On Fredholm type integral equations in H\"oder spaces: Existence of solutions

In this study, we investigate the existence of solutions for a quadratic integral equation of the Fredholm type. The analysis is conducted on a finite interval bounded by two arbitrary constants, where the unknown function is associated with a prescribed kernel and two operators satisfying specific regularity and boundedness conditions. The methodological framework of this work is built upon the classical Schauder fixed point theorem, complemented by the structural properties of Hölder spaces. The results obtained not only encompass but also extend several previous contributions in the literature, as the discussion is carried out within a generalized setting over the entire interval. Moreover, an illustrative example is provided to highlight the effectiveness and applicability of the proposed approach.