. Math. Anal. Appl. 314, No. 2, 631–643 (2006; Zbl 1086.47060)] considered the iterative method for approximation to zeros of an \(m\)-accretive operator \(A\) in a Banach space \(X\). In this paper, we propose a hybrid viscosity approximation method with perturbed mapping that generates the sequence \(\{x_n\}\) by the algorithm \(x_{n+1}= \alpha_n(u+f(x_n))+ (1-\alpha_n)[J_{r_n}x_n-\lambda_nF( J_{r_n}x_n)]\), where \(\{a_n\}\), \(\{r_n\}\) and \(\{\lambda_n\}\) are three sequences satisfying certain conditions, \(f\) is a contraction on \(X\), \(J_r\) denotes the resolvent \((I+rA)^{-1}\) for \(r>0\), and \(F\) is a perturbed mapping which is both \(\delta\)-strongly accretive and \(\lambda\)-strictly pseudocontractive with \(\delta+\lambda\geq 1\). Under the assumption that \(X\) either has a weakly continuous duality map or is uniformly smooth, we establish some strong convergence theorems for this hybrid viscosity approximation method with perturbed mapping.

MSC:
47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
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