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Viscosity approximation methods for pseudocontractive mappings in Banach spaces. (English) Zbl 1178.47049
Let K be a closed convex subset of a Banach space E and let T:KE be a continuous weakly inward pseudocontractive mapping. Then for t(0,1), there exists a sequence {y t }K satisfying y t =(1-t)f(y t )+tT(y t ), where fΠ K :={f:KK,acontractionwithasuitablecontractiveconstant}. Suppose further that F(T) and E is reflexive and strictly convex which has uniformly Gâteaux differentiable norm. Then it is proved that {y t } converges strongly to a fixed point of T which is also a solution of certain variational inequality. Moreover, an explicit iteration process which converges strongly to a fixed point of T and hence to a solution of certain variational inequality is constructed, provided that T is Lipschitzian.

47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
65J15Equations with nonlinear operators (numerical methods)