Common fixed point in $G$-metric spaces via generalized $\Gamma$-$C_F$-simulation function

We present the generalized $\Gamma$-$C_F$-simulation function and establish the common fixed point result for weak $(\eta_F, g)$-contraction in complete $G$-metric space. The exploration extends to its ramifications on both quasi-metric spaces and metric spaces. The study explores the existence of a solution for a non-linear integral equation as an application of these results.