Fixed Point Methods for Rational-Type Contractions: Applications to Nonlinear Integral and Boundary Value Problems

This paper presents new fixed point results for a rational-type \((\alpha \text {-}F)\) -contraction in super metric spaces, a natural generalization of b -metric spaces. The proposed theorems extend classical fixed point principles and are applied to investigate the existence and uniqueness of solutions to nonlinear integral equations and Volterra-type integral inclusions, which frequently arise in boundary value problems with nonlocal conditions. By employing the super metric space framework, the results allow the treatment of more complex systems exhibiting nonlinear and multiscale behaviors. Illustrative examples are provided to demonstrate the applicability and effectiveness of the theoretical findings. These results contribute to the broader understanding of abstract functional structures and their applications in nonlinear analysis and mathematical physics.