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Convergence theorems of iterative algorithms for continuous pseudocontractive mappings. (English) Zbl 1126.47054
In the setting of a reflexive and strictly convex Banach sapce with a uniformly Gâteaux differentiable norm, the autors discuss the convergence of the viscosity iteration process to a solution of the following variational inequality problem: find $x^*\in F(T)$ such that $$\langle (f(x^*)-x^*,j(x-x^*)\rangle\le 0\quad\forall x\in F(T),$$ where $j$ is the normlized dual mapping, $f$ is a Lipschitz strongly pseudoconstractive mapping, and $F(T)$ is the fixed point set of a continuous pseudocontractive mapping. The authors also discuss the convergence of a modified implicit iteration to solve the above problem.

MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H05 Monotone operators (with respect to duality) and generalizations 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces
Full Text:
References:
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