Abstract
Based on recent works by Byrne-Censor-Gibali-Reich [C. Byrne, Y. Censor, A. Gibali and S. Reich, The split common null point problem, J. Nonlinear Convex Anal. 13 (2012), 759-775] and the second author [W. Takahashi, Strong convergence theorems for maximal and inverse-strongly monotone mappings in Hilbert spaces and applications, J. Optim. Theory Appl. 157 (2013), 781-802], we study the split common null point problem for maximal monotone mappings in Hilbert spaces which is related to the split feasibility problem by Censor and Elfving [Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms 8 (1994), 221-239]. We first obtain some properties for resolvents of maximal monotone operators in Hilbert spaces. Then using these properties, we establish a strong convergence theorem for finding a solution of the split common null point problem which is characterized as a unique solution of the variational inequality of a nonlinear operator. As applications, we get two new strong convergence theorems which are connected with the split feasibility problem and an equilibrium problem.