On some weak contractive mappings of integral type and fixed point results in $b$-metric spaces

In the article, we considered the fixed point problem for contractive mappings of integral type in the setting of $b$-metric spaces for the first time. First, we introduced the concepts of $\theta$-weak contraction and $\theta$- $\psi$-weak contraction. Second, the existence and uniqueness of fixed points of contractive mappings of integral type in $b$-metric spaces were studied. Meanwhile, two examples were given to prove the feasibility of our results. As an application, we proved the solvability of a functional equation arising in dynamic programming.