Projection Methods for Solving a Class of Split Common Solution Problems

We study the split common solution problem with multiple output sets in real Hilbert spaces. To solve this problem, we propose two new self-adaptive iterative algorithms based on hybrid and shrinking projection methods. Unlike existing approaches, the proposed methods require no prior knowledge of the norms of the transfer operators or the inverse strong monotonicity constants of the associated operators. Under suitable conditions, we establish strong convergence of the generated sequences to a solution of the problem. As applications, we show that our results can be used to solve several related problems, including the split common fixed point problem and the split minimum point problem with multiple output sets. Numerical experiments are presented to illustrate the effectiveness and computational performance of the proposed algorithms.