In this paper, we introduce $\alpha$-admissible extended $\mathcal{Z}$-contraction in the extended rectangular $b$-metric spaces, then we provide some other conditions in Theorem 3.1, which are different from that in Chifu et al. ^[1], and obtain the existence and uniqueness of fixed point in such spaces. Moreover, some examples are given to show the validity of our main theorems, and we give some corollaries related to our main results. As an application, we apply our main results to solve the existence of solutions for a class of boundary value problems of second order ordinary differential equations.