Convergence theorems of iterative algorithms for continuous pseudocontractive mappings. (English) Zbl 1126.47054

Song, Yisheng; Chen, Rudong

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 67, No. 2, 486-497 (2007).

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 \[ \langle (f(x^*)-x^*,j(x-x^*)\rangle\leq 0\quad\forall x\in F(T), \] 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.

Reviewer: Ya-Ping Fang (Chengdu)

MSC:

47J25	Iterative procedures involving nonlinear operators
47H05	Monotone operators and generalizations
47H10	Fixed-point theorems

Keywords:

pseudocontractive mapping; modified implicit iteration; viscosity iteration; strong convergence; reflexive and strictly convex Banach space; uniformly Gâteaux differentiable norm

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