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A new hybrid iterative method for mixed equilibrium problems and variational inequality problem for relaxed cocoercive mappings with application to optimization problems. (English) Zbl 1221.49010
Summary: We introduce and study a new hybrid iterative method for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of variational inequalities for a \(\xi \)-Lipschitz continuous and relaxed \((m,v)\)-cocoercive mappings in Hilbert spaces. Then, we prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm which solves some optimization problems under some suitable conditions. 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), doi:10.1016/j.jmaa.2008.12.005] and X. Gao and Y. Guo [J. Inequal. Appl. 2008, Article ID 454181, 23 p. (2008; Zbl 1153.49011), doi:10.1155/2008/454181] and many others.

49J40 Variational inequalities
47J25 Iterative procedures involving nonlinear operators
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
Full Text: DOI
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