New fixed point results in double controlled metric type spaces with applications

The concept of an $\mathcal{F}$-contraction was introduced by Wardowski, while Samet et al. introduced the class of $\alpha$-admissible mappings and the concept of ( $\alpha$- $\psi$)-contractive mapping on complete metric spaces. In this paper, we study and extend two types of contraction mappings: ( $\alpha$- $\psi$)-contraction mapping and ( $\alpha$- $\mathcal{F}$)-contraction mapping, and establish new fixed point results on double controlled metric type spaces. Moreover, we demonstrate some examples and present an application of our result on the existence and uniqueness of the solution for an integral equation.