Viscosity approximation methods for nonexpansive mappings and generalized variational inequalities. (Chinese. English summary) Zbl 1223.47109

Zhang, Lijuan; Chen, Junmin; Hou, Zhibin

Acta Math. Sin., Chin. Ser. 53, No. 4, 691-698 (2010).

Summary: Viscosity approximation methods for nonexpansive mappings are studied. Consider the general iteration process \(\{x_n\}\), where \(x_0\in C\) is arbitrary and \(g(x_{n+1})=\alpha_nf(x_n)+(1-\alpha_n)SP_C(g(x_n)-\lambda_n Ax_n)\), \(S\) is a nonexpansive self-mapping of a closed convex subset \(C\) of a Hilbert space \(H\). It is shown that \(\{x_n\}\) converges strongly to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the Noor variational inequality for an inverse strongly monotone mapping which solves some variational inequalities.

MSC:

47J25	Iterative procedures involving nonlinear operators
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J20	Variational and other types of inequalities involving nonlinear operators (general)
49J40	Variational inequalities

Keywords:

viscosity approximation; nonexpansive mapping; variational inequalities; strong convergence