| 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.
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