New Fixed Point Theorems for Mappings in Metric and Fuzzy Metric Spaces: An Analytical Approach

Abstract

The Banach Contraction Principle remains one of the most powerful tools of nonlinear functional analysis, and its extensions to more general contractive conditions and to fuzzy metric settings continue to attract active research interest. This study develops a unified analytical framework for fixed point theory in complete metric spaces and complete fuzzy metric spaces (in the sense of George and Veeramani). A new generalized rational-type contractive condition is introduced, combining a linear term, a rational term, and a mixed term, and existence and uniqueness of a fixed point is established together with convergence of the associated Picard iteration. An analogous condition is then formulated in the fuzzy metric setting through a monotone transformation of the membership function, and a parallel existence-uniqueness theorem is proved. A common fixed point theorem for a weakly compatible pair of self-mappings is further obtained as an extension. The theoretical results are illustrated through worked numerical examples, iteration tables, and graphical depictions of convergence, and are compared against classical conditions of Banach, Kannan, and Chatterjea in a summary table. Potential applications to integral equations and to fuzzy-uncertainty models are discussed, and directions for further generalization are indicated.