## Iterative method for fixed point problem, variational inequality and generalized mixed equilibrium problems with applications.(English)Zbl 1247.47073

Summary: We introduce a new iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of a generalized mixed equilibrium problem, and the set of solutions of the variational inequality problem for a co-coercive mapping in a real Hilbert space. Then, strong convergence of the scheme to a common element of the three sets is proved. Furthermore, new convergence results are deduced, and finally, we apply our results to solving optimization problems, and present other applications.

### MSC:

 47J25 Iterative procedures involving nonlinear operators 47H06 Nonlinear accretive operators, dissipative operators, etc. 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47J20 Variational and other types of inequalities involving nonlinear operators (general) 47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
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