On The Split Problem In Hilbert Space

In 2011, A. Moudafi proposed the split variational inclusion problem, which generalizes the split feasibility problem. In 2012, A. Moudafi proposed the split e-quality problem, which also generalizes the split feasibility problem. A. Moudafi in-troduced the alternating CQ-algorithms and the relaxed alternating CQ-algorithms for solving split feasibility problem. Recently, S-S. Chang proposed the split e-quality variational inclusion problem by combining the split variational inclusion problem and the split equality problem.In this article we studied the split problem of Hilbert mainly from three aspect-s:split equality problem, split variational inclusion problem and split equality vari-ational inclusion problem. Firstly, we propose a new iterative algorithm for solving split equality problem, trying to find a common solution of split equality problem and the fixed point set of a family of k-asymptotically strictly pseudo-nonspreading mapping, which is more general than that of k-strictly pseudo-nonspreading map-ping and k-asymptotically strictly pseudo-contractions. Then, we introduced the multiple-sets split variational inclusion problems, moreover the algorithms were extended to the original split feasibility problem. Finally we proposed two algo-rithms for solving split equality variational inclusion problem. We also prove the sequences generated by the proposed algorithms converge strongly to a common solution of the split equality variational inclusion problem and fixed points of a family of nonexpansive mappings, which is also an unique solution of a variation-al inequality as an optimality condition for a minimization problem, we gave an example and numerical results to illustrate the convergence of our algorithms.