Research On The Properties Of The Interval-valued Metric Spaces And Approximations Of Fuzzy Numbers

Interval-valued metric spaces, as a special kind of fuzzy metric spaces, have many good properties. In this dissertation we discuss topological properties of interval-valued metric spaces, completions of interval-valued metric spaces, char-acterizations of continuous mappings on interval-valued metric spaces, and related fixed point theorems. Fuzzy numbers, as a special kind of fuzzy sets, play an im-portant role in the researches on fuzzy mathematics and its applications. There are many research aspects about fuzzy numbers, such as the distance between two fuzzy numbers, sorting and the approximation problems of them, and so on. In this paper, some relative properties of an interval-valued metric space are discussed. Three metrics on the fuzzy number set are given according to the existing defini-tions of distance, and two types of interval-valued metrics are defmed via their sizes, and their properties are discussed. Further, two types of special fuzzy numbers’ center of gravity and determinate relationship each other are discussed, and the approximation problems between them are also considered. Specific content is as follows:Chapter1This chapter concentrates on the preliminary knowledge. Some basic concepts, the related algorithms and some basic properties about fuzzy set, fuzzy numbers and interval numbers are reviewed.Chapter2Firstly, the definition of interval-value metric space is given. Sec-ondly, a set of equivalent conditions of mapping-continuity, the topological prop-erties and the completion of this kind of space are discussed, where topological properties contain T2separability, regular separability, normal separability, com-pletely regular separability, completely normal separability, the first countability and para-compactness. Finally, several interval-valued measures are defined in the real number space.Chapter3On the basis of algorithms and the related properties of interval numbers, three kinds of fixed point theorems in the interval-valued metric space are proved. They are fixed point theorem of a single mapping, common fixed point theorem of two interchangeable mappings and un-exchange mappings, and fixed point theorem of set-valued weakly compression mapping, respectively. Chapter4The purpose of this chapter is to discuss the center of gravity of two special classes of fuzzy numbers and their determined relationships each other. The first one is the generalized trapezoidal fuzzy numbers (in which the real numbers, closed interval, triangular fuzzy numbers and trapezoidal fuzzy numbers as its special cases). Another one is obeying middle type T-distributed fuzzy numbers. We intend to solve the following two questions:How can we determine the center of gravity of a given fuzzy number and consider its special cases? How can we identify this unknown fuzzy number which satisfies the given center of gravity, width and left/right extension condition? Finally, we discuss the approximation problem of these fuzzy numbers when they having the same as center of gravity and kernel width based on the distance between fuzzy numbers.Chapter5Three kinds of measures pi (i=1,2,3) and their sizes on the fuzzy number set are given according to the existing distances between fuzzy num-bers, and then two types of interval-valued metrics are defined and their properties are discussed. Finally, several metrics (weakly metrics) and interval-valued metrics (weakly interval-valued metrics) are discussed based on the degree of approximation on the fuzzy sets.At last, some future works are proposed.