In partial ♭-metric spaces, we first define $F_{\rho \flat}$ -weak contraction mappings and develop fixed point theorems in these mappings. In the context of ♭-metric and partial ♭-metric spaces, this study aims to establish the concept of proximally $F_{\rho \flat}$ -weakly dominated pair of mappings and derive common best proximity point theorems using this pair of mappings. The best proximity point and associated fixed point theorems in the literature are generalized by our new findings. Furthermore, we illustrate our findings with examples. Finally, as evidence for our conclusion, we demonstrate that an integral equation has a solution.
