Controlled 4-dimensional metric type spaces with fixed point results

In this paper, we introduce the concept of controlled four–dimensional metric type spaces, which generalizes the structure of four-dimensional metric spaces by incorporating a control function into the generalized triangle inequality. We investigate several fundamental properties of this newly defined space and present illustrative examples to demonstrate its structure. Moreover, we introduce the concept of ${\alpha }_q$ –admissible mappings and generalize Wardowski's contraction principle by formulating $({\alpha }_q\text{-}F)$ –contractive mappings within the framework of controlled four–dimensional metric type spaces. We also propose a contraction condition defined via an iterative function $P$, thereby extending classical contraction principles to this setting. We prove the existence and uniqueness of fixed points in complete controlled four–dimensional metric type spaces, thereby extending and enhancing several related results in the literature. In addition, illustrative examples are provided for each main theorem to demonstrate the applicability and validity of the obtained results. Furthermore, a non-trivial application to a four-component boundary value problem is presented in Section 5.