(αη)-contractive and (βχ)-contractive mapping based fixed point theorems in fuzzy bipolar metric spaces and application to nonlinear Volterra integral equations

In this paper, we introduce some novel concepts within the realm of fuzzy bipolar metric spaces, namely ($\alpha \eta$)-contractive type covariant mappings and contravariant mappings, and ($\beta \chi$)-contractive type covariant mappings. We establish some fixed point theorems that demonstrate both the existence and uniqueness of fixed points for ($\alpha \eta$)-contractive type covariant mappings and contravariant mappings, and for ($\beta \chi$)-contractive type covariant mappings in complete fuzzy bipolar metric spaces utilizing the triangular property. Additionally, to substantiate the findings, some illustrative examples and consequential outcomes are presented. Furthermore, the proven results serve to extend, generalize, and enhance the corresponding outcomes documented in existing literature. A practical application of these findings in the context of non-linear Volterra integral equations is demonstrated, solidifying and reinforcing the credibility of the established results. Overall, this paper contributes to the understanding of fixed point theory in the context of fuzzy bipolar metric spaces and highlights the significance of ($\alpha \eta$)-contractive mappings and ($\beta \chi$)-contractive mappings in this domain.