Strong convergence results of split equality hierarchical fixed point problems

The primary objective of this study is to approximate a nontrivial solution to a significant extension of the hierarchical fixed point problem, known as the split equality hierarchical fixed point problem, in the context of real Hilbert spaces. We present an accelerated simultaneous Halpern-type iterative algorithm with a step size independent of the operator norms. This method combines accelerated extrapolation and the Halpern-type iterative algorithm. Moreover, we analyze that the sequence pair induced by the suggested algorithm strongly converges to some solution to the split equality hierarchical fixed point problem under some simple control sequence conditions. Furthermore, we provide numerical experiments to demonstrate the rationality and superiority of our algorithm. Some notable recent findings in this field have been improved, generalized, and extended as an outcome of this paper.