Cloud hybrid methods for solving split equilibrium and fixed point problems for a family of countable quasi-Lipschitz mappings and applications

The purpose of this article is to introduce a new multidirectional hybrid shrinking projection iterative algorithm (or called cloud hybrid shrinking projection iterative algorithm) for solving the common element problems which consist of a generalized split equilibrium problems and fixed point problems for a family of countable quasi-Lipschitz mappings in the framework of Hilbert spaces. It is proved that under appropriate conditions, the sequence generated by the multidirectional hybrid shrinking projection method, converges strongly to some point which is the common fixed point of a family of countable quasi-Lipschitz mappings and the solution of the generalized split equilibrium problems. This iteration algorithm can accelerate the convergence speed of iterative sequence. The main results were also applied to solve split variational inequality problem and split optimization problems. Meanwhile, the main results were also used for solving common problems which consist of a generalized split equilibrium problems and fixed point problems for asymptotically nonexpansive mappings. The results of this paper improve and extend the previous results given in the literature.