Abstract
In this paper, we prove a strong convergence theorem for a new hybrid method, using shrinking projection method introduced by Takahashi and a fixed point method for finding a common element of the set of solutions of mixed equilibrium problem and the set of common fixed points of a countable family of multivalued nonexpansive mappings in Hilbert spaces. We also apply our main result to the convex minimization problem and the fixed point problem of a countable family of multivalued nonexpansive mappings.
Highlights
The mixed equilibrium problem (MEP) includes several important problems arising in optimization, economics, physics, engineering, transportation, network, Nash equilibrium problems in noncooperative games, and others
We use the shrinking projection method defined by Takahashi [ ] and our new method to define a new hybrid method for MEP and a fixed point problem for a family of nonexpansive multivalued mappings
We introduce a new hybrid method for finding a common element of the set of solutions of a mixed equilibrium problem and the set of common fixed points of a countable family of multivalued nonexpansive mappings in Hilbert spaces
Summary
The mixed equilibrium problem (MEP) includes several important problems arising in optimization, economics, physics, engineering, transportation, network, Nash equilibrium problems in noncooperative games, and others. In , Wangkeeree and Wangkeeree [ ] proved a strong convergence theorem of an iterative algorithm based on extragradient method for finding a common element of the set of solutions of a mixed equilibrium problem, the set of common fixed points of a family of infinitely nonexpansive mappings and the set of the variational inequality for a monotone Lipschitz continuous mapping in a Hilbert space.
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