We introduce iterative scheme generating methods (ISGMs) to find common fixed points of nonlinear mappings through shrinking projection methods, leading to strong convergence theorems. These findings extend a recent study by Kondo [A. Kondo, Math. Ann. 391 (2025), 2007-2028], which only demonstrates weak convergence. Although ISGMs combined with shrinking projection methods were explored in a prior study [A. Kondo, Carpathian J. Math. 40 (2024), 819-840], that work depended on mean-valued sequence properties. This study develops ISGMs without relying on mean-valued sequences, yielding infinitely many strong convergence theorems. An application to a common split feasibility problem is also presented.

The author would like to thank the Ryousui Gakujutsu Foundation and the Institute for Economics and Business Research of Shiga University for financial support. The author is also grateful to the anonymous referees for their attentive reading and helpful comments.