【作者】 邱东；

【导师】 舒兰；

【作者基本信息】 电子科技大学 ， 应用数学， 2009， 博士

【摘要】 模糊分析学是模糊数学的重要组成部分，一直以来都是学术界最为重视的课题之一．本文对模糊分析学中几个相互联系十分紧密的问题进行了研究．此外，作为模糊分析学的应用，也给出了模糊分析与粗糙集理论的结合研究．全文共六章，主要研究工作分四个部分：第一部分研究了模糊实数的推广和模糊复分析．首先给出了模糊实数的一些性质．接下来，由反例说明模糊复分析中最基本概念的定义(模糊复数和一般模糊复数)存在的一些问题．重新修正模糊复分析中这两个基本概念后，一些重要的结论再一次被得到．最后，分析了现有的几种模糊复变函数的导数定义，改进了一些已有的重要结论．所有的这些事实说明尽管对模糊复分析Buckley1989年就开始了研究，但是其仍然处于起步阶段，需要进一步的深入研究．第二部分研究了模糊凸分析及其推广．利用模糊单形，给出了模糊凸包的几个等价刻画．作为模糊凸性的推广，从模糊拟凸集和模糊凸集中衍生出一般模糊星状集和模糊伪星状集这两个新的概念．接着讨论了模糊星状集、模糊拟星状集、模糊伪星状集和一般模糊星状集之间的关系．同时，详细地给出了这些不同类型的星状性的基本性质．讨论了模糊星状集的落影问题并得到了一些重要结论．澄清了两个关于模糊凸过程的定义并得到了模糊凸过程的一些基本性质．这些性质改进了关于这两类模糊映射与其图像之间的关系的已有结论．最后，指出了两种s-凸模糊过程之间的联系和区别并给出了s-凸模糊过程和s-凸模糊映射的一些充要条件．第三部分研究了模糊集组成的距离空间和模糊映射族的公共不动点理论．首先推广了关于全体非空紧集在Hausdorff距离下形成的空间中的一个经典结果，并运用此结果建立了紧距离空间的全体具有非空紧截集的模糊集合以d_∞-距离形成的空间(?)(X)的完备性．为了进一步推广这一结果，在推广了关于全体非空有界闭集在Hausdorff距离下形成的空间中的一个经典结果后，用类似的方法得到对完备距离空间X的全体具有非空有界闭截集的模糊集合引入由Hausdorff距离诱导出的上确界距离，即d_∞-距离形成的空间(?)(X)的完备性．此外，也分别讨论了这两个空间中模糊映射族的公共不动点问题，并举例说明了不动点理论的有效性．第四部分是模糊分析和粗糙集理论的结合研究．一方面，运用粗糙集理论推广粗糙函数为粗糙映射并给出了粗糙映射的各种理论性质；另一方面，从映射的角度出发，考虑对粗糙集理论的推广使其能解决在信息变化下对集合的近似问题．最后给出了新近似算子的一些基本性质．更多还原

【Abstract】 Fuzzy analysis is an important part in the theory of fuzzy mathematics.It is one ofthe most focused problems in the fuzzy research community.This doctoral dissertationstudies several problems which are closely related with each other in fuzzy analysis.Inaddition,a cross-disciplinary research between fuzzy analysis and rough set theory is alsogiven as the application of fuzzy analysis.The dissertation consists of four parts with sixchapters:Part 1 contributes to the generalizations of fuzzy real numbers and fuzzy complexanalysis.Firstly some properties of fuzzy real numbers are given.Next,by counterexam-ples,it is shown that there are some errors in the definitions of the fundamental concepts offuzzy complex analysis,fuzzy complex number and generalized complex fuzzy number.After modifying the two definitions,some important results of fuzzy complex analysis areobtained again.At last,the existing definitions of derivatives for fuzzy complex functionsare investigated,and some main results in literature are improved.All of these facts showthat although the study of fuzzy complex analysis was launched by Buckley in 1989,it isstill in the initiatory phase,and needs further study.Part 2 is to study fuzzy convex analysis and its generalizations.By using fuzzysimplex,several equivalent descriptions for the fuzzy convex hull are presented.As thegeneralizations of the fuzzy convexity,fuzzy general starshaped sets and fuzzy pseudo-starshaped sets are derived from the concepts of quasi-convex fuzzy sets and convex fuzzysets.Then,the exact relationships among the concepts of fuzzy starshaped sets,fuzzyquasi-starshaped sets,fuzzy pseudo-starshaped sets and fuzzy general starshaped sets areclarified.In the meantime,a detailed study on the basic properties of these different typesof starshapedness is also given.The shadows of fuzzy sets are investigated and severalimportant results on the shadows of starshaped fuzzy sets are yielded.Two conceptsconcerning convex fuzzy process are clarified,and some basic properties of convex fuzzyprocesses are presented.These results improve some known results about the connectionbetween these fuzzy mappings and their graphs.Finally,the correlation and difference oftwo definitions of s-convex fuzzy process are investigated.Some necessary and sufficientconditions for s-convex fuzzy processes and for s-convex fuzzy mappings are given. Part 3 is devoted to investigating the metric spaces of fuzzy sets and the commonfixed point theory for fuzzy mappings.Firstly,a classical result about the space ofnonempty compact sets with the Hausdorff metric is generalized.By using this result,the completeness of(?)(X) with respect to the compactness of the metric space X isestablished,where(?)(X) is the class of fuzzy sets with nonempty compactα-cut sets,equipped with the supremum metric d_∞which takes the supremum on the Hausdorff dis-tances between the correspondingα-cut sets.To extend the result,after generalizing aclassical result about the space of nonempty bounded closed sets with the Hausdorff met-ric,the completeness of(?)(X) with respect to the completeness of the metric space Xis established in a similar way,where(?)(X) is the class of fuzzy sets with nonemptybounded closedα-cut sets,equipped with the supremum metric d_∞which takes the supre-mum on the Hausdorff distances between the correspondingα-cut sets.In addition,somecommon fixed point theorems for fuzzy mappings in the two spaces are proved and anexample is given to illustrate the validity of the main results in fixed point theory.Part 4 is a cross-disciplinary research between fuzzy analysis and rough set theory.On the one hand,by using rough set theory,the concept of rough function is generalizedto rough mapping and various theoretic properties are exploited to characterize the roughmappings.On the other hand,from the mapping view,the rough set theory is extended inorder to solve the approximation problem under the changing information.Finally someproperties about the approximation operators are discussed.更多还原