Convexity And Its Application Of Fuzzy Mappings

Fuzzy mapping is an important part in the theory of fuzzy analysis. People pay more and more attention to the fuzzy mapping. Along with the development of convex fuzzy set, people investigate the convexity of fuzzy mapping on convex sets and its application in fuzzy programming. However, many fuzzy mappings are not convex. People start to generalize the convexity of fuzzy mapping in different ways. Based on this, in this paper we explore properties of well-known convex fuzzy mappings and some generalized ones, propose some new concepts of generalized convex fuzzy mappings, and apply the results to fuzzy programming problems. We summarize as following.1. Introduce a concept of semi-continuous fuzzy mapping, discuss the semi-continuous of sum, difference and product of two such mappings and some properties, and show the semi-boundedness of the mappings.2. Give some sufficient conditions for semi-continuous fuzzy mapping being convex; discuss the relationship between convexity and strict convexity, convexity and B - convexity, convexity and weak convexity, as well as the equivalence under some conditions.3. Introduce the definition of semi-strict convex fuzzy mapping, discuss the relationship between semi-strict convexity and convexity, semi-strict convexity and strict convexity, show that a local minimum is also a global minimum for semi-strict convex fuzzy mappings.4. Introduce a concept of E - convex fuzzy mapping. Give some theorems which describe E - convex fuzzy mapping are obtained. As application, we discuss the application of E - convex fuzzy mapping in fuzzy programming. Give the condition that a point is global optimum solution. In topological vector space (V ,d) , discuss E - convexity of differential fuzzy mapping, and obtain a sufficient necessary condition that E - convex differential fuzzy mapping reach global minimum.5. Introduce a concept of E - quasiconvex fuzzy mapping. An example is given to show that E - quasiconvex fuzzy mapping may not be E - convex fuzzy mapping. We obtain a sufficient necessary condition that a fuzzy mapping is an E - quasiconvex fuzzy mapping on E - convex fuzzy set and several applications in fuzzy programming are given.6. Introduce the definition of B - E-convex fuzzy mapping. It is not only a generalization of B - convex fuzzy mapping, but also a generalization of E - convex fuzzy mapping. We obtain that a fuzzy mapping is B - E- convex fuzzy mapping if and only if its epigraph is a B - E - convex set. The properties of differential B - E- convex fuzzy mapping are discussed and some results concerning with global minimum are obtained.