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Zbl 1142.47326
Ceng, Lu-Chuan; Xu, Hong-Kun; Yao, Jen-Chih
The viscosity approximation method for asymptotically nonexpansive mappings in Banach spaces. (English)
[J] Nonlinear Anal., Theory Methods Appl. 69, No. 4, A, 1402-1412 (2008). ISSN 0362-546X

Summary: A recent trend in the iterative methods for constructing fixed points of nonlinear mappings is to use the viscosity approximation technique. The advantage of this technique is that one can find a particular solution to the associated problems, and in most cases this particular solution solves some variational inequality. In this paper, we try to extend this technique to find a particular common fixed point of a finite family of asymptotically nonexpansive mappings in a Banach space which is reflexive and has a weakly continuous duality map. Both implicit and explicit viscosity approximation schemes are proposed and their strong convergence to a solution to a variational inequality is proved.

MSC 2000:: *47H06 Accretive operators, etc. (nonlinear)
47H09 Mappings defined by "shrinking" properties
47J05 Equations involving nonlinear operators (general)
47J25 Methods for solving nonlinear operator equations (general)

Keywords: viscosity approximation; asymptotically nonexpansive mapping; variational inequality; weak continuous duality map

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