Expansion mapping in controlled metric space and extended B-metric space

This paper delves into the intricate study of expansion mappings within the frameworks of controlled metric spaces and extended B-metric spaces. Expansion mappings, known for their crucial role in fixed-point theory and iterative processes, are examined under the lens of these generalized metric spaces to uncover their distinct properties and extended applicability. Controlled metric spaces, which incorporate a dynamic control function to modulate the distance measurements, offer a refined approach to traditional metric space concepts. This flexibility allows for a more nuanced understanding of spatial relationships and convergence behaviors. Extended B-metric spaces, by relaxing the stringent requirements of the triangle inequality, open new avenues for theoretical exploration and practical application, accommodating broader classes of functions and sequences. In this study, we aim to provide a comprehensive analysis of the behavior of expansion mappings in these sophisticated metric frameworks. We will present new theoretical results that extend and generalize existing principles, highlighting the interplay between the control functions in controlled metric spaces and the relaxed conditions in extended B-metric spaces. Additionally, we explore practical applications of these findings in various fields, including optimization, computational mathematics, and the analysis of iterative methods. By bridging the gap between classical metric spaces and their generalized counterparts, this paper contributes to a deeper understanding of the mathematical foundations and potential applications of expansion mappings. The results presented herein pave the way for further research and development in this vibrant area of mathematical analysis.