基于三角模和余模的模糊值函数的微分理论研究

项目名称： 基于三角模和余模的模糊值函数的微分理论研究

项目编号： No.11201512

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 邱东

作者单位： 重庆邮电大学

项目金额： 21万元

中文摘要： 模糊分析学发展至今虽然已取得了长足进步，但仍然存在许多厄待解决的问题：各理论分支独立发展缺乏联系，体系林立，而且发展极不均衡，有的长足发展而有的仍然停留在概念的改进与推广上，特别是模糊微分理论已经成了模糊微分方程、模糊优化等理论研究的瓶颈。上述问题严重影响了模糊分析学理论交叉融合，也影响其应用价值。 本项目将基于三角模和余模探寻模糊分析学分支理论之间联系（主要集中在模糊测度理论与模糊距离空间理论之间），融合模糊分析学的各理论分支，并用其研究模糊微分问题，通过模糊测度与积分中的Radon-Nikodym理论建立模糊微分理论，进而建立相应的模糊微分方程理论，研究解的性质。从而避免孤立地研究微分问题，这将会使在解决模糊微分问题的同时，建立模糊微分与模糊积分之间的联系，进而建立模糊分析学的各个理论分支之间联系，对模糊分析学的理论统一起到积极的推动作用。

中文关键词： 三角模；三角余模；模糊微分；模糊函数；

英文摘要： The development of fuzzy analysis has achieved great progress; yet there are many problems remained to be solved. Several branches of fuzzy analysis are now developing independently. Some of them have been deeply investigated, while others' focuses are still on the generalization of concepts. In particular, the fuzzy differential theory has become the bottleneck of the development of fuzzy differential equation theory and fuzzy optimization. This deficiency has negatively influenced the interdisciplinary study of fuzzy analysis, and also limited its application scope. In this project, we will find connections among the several branches of fuzzy analysis (the main focus will be fuzzy measure theory and fuzzy metric theory) based on triangular-norms and triangular conorms. These connections will be used in the study of fuzzy differential. We will develop the fuzzy differential theory based on the Radon-Nikodym theorem in theory of fuzzy measure and integral. Also, we will develop fuzzy differential equation theory and study the properties of solutions of difference equation. In this way, we consider the fuzzy differential and fuzzy integral integratedly, rather than independently. Our study will strengthen the connections among the several branches of fuzzy analysis and contribute to the unified theory of fuzzy

英文关键词： triangular norms；triangular conorms；fuzzy differential；fuzzy functions；