Generalized $({\alpha }_s,\xi ,\hslash ,\tau )$-Geraghty contractive mappings and common fixed point results in partial $b$-metric spaces

In this paper, we introduce two new classes of mixed $(\mathcal{S}\mathcal{,}\mathcal{T})$- $\alpha$-admissible mappings and interspersed $(\mathcal{S},\mathfrak{g},\mathcal{T})$- $\alpha$-admissible mappings and study the sufficient conditions for the existence and uniqueness of a common fixed point of generalized $({\alpha }_s,\xi ,\hslash ,\tau )$-Geraghty contractive mapping in the framework of partial $b$-metric spaces. We also provide two examples to show the applicability and validity of our results. Moreover, we present an application to the existence of solutions to an integral equation by means of one of our results.