Modified hybrid iterative methods for generalized mixed equilibrium, variational inequality and fixed point problems

Volume 10, Issue 7, pp 3732--3754 http://dx.doi.org/10.22436/jnsa.010.07.30

Publication Date: July 25, 2017 Submission Date: December 02, 2016

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Authors

Jong Soo Jung - Department of Mathematics, Dong-A University, Busan 49315, Korea.

Abstract

In this paper, we introduce two modified hybrid iterative methods (one implicit method and one explicit method) for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of solutions of a variational inequality problem for a continuous monotone mapping and the set of fixed points of a continuous pseudocontractive mapping in Hilbert spaces, and show under suitable control conditions that the sequences generated by the proposed iterative methods converge strongly to a common element of three sets, which solves a certain variational inequality. As a direct consequence, we obtain the unique minimum-norm common point of three sets. The results in this paper substantially improve upon, develop and complement the previous well-known results in this area.

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Keywords

• Hybrid iterative method
• generalized mixed equilibrium problem
• continuous monotone mapping
• continuous pseudocontractive mapping
• variational inequality
• fixed point
• \(\rho\)-Lipschitzian and \(\eta\)-strongly monotone mapping
• metric projection.

MSC

•  49J30
•  49J40
•  47H09
•  47H10
•  47J20
•  47J25
•  47J05
•  49M05

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