In this study, the authors verify fixed-point results for Geraghty contractions with a restricted co-domain of the auxiliary function in the context of generalized metric structure, namely the $S_b$-metric space. This new idea of defining Geraghty contraction for self-operators generalizes a large number of previously published, closely related works on the presence and uniqueness of a fixed point in $S_b$-metric space. Also, the outcomes are achieved by removing the continuity constraint of self-operators. We also provide examples to elaborate on the obtained results and an application to the integral equation to illustrate the significance in the literature.