THE SPLIT COMMON NULL POINT PROBLEM AND HALPERN-TYPE STRONG CONVERGENCE THEOREM IN HILBERT SPACES

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 third 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 obtain a Halpern-type strong convergence theorem for finding a solution of the split common null point problem for three maximal monotone mappings which is related to the split feasibility problem by Censor and Elfying [Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms 8 (1994), 221-239]. The solution of the split common null point problem 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 common null point problem and an equilibrium problem.