In this paper, we study the existence and uniqueness of common fixed point of ${\alpha }_{i,j}$- ${\phi }_{E_{M,N}}$-Geraghty contraction mappings and the existence of coincidence point of ${\alpha }_{i,j}$- ${\phi }_{E_N}$-Geraghty contraction mapping in the framework of $b$-metric spaces. We also give two examples to support our results.

In this paper, we mainly studied the existence and uniqueness of fixed points of $\alpha$- $\psi$-Geraghty contractive mappings of integral type in quasi-partial metric spaces. We also gave examples to support our results.

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.