The convergence and numerical analysis of shrinking projection algorithms with two iterative sequences and hybrid midpoint rules in Hilbert spaces

In this paper, firstly, based on Halpern iterative algorithm, Mann iterative algorithm, shrinking projection algorithm and hybrid midpoint rules, we introduce two modified strong convergence algorithms for nonexpansive mappings. In theory, this idea for hybrid midpoint rules is meaningful and it has a certain positive effect on the construction of the algorithm and the improvement of its corresponding properties. Next, two strong convergence theorems of the proposed algorithms are established. Then, we present the theoretical application of the main conclusions to the splitting feasibility problem. Finally, we demonstrate the iterative performance of the proposed algorithms through two numerical examples.