In this manuscript, we induced several vivid common fixed point theorems for four maps in the setting of complete multiplicative $G_{\mathit{M}}-$metric space via various contractive conditions such as $\mathrm{\Delta }$-implicit contractions. Some new definitions and results are introduced in multiplicative $G_{\mathit{M}}-$ metric space.Moreover, illustrative examples are given to validate our obtained results and an application to a system of nonlinear integral equations are provided to show the novelty of our new results.