Abstract

This paper discusses contractive mappings in controlled metric spaces and extended b-metric spaces. It starts with setting up the foundational definitions and properties of the mappings of contractive mappings and associated aspects emphasized in terms of their crucial role in ensuring convergence in iterative processes. Investigation is made in controlled metric spaces on how a controlled approach may increase the flexibility and applicability of contractive mappings, specifically in non-standard metrics. This paper extends the ideas developed here further to b-metric spaces where we discuss what aspects of extending the traditional metric framework are affected. Examples and applications show that such mappings have much importance in various mathematical as well as computational contexts. Results obtained contribute to a broader understanding of fixed-point theory and its potential applications in solving complex problems on different areas that include optimization and dynamical systems.