The modified split generalized equilibrium problem for quasi-nonexpansive mappings and applications

In this paper, we introduce a new problem, the modified split generalized equilibrium problem, which extends the generalized equilibrium problem, the split equilibrium problem and the split variational inequality problem. We introduce a new method of an iterative scheme [Formula: see text] for finding a common element of the set of solutions of variational inequality problems and the set of common fixed points of a finite family of quasi-nonexpansive mappings and the set of solutions of the modified split generalized equilibrium problem without assuming a demicloseness condition and [Formula: see text] , where T is a quasi-nonexpansive mapping and [Formula: see text] ; a difficult proof in the framework of Hilbert space. In addition, we give a numerical example to support our main result.