Relaxed contractions in suprametric spaces: A unified framework with applications to nonlinear differential models

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.