Simulation functions on metric fixed point theory

This paper aims to generalize the primary result of Khojasteh et al. by introducing and expanding the concept of simulation functions and weak ζ-contractions under weaker conditions. The Banach contraction principle is extended using the Kummer test, which generalizes this principle to weak ζ-contractions, providing error estimates for various types of contractions, including Boyd and Wong’s contraction. The results not only offer new findings but also improve and complete several earlier works. Additionally, practical examples are provided to illustrate the applications and demonstrate the necessity of certain assumptions in the theory of metric fixed points.