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 $\left\{x_n\right\}$ 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 $T_{\omega }:=(1-\omega )I+\omega T$ , where T is a quasi-nonexpansive mapping and $\omega \in (0,\frac{1}{2})$ ; a difficult proof in the framework of Hilbert space. In addition, we give a numerical example to support our main result.