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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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