Fixed point solutions of variational inequality and generalized equilibrium problems with applications

Abstract

In this paper, we introduce a new iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solution of generalized 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 obtaining zeroes of maximal monotone operators and co-coercive mappings.

A new hybrid algorithm for a system of equilibrium problems and variational inclusion

“…It is easy to see that for any λ constant is in (0, 2β], then the mapping I − λB is nonexpansive, where I is the identity mapping on H. The generalized mixed equilibrium problem include fixed point problems, optimization problems, variational inequalities problems, Nash equilibrium problems, noncooperative games, economics and the equilibrium problems as special cases (see, e.g., [2,8,9,20,22,31]). Some methods have been proposed to solve the equilibrium problem; see, for instance [6,10,[12][13][14][15][16]19,25,29,30].…”