generalized dynamic process for generalized $(psi, s,f)$-contraction with applications in $b$-metric spaces

In this paper, we develop the notion of $(psi, f)$-contraction mappings introduced in [49] in $b$-metric spaces. to achieve this, we introduce the notion of generalized multi-valued $(psi, s, f)$-contraction type i mapping with respect to generalized dynamic process $d(s, t, x_0),$ generalized multi-valued $(psi, s, f)$-contraction type ii mapping with respect to generalized dynamic process $d(s, t, x_0),$ and establish common fixed point results for these classes of mappings in complete $b$-metric spaces. as an application, we obtain the existence of solutions of dynamic programming and integral equations. the results presented in this paper extends and complements some related results in the literature.