Fixed-point results via $ \alpha_{i}^{j} $-$ \left({\bf D}_{{\mathscr{C}}}\left(\mathfrak{P}_{\hat E}\right)\right) $-contractions in partial $ \flat $-metric spaces

In this study, we characterize a novel contraction mapping referred to as $ \alpha_{i}^{j} $-$ \left({\bf D}_{{\mathscr{C}}}\left(\mathfrak{P}_{\hat E}\right)\right) $-contraction in light of $ {\bf D}_{\mathscr{C}} $-contraction mappings associated with the Geraghty-type contraction and $ E $-type contraction. Besides, a novel common fixed-point theorem providing such mappings is demonstrated in the context of partial $ \flat $-metric spaces. It is stated that the main theorem is a generalization of the existing literature, and its comparisons with the results are expressed. Additionally, the efficiency of the result of this study is demonstrated through some examples and an application to homotopy theory.