Simultaneous iteration for variational inequalities over common solutions for finite families of nonlinear problems

In this paper, we apply Theorem 3.2 of [G. M. Lee, L.-J. Lin, J. Nonlinear Convex Anal., 18 (2017), 1781–1800] to study the variational inequality over split equality fixed point problems for three finite families of strongly quasi-nonexpansive mappings. Then we use this result to study variational inequalities over split equality for three various finite families of nonlinear mappings. We give a unified method to study split equality for three various finite families of nonlinear problems. Our results contain many results on split equality fixed point problems and multiple sets split feasibility problems as special cases. Our results can treat large scale of nonlinear problems by group these problems into finite families of nonlinear problems, then we use simultaneous iteration to find the solutions of these problems. Our results will give a simple and quick method to study large scale of nonlinear problems and will have many applications to study large scale of nonlinear problems.