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Hybrid iterative scheme by a relaxed extragradient method for solutions of equilibrium problems and a general system of variational inequalities with application to optimization. (English) Zbl 1175.49009
Summary: We introduce a new iterative process for finding the common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem and the solutions of the variational inequality problem for two inverse-strongly monotone mappings. We introduce a new viscosity relaxed extragradient approximation method which is based on the so-called relaxed extragradient method and the viscosity approximation method. We show that the sequence converges strongly to a common element of the above three sets under some parametric controlling conditions. Moreover, using the above theorem, we can apply to finding solutions of a general system of variational inequality and a zero of a maximal monotone operator in a real Hilbert space. The results of this paper extend, improve the results of L.-C. Ceng, C.-Y. Wang and J.-C. Yao [Math. Meth. Oper. Res. 67, No. 3, 375–390 (2008; Zbl 1147.49007)], S. Plubtieng and R. Punpaeng [Appl. Math. Comput. 197, No. 2, 548–558 (2008; Zbl 1154.47053)] Y. Su, M. Shang and X. Qin [Nonlinear Anal., Theory Methods Appl. 69, No. 8 (A), 2709–2719 (2008; Zbl 1170.47047)], L. Li and W. Song [Nonlinear Anal., Hybrid Syst. 1, No. 3, 398–413 (2007; Zbl 1117.49011)] and many others.

MSC:
49J40 Variational inequalities
47J25 Iterative procedures involving nonlinear operators
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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