Some Best Proximity Theorems for Generalized Proximal $\mathcal{Z}$-Contraction Maps in $b$-Metric Spaces with Applications

The simulation function and the notion of $\mathcal{Z}$-contraction concerning $\zeta$ which generalizes the Banach contraction principle and unifies several known types of contractions involving the combination of $d(fx, fy)$ and $d(x, y)$. In this paper, we introduce generalized proximal $\mathcal{Z}$-contraction maps and prove the existence and uniqueness of the best proximity points in complete $b$-metric spaces. Our results extend and generalize the results of Olgun et al. \cite{Olgun}, Abbas et al. \cite{Abbas} and Goswami et al. \cite{Goswami2024} from the metric space setting to b-metric spaces. We provide examples in support of our results. We provide applications to demonstrate the significance of our theoretical conclusions by solving nonlinear integral equations of Fredholm type and functional equations that arise in dynamic programming.