In this paper, we introduce some novel concepts within the realm of fuzzy bipolar metric spaces, namely ()-contractive type covariant mappings and contravariant mappings, and ()-contractive type covariant mappings. We establish some fixed point theorems that demonstrate both the existence and uniqueness of fixed points for ()-contractive type covariant mappings and contravariant mappings, and for ()-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 ()-contractive mappings and ()-contractive mappings in this domain.

中文翻译：

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模糊双极度量空间中基于([公式省略])-收缩和([公式省略])-收缩映射的不动点定理及其在非线性Volterra积分方程中的应用


在本文中，我们介绍了模糊双极度量空间领域内的一些新概念，即()-收缩型协变映射和逆变映射，以及()-收缩型协变映射。我们建立了一些不动点定理，证明了 ()-收缩型协变映射和逆变映射以及利用三角性质的完全模糊双极度量空间中 ()-收缩型协变映射的不动点的存在性和唯一性。此外，为了证实研究结果，还提供了一些说明性示例和相应结果。此外，经过验证的结果有助于扩展、概括和增强现有文献中记载的相应结果。论证了这些发现在非线性 Volterra 积分方程中的实际应用，巩固并增强了既定结果的可信度。总的来说，本文有助于理解模糊双极度量空间背景下的不动点理论，并强调了 ()-收缩映射和 ()-收缩映射在该领域的重要性。