Fuzzy General Grey Cognitive Map (FGGCM) and Fuzzy Grey Cognitive Map (FGCM) are extensions of Fuzzy Cognitive Map (FCM) in terms of uncertainty. FGGCM allows for the processing of general grey number with multiple intervals, enabling FCM to better address uncertain situations. Although the convergence of FCM and FGCM has been discussed in many literature, the convergence of FGGCM has not been thoroughly explored. This paper aims to fill this research gap. First, metrics for the general grey number space and its vector space is given and proved using the Minkowski inequality. By utilizing the characteristic that Cauchy sequences are convergent sequences, the completeness of these two space is demonstrated. On this premise, utilizing Banach fixed point theorem and Browder-Gohde-Kirk fixed point theorem, combined with Lagrange's mean value theorem and Cauchy's inequality, deduces the sufficient conditions for FGGCM to converge to a unique fixed point when using tanh and sigmoid functions as activation functions. The sufficient conditions for the kernels and greyness of FGGCM to converge to a unique fixed point are also provided separately. Finally, based on Web Experience and Civil engineering FCM, designed corresponding FGGCM with sigmoid and tanh as activation functions by modifying the weights to general grey numbers. By comparing with the convergence theorems of FCM and FGCM, the effectiveness of the theorems proposed in this paper was verified. It was also demonstrated that the convergence theorems of FCM are special cases of the theorems proposed in this paper. The study for convergence of FGGCM is of great significance for guiding the learning algorithm of FGGCM, which is needed for designing FGGCM with specific fixed points, lays a solid theoretical foundation for the application of FGGCM in fields such as control, prediction, and decision support systems.

翻译：模糊广义灰色认知图（FGGCM）与模糊灰色认知图（FGCM）是模糊认知图（FCM）在不确定性方面的扩展。FGGCM能够处理具有多个区间的广义灰数，使得FCM能够更好地应对不确定情境。尽管已有大量文献探讨了FCM与FGCM的收敛性，但FGGCM的收敛性尚未得到深入研究。本文旨在填补这一研究空白。首先，利用闵可夫斯基不等式给出并证明了广义灰数空间及其向量空间的度量。通过利用柯西序列为收敛序列的特性，证明了这两个空间的完备性。在此基础上，运用巴拿赫不动点定理与布劳德-戈德-柯克不动点定理，结合拉格朗日中值定理与柯西不等式，推导出当采用tanh与sigmoid函数作为激活函数时，FGGCM收敛至唯一不动点的充分条件。同时，分别给出了FGGCM的核与灰度收敛至唯一不动点的充分条件。最后，基于Web体验与土木工程FCM，通过将权重修改为广义灰数，设计了以sigmoid与tanh为激活函数的对应FGGCM。通过与FCM及FGCM的收敛定理进行比较，验证了本文所提定理的有效性，并证明了FCM的收敛定理是本文定理的特例。对FGGCM收敛性的研究，对指导FGGCM的学习算法——这是设计具有特定不动点的FGGCM所必需的——具有重要意义，为FGGCM在控制、预测及决策支持系统等领域的应用奠定了坚实的理论基础。