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A general iterative method for addressing mixed equilibrium problems and optimization problems. (English) Zbl 1205.49011
Summary: We introduce a new general iterative method for finding a common element of the set of solutions of a Mixed Equilibrium Problem (MEP), the set of fixed points of an infinite family of nonexpansive mappings \(\{T_n\}^\infty_{n=1}\) and the set of solutions of variational inequalities for a \(\xi \)-inverse-strongly monotone mapping in Hilbert spaces. Furthermore, we establish a strong convergence theorem for the iterative sequence generated by the proposed iterative algorithm under some suitable conditions, which solves some optimization problems. Our results extend and improve the recent results of Y. Yao, M. A. Noor, S. Zainab and Y.-C. Liou [J. Math. Anal. Appl. 354, No. 1, 319–329 (2009; Zbl 1160.49013), Y. Yao, M. A. Noor and Y.-C. Liou [On iterative methods for equilibrium problems, Nonlinear Anal. 70, No. 1, 479–509 (2009)] and many others.

MSC:
49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
49J40 Variational inequalities
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J25 Iterative procedures involving nonlinear operators
49M05 Numerical methods based on necessary conditions
90C99 Mathematical programming
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