Fuzzy variational inequalities (FVIs) are modeling tools used to characterize a variety of fuzzy decision making problems arising in mathematical optimization, control theory, operations research, and game theory. Fractional order delay differential equations are also finding applications in all disciplines including chemistry, physics, and finance. The aim of this paper is to introduce and study a new dynamical fuzzy system, named the fractional differential fuzzy variational inequality (FDFVI) with delay, which bridges these two areas of research and broadens the class of problems amenable to be studied under the fuzzy environments. By using the KKM theorem and monotonicity arguments, we show that the solution set of the FVI is nonempty, convex and compact. We also establish the upper semicontinuity of the solution mapping U of the FVI involved in the FDFVI with delay. Moreover, we obtain an existence of the mild solution for the FDFVI with delay by employing Bohnenblust-Karlin's fixed point theorem. In addition, we provide an approximating algorithm to find a solution of the FDFVI with delay. Finally, we give two numerical examples to illustrate our main results.

模糊变分不等式 (FVI) 是用于表征数学优化、控制理论、运筹学和博弈论中出现的各种模糊决策问题的建模工具。分数阶延迟微分方程也在包括化学、物理和金融在内的所有学科中得到应用。本文的目的是介绍和研究一种新的动态模糊系统，称为带延迟的分数微分模糊变分不等式 (FDFVI)，它连接了这两个研究领域并拓宽了在模糊环境下可以研究的问题类别. 通过使用 KKM 定理和单调性论证，我们证明了 FVI 的解集是非空的、凸的和紧的。我们还建立了解映射的上半连续性FVI 的U延迟参与 FDFVI。此外，我们利用 Bohnenblust-Karlin 不动点定理，得到了 FDFVI 具有延迟的温和解的存在性。此外，我们提供了一种近似算法来找到具有延迟的 FDFVI 的解决方案。最后，我们给出两个数值例子来说明我们的主要结果。