Applications in Computational Mathematics of Fixed Points in Partial and Fuzzy Metric Spaces

The theoretical underpinnings and computational applications of fixed point theory in partial and fuzzy metric spaces are the main topics of this paper.While fuzzy metric spaces use fuzzy sets to express uncertainty with distances as fuzzy numbers ((,,) [0,1]), partial metric spaces generalize classical metrics by including nonzero self-distances ((,)0).To prove the existence and uniqueness of fixed points under extended contractive conditions, we extend Banach's Contraction Mapping Theorem to these spaces.For example, (( * ), * )=( * , * ) and ((),(),)(,,/).In these contexts, we examine convergence behavior and show that fixed point iterations converge more quickly under fuzzy and partial metrics than in classical metric spaces.Specifically, using Picard iteration to solve nonlinear equations such as -2+1=0 in partial metric space resulted in 30% fewer iterations to convergence.Fuzzy K-means clustering of asymmetric datasets increased intra-cluster similarity by 18% in fuzzy metric applications.Numerical simulations in partial metric space that employed finite element techniques also shown improved accuracy and stability.The practical advantages of partial and fuzzy metric frameworks in enhancing convergence, resilience, and managing imprecise or asymmetric data in real-world computer tasks are validated by these studies.The significance of these generalized metric spaces in improving algorithm design, data analysis, and intelligent systems is further supported by this paper, which connects traditional fixed point theory with current computational issues.