Quintuple Fixed Point Theorems for Contractive-Type Mixed Monotone Mappings in Cauchy Spaces with Applications to Fuzzy Metric Spaces

Fixed point theory continues to be a fundamental tool in nonlinear analysis, particularly in the study of the existence and uniqueness of solutions under contractive conditions. This work presents new quintuple fixed point theorems for mixed monotone mappings satisfying generalized contractive-type conditions in Cauchy spaces. By extending fixed point theory to five variables within a partially ordered metric structure, the results generalize and unify several earlier findings on coupled and tripled fixed points. The approach accommodates mappings that are neither necessarily continuous nor symmetric, under relatively mild contractive conditions. A major contribution of this work is to fuzzy metric spaces, where the induced metric function from a fuzzy setting is shown to form a Cauchy space, thereby establishing that the main theorems hold under fuzzy uncertainty. A supporting numerical scheme is provided to illustrate the convergence of iterative sequences to a unique quintuple fixed point in the fuzzy spaces. These findings provide a mathematically robust and computationally viable framework for analysing nonlinear problems in uncertain environments, with potential applications in fuzzy systems. These results establish a foundational step toward higher-dimensional fixed point theory, strengthening its applicability across mathematics, computational sciences, and decision models involving uncertainty and complex interdependencies.