A fresh look at Volterra integral equations: a fixed point approach in $\mathfrak{F}$-bipolar metric spaces

Our aim of the research article was to obtain fixed point results for generalized contractions equipped with precisely defined control functions in the framework of $\mathfrak{F}$-bipolar metric spaces. As a consequence, some well-known results from the literature were derived as special cases of our leading theorems. To validate the theoretical findings and demonstrate their practical relevance, a non-trivial illustrative example is included. Furthermore, we explored the applicability of our major results by addressing the existence and uniqueness of solutions to Volterra integral equations, a class of problems frequently arising in mathematical modeling of real-world systems. In addition, we extended the utility of our fixed point theorems to homotopy theory, where they served as a foundational tool in analyzing the existence of continuous deformations between mappings, thereby contributing to the study of topological invariants and the structural properties of function spaces.