Hybrid viscosity methods for equilibrium problems, variational inequalities, and fixed point problems

Abstract

In this paper, we introduce a hybrid viscosity iterative algorithm for finding a common element of the set of solutions of a general mixed equilibrium problem, the set of solutions of general system of variational inequalities, the set of common fixed points of one finite family of nonexpansive mappings, and another infinite family of nonexpansive mappings in a real Hilbert space. This hybrid viscosity iterative algorithm is based on viscosity approximation method, Mann’s iterative method, projection method, strongly positive bounded linear Operator, and -mapping approaches. We study the strong convergence of the proposed algorithm to a common element under appropriate assumptions, which also solves an optimization problem. The result presented in this paper improves and extends some known results in the literature.

Keywords:

• hybrid viscosity iterative algorithm
• general mixed equilibrium problem
• general system of variational inequalities
• nonexpansive mapping
• inverse strongly monotone mapping

AMS Subject Classifications:

• Primary: 49J30
• 47H09
• Secondary: 47J20
• 49M05

View correction statement:

Erratum

Acknowledgements

The authors thank the learned reviewers for their valuable comments and appreciation of the work done in the manuscript.

Notes

No potential conflict of interest was reported by the authors.

All authors contributed equally to this work. This article was originally published with errors. This version has been corrected. Please see Erratum (http://dx.doi.org/10.1080/00036811.2015.1061743).