best proximity point theorems by $k$, $c$ and $mathcal{mt}$ types in $b$-metric spaces with an application

In this paper, we introduce the concept of weak $mathcal{mt}-k$ rational cyclic and weak $mathcal{mt}-c$  rational cyclic conditions and a combination of both conditions in what we call weak $mathcal{mt}-kc$ rational cyclic condition. we investigate some best proximity points theorems for a pair of mappings that satisfy these conditions that have been established in $b$-metric spaces. our results include an application to the nonlinear integral equation as well.