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The viscosity approximation method for asymptotically nonexpansive mappings in Banach spaces. (English) Zbl 1142.47326
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:
47H06Accretive operators, dissipative operators, etc. (nonlinear)
47H09Mappings defined by “shrinking” properties
47J05Equations involving nonlinear operators (general)
47J25Iterative procedures (nonlinear operator equations)