In this study, we characterize a novel contraction mapping referred to as ${\alpha }_i^j$- $\left({\mathbf{D}}_{\mathcal{C}}\left({\mathfrak{P}}_{\hat{E}}\right)\right)$-contraction in light of ${\mathbf{D}}_{\mathcal{C}}$-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 $\mathrm{♭}$-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.

This paper develops a relaxed fixed-point framework for nonlinear operators acting on suprametric spaces. A new class of control functions ${\mathrm{\Theta }}^R$ is introduced, allowing strictly increasing but possibly discontinuous behaviors that go beyond the classical $\vartheta$-contraction structure. Within this setting, several relaxed ${\vartheta }_{\mathcal{R}}$-type contractive conditions are formulated through max-based suprametric functionals and on complete suprametric spaces. These conditions guarantee the existence and uniqueness of fixed-points under a suitable jump requirement on the control function. The theory is supported by explicit examples showing how discontinuities and nonlinear growth patterns influence convergence. Finally, two differential models, namely a second-order particle motion problem and a fourth-order beam equation, are used to demonstrate that their associated integral operators admit unique solutions within the proposed relaxed suprametric framework.