Common fixed point ($\alpha_*$-$\psi$-$\beta_{i}$)-contractive set-valued‎ ‎mappings on orthogonal Branciari $S_{b}$-metric space

In [24], Khan et al. established some fixed point theorems in complete and compact metric spaces by using altering distance functions. In [16] Gordji et al. described the notion of orthogonal set and orthogonal metric spaces. In [18] Gungor et al. established fixed point theorems on orthogonal metric spaces via altering distance functions. In [25] Lotfy et al introduced the notion of $\alpha_{*}$-$\psi$-common rational type mappings on generalized metric spaces with application to fractional integral equations. In [28] K. Royy et al. described the notion of Branciari $S_b$-metric space and related fixed point theorems with an application. In this paper, we introduce the notion of the common fixed point ($\alpha_*$-$\psi$-$\beta_{i}$)-contractive set-valued mappings on orthogonal Branciari $S_{b}$-metric space with the application of the existence of a unique solution to an initial value problem.