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A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. (English) Zbl 1143.65049

The authors investigate the problem of finding a common element of the set of solutions of a mixed equilibrium problem (MEP) of the form
find \(x^*\in C\) such that \(\Theta(x^*,y)+ \varphi(y)- \varphi(x^*)\geq 0\), \(\forall y\in C\), where \(\Theta\) is an equilibrium bifunction, i.e., \(\Theta(u, u)= 0\) for each \(u\in C\)
and the set of common fixed points of finitely many nonexpansive mappings in a real Hilbert space.
First, by using the well-known KKM technique the existence and uniqueness of solutions of the auxiliary problems for the MEP is derived. Second, by virtue of this result a hybrid iterative scheme for finding a common element of the set of solutions of MEP and the set of fixed points of finitely many nonexpansive mappings is introduced. Furthermore, the authors prove that the sequences generated by the hybrid iterative scheme converge strongly to a common element of the set of solutions of MEP and the set of common fixed points of finitely many nonexpansive mappings.

MSC:

65K10 Numerical optimization and variational techniques
49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
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