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Existence and iteration for a mixed equilibrium problem and a countable family of nonexpansive mappings in Banach spaces. (English) Zbl 1221.49003

Summary: We prove the existence of a solution of the mixed equilibrium problem (MEP) by using the KKM mapping in a Banach space setting. Then, by virtue of this result, we construct a hybrid algorithm for finding a common element in the solutions set of a mixed equilibrium problem and the fixed points set of a countable family of nonexpansive mappings in the frameworks of Banach spaces. By using a projection technique, we also prove that the sequences generated by the hybrid algorithm converge strongly to a common element in the solutions set of MEP and common fixed points set of nonexpansive mappings. Moreover, some applications concerning the equilibrium and the convex minimization problems are obtained.

MSC:

49J27 Existence theories for problems in abstract spaces
47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
47J20 Variational and other types of inequalities involving nonlinear operators (general)
65J15 Numerical solutions to equations with nonlinear operators
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