Some Results On Interval - Valued Fuzzy Quasi - Metric Spaces And Fuzzy Numbers

In order to deal with the fuzzy information more flexible, interval-valued fuzzy metric space was defined and investigated. In this paper, some properties of the interval-valued fuzzy quasi metric space (i.e., the interval-valued weak form of fuzzy metric space), includ-ing completion, fixed point, bicompletion, uniformity, metrizable, uniformly continuous expansion will be studied. And the properties of fuzzy number which is related to Kaleva metric space will be discussed. The method of using trapezoid fuzzy number to approxi-mate fuzzy number and related examples which are specific, intuitive will be mainly given. The main contents are as follows:Chapter1Some basic knowledge, such as fuzzy metric space, interval-valued fuzzy metric space, fuzzy dynamical system and fuzzy numbers are introduced.Chapter2The concept of fuzzy quasi metric space is defined and some of its proper-ties are studied. Firstly, the definition of interval-valued fuzzy quasi metric spaces is given, and it is proved that topology induced by quasi metric is consistent with the one induced by the related standard interval-valued fuzzy quasi metric, every quasi-metrizable topol-ogy space admits a compatible interval-valued fuzzy quasi metric, the topology induced by interval-valued fuzzy quasi metric is quasi metrizable. Secondly, the concept of balanced in-terval valued fuzzy quasi metric space is defined, and it is proved that every interval-valued fuzzy metric space is balanced. Next, the bicompletion of interval-valued fuzzy quasi met-ric space is discussed; and it is proved that the bicompletions of an interval-valued fuzzy quasi metric space are isometric to each other. Finally, the theorem of interval-valued quasi metric space isometric isomorphism extension is given, the method of constructing a bi-completion and the sufficient and necessary condition of interval-valued fuzzy quasi metric space can be bicompletion are obtained. In addition, the metrizable of an interval-valued fuzzy quasi metric space is also discussed.Chapter3Several problems of a kind of special interval-valued fuzzy quasi metric space (i.e., the interval-valued fuzzy metric space) are studied. Firstly, interval-valued fuzzy metric spaces completion is discussed, and an interval-valued fuzzy metric space which can not be completed is constructed, then the uniformly continuous extension theorem of interval-valued fuzzy metric space is obtained. Secondly, the relationship between interval-valued fuzzy metric space uniformity and its induced favorable space is discussed; it is proved that the standard interval-valued fuzzy metric space has only one completion. Next, the equivalent conditions of two interval-valued metric space uniformity is gotten and the fixed point theorem of interval-valued fuzzy metric space (the famous Banach and Edelstein fixed point theorem is generalized to the interval-valued fuzzy metric space) is obtained. Finally, the application of special interval value fuzzy quasi metric space in dynamical system is given, that improve the result in a kind of more special interval value fuzzy quasi metric space (i.e., metric space).Chapter4The trapezoidal fuzzy number (it is related with Kaleva fuzzy metric space) approximation problem is studied. The triangular approximation formula of fuzzy numbers is given, some properties of trapezoidal fuzzy numbers are generalized, more specific and intuitive method for calculation trapezoidal approximate is gotten.Finally, some problems which will be studied in the future are displayed.