This study investigates quadruple fixed point theorems for contractive-type mixed monotone mappings in fuzzy metric spaces. The results establish general conditions ensuring the existence and uniqueness of quadruple fixed point, thereby extending the scope of multi-tupled fixed point theory. The framework is shown to be applicable not only within fuzzy structures but also in complete Cauchy spaces, which strengthens the connection between classical and fuzzy analyses. As a practical application, a nonlinear integral system involving four interdependent variables is examined. The findings demonstrate that the proposed theorems provide a rigorous foundation for proving solvability and uniqueness of such systems. Numerical illustrations are also presented to validate convergence and highlight the practical relevance of the theory. This work contributes to the broader development of fixed point analysis and its applications in solving complex problems characterized by interdependence and uncertainty.
Aniki et al. (Thu,) studied this question.