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

Abstract 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 {xn} $\{x_{n}\}$ 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 Tω:=(1−ω)I+ωT $T_{\omega }:= (1-\omega)I+ \omega T$, where T is a quasi-nonexpansive mapping and ω∈(0,12) $\omega \in ( 0,\frac{1}{2} ) $; a difficult proof in the framework of Hilbert space. In addition, we give a numerical example to support our main result