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Zbl 1206.47067
Chantarangsi, Wanpen; Jaiboon, Chaichana; Kumam, Poom
A viscosity hybrid steepest descent method for generalized mixed equilibrium problems and variational inequalities for relaxed cocoercive mapping in Hilbert spaces. (English)
[J] Abstr. Appl. Anal. 2010, Article ID 390972, 39 p. (2010). ISSN 1085-3375; ISSN 1687-0409/e

Summary: We present an iterative method for fixed point problems, generalized mixed equilibrium problems, and variational inequality problems. Our method is based on the so-called viscosity hybrid steepest descent method. Using this method, we can find the common element of the set of fixed points of a nonexpansive mapping, the set of solutions of generalized mixed equilibrium problems, and the set of solutions of variational inequality problems for a relaxed cocoercive mapping in a real Hilbert space. Then, we prove the strong convergence of the proposed iterative scheme to the unique solution of variational inequality. The results presented in this paper generalize and extend some well-known strong convergence theorems in the literature.

MSC 2000:: *47J25 Methods for solving nonlinear operator equations (general)
47J20 Inequalities involving nonlinear operators
47H09 Mappings defined by "shrinking" properties

Keywords: viscosity hybrid steepest descent method; nonexpansive mapping; relaxed cocoercive mapping; real Hilbert space; strong convergence

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