Fixed Point Theorems Of Fuzzy Integer Value Mappings And Optimization Management To Balance Problem

When the fuzzy number was introduced,they are used in various engineering fields,and has attracted considerable attention.In engineering or real life,we not only deal with the continuous uncertain or imprecise quantities,but also focus on the discrete type of quantities,such as ”a group of people”,”a bus of people”,”a flock of sheep” and”the grade of the public life in some city(like medium sized city)”.Therefore,studying discrete fuzzy numbers and using its theories to deal with uncertain or imprecise discrete type quantities are important.Approximation theory is an important research field in mathematics,and has strong application background.In this research field,many important results have been made.In 2001,Voxman defined(1-dimensional)discrete fuzzy numbers and obtained some results about them in [16];Then the authors studied some characters of discrete fuzzy numbers in [1]-[5];We defined a special kind of discrete fuzzy numbers which is called trapezoid fuzzy integers,investigate the operations of trapezoid fuzzy integers,and gave a practical example to show their application in [20].So studying fuzzy integer numbers and using fuzzy integer numbers to represent uncertain or imprecise integer type quantities(so that we can use the fuzzy integer number theories to solve engineering problems or practical problems related to uncertain or imprecise discrete type quantities)are important and meaningful.Although we studied trapezoid fuzzy integers,the concept of trapezoid fuzzy integers has a big limitation,and they are only suitable to represent the uncertain or imprecise type quantities which are Two-side type(see figure 3.1(a)),but not suitable to represent the uncertain or imprecise type quantities which are Right-side type(see figure3.1(b))or Left-side type(see figure 3.1(c)),or possess some characteristics.In addition,fixed point theory of fuzzy mapping is also an important research areas,and many researchers are interested in it.For example,recently,[43] proved an existence of a fixed point for a fuzzy mapping satisfying a generalized contractive condition;and [27] obtained some common fixed point theorems for four fuzzy mappings in completing ordered metric linear spaces;[38] verified some frequent fixed point theorems for some different mappings.We establish some theories on fuzzy integers(one kind of discrete fuzzy numbers)and fuzzy integer value mappings,and apply the obtained theory to optimal management for balance problem.In this paper,based on previous studies,the definition of fuzzy integer,the expression in the form of level set and its properties and operation properties,in order to facilitate the application,a special fuzzy integer is introduced —trapezoidal fuzzy integers and how to construct trapezoidal fuzzy integers to represent uncertain or inaccurate integer quantities.Then we give the concepts of maximum and minimum values,and get the fixed point theorem of fuzzy integer value mapping and the model framework of the balance problem,and use the theorem of fuzzy integer and fuzzy integer value mapping to establish the optimal management of the balance problem.The main work and results of this paper lie in the following:1.Chapter 1 introduces the research background,purpose and meaning.2.Chapter 2 provides some basic concepts of fuzzy set theories and fuzzy numbers.3.In Chapter 3,we give the definition of fuzzy integral which is based on discrete fuzzy number,and some theorems are given in this chapter.Then,we discuss the closed property of fuzzy integer operation,and the results show that the general addition operation is still closed,and the general multiplication and scalar multiplication can not be closed.We defined a new multiplication and scalar multiplication to represent the closed property of the fuzzy integer operations.At the end of this chapter,we give a new fuzzy integer—trapezoidal fuzzy integer,and give two examples to show how to construct a trapezoidal fuzzy integer to express uncertain or imprecise integer numbers.4.In Chapter 4,the concepts of supremum and infimum of bounded fuzzy integer set are given,their existence are proved,and their expressions formula in level set form are set up.Then some fixed point theorems for fuzzy integer value mapping are obtained,and a balance problem is proposed.And then,using the new theories established by us about fuzzy integers and fuzzy integer value mappings,we give two examples to show the optimal management method for the proposed balance problem.5.In Chapter 5,we summarize the dissertation and give some suggestions for future research work.