Analyzing Uncertainty through Nonlinear Rational Contractions in Perturbed Metric Spaces

In the framework of perturbed metric spaces introduced by Jleli and Samet, where the effective distance is determined by the exact metric d = \(\mathcal{D}\) − \(\mathcal{P}\), we investigate rational-type contractive conditions that incorporate the perturbation component. First, we introduce a perturbed rational contraction formulated directly in terms of the exact metric d and establish a Banach-type fixed point theorem guaranteeing the existence and uniqueness of a fixed point together with convergence of the Picard iteration. Next, we propose a nonlinear perturbed rational contraction in which the contractive control is expressed through a rational inequality involving the perturbation functional \(\mathcal{P}\). Under the parameter restriction λ(1 + 2a) < 1, we prove a corresponding Banach-type fixed point theorem and derive an explicit linear convergence estimate for the iterative sequence. Several examples are presented to illustrate the applicability of the proposed contractive conditions. As an application, we study a nonlinear Volterra fractional integral equation and show that the associated solution operator admits a unique fixed point under a natural Lipschitz-type condition on the kernel, leading to existence and uniqueness of the solution in an appropriate function space. These results extend classical rational contraction principles to perturbed metric structures and contribute to the development of fixed point theory in generalized metric frameworks involving perturbations.